Research ArticleNumerical Optimal Control for Problems withRandom Forced SPDE Constraints

R Naseri and A Malek

Department of Applied Mathematics Faculty of Mathematical Sciences Tarbiat Modares University PO Box 14115-134 Tehran Iran

Correspondence should be addressed to A Malek malamodaresacir

Received 22 September 2013 Accepted 10 December 2013 Published 20 February 2014

Academic Editors H Homeier and F Zirilli

Copyright copy 2014 R Naseri and A Malek This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A numerical algorithm for solving optimization problems with stochastic diffusion equation as a constraint is proposed Firstseparation of randomanddeterministic variables is done viaKarhunen-Loeve expansionThen the problem is discretized in spatialpart using the finite element method and the polynomial chaos expansion in the stochastic part of the problemThis process leadsto the optimal control problem with a large scale system in its constraint To overcome these difficulties the adjoint technique forderivative computation to implementation of the optimal control issue in preconditioned Newtonrsquos conjugate gradient method isused By some numerical simulation it is shown that this hybrid approach is efficient and simple to implement

1 Introduction

Physical problems in many cases can be formulated asoptimization problems These problems are utilized to gaina more widely understanding of physical systems Typicallythese problems depend on some models which in manycases are deterministic Uncertainty might plague every-thing from modeling assumptions to experimental dataAs such in order to accommodate for these uncertaintiesmany practitioners have developed stochastic models Inorder to make sense of and solve these models in additionto randomness of the models some additional theory isrequired to the resulting optimization problems This paperproposes an adjoint based approach for solving optimizationproblems governed by stochastic diffusion equations Inorder to deal with the stochastic partial differential equations(SPDE) as a constraint in an optimization problem one mayfirst solve the SPDE Since the provided systems by thisapproach are random and nonlinear such kind of problemsare difficult to handle and very challenging One of the mostchallenging examples in this area is the control of stochasticdiffusion equations with random forcing [1] Polynomialchaos expansion (PCE) [2] provides a good direction forsolution of nonlinear SPDEs numerically In the presence ofthe random forcing PCE seems to be more accurate and

efficient numerical method than Monte Carlo simulation Infact the PCE can be interpreted as the Fourier expansionin the probability space Particularly the aim of this work isthe numerical solution for the distributed control problemsinvolving the stochastic diffusion equations in the form

minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω (1)

119906 (x 120596) = 0 on 120597119863 times Ω (2)

where 119863 is the spatial domain 120597119863 is the boundary of thespatial domain Ω is probability space x isin 119863 120596 isin Ω119906 is the solution of the SPDE 119911 is the source functionand 119886(x 120596) is the permeability field of the problem As animportant assumption for the stochastic diffusion equation(1) it is assumed that the random coefficient 119886(x 120596) satisfiesthe elliptic condition [3] that is there exists a constant 119886minsuch that

0 lt 119886min le 119886 (x 120596) forall (x 120596) isin 119863 times Ω (3)

Karhunen-Loeve expansion (KLE) of correlated randomfunctions is used to separate the random and deterministicparts in random coefficient [4 5] Therefore a finite dimen-sional approximation 119886

119872(x 120596) is performed by truncating the

Hindawi Publishing CorporationISRN Applied MathematicsVolume 2014 Article ID 974305 11 pageshttpdxdoiorg1011552014974305

2 ISRN Applied Mathematics

Karhunen-Loeve expansion of the permeability field 119886(x 120596)Then (1) is approximated by

nabla (119886

119872 (x 120596) nabla119906) = 119911 (4)

For approximation of the function 119906 with 119873randomvariables and the Gaussian random variables of the highestdegree119870 the number of PCE coefficients are(119873+119870)119873119870[6] Now weak formulation of (4) and then Galerkinfinite element method are used to solve the SPDE problemapproximately Generally in nature optimization with SPDE-constraint is infinite dimensional large and complex Thereare two numerical approaches for solving this problem [2 7]

(i) discretize then optimize (DO) for problems that canbe trivially discretized first

(ii) optimize then discretize (OD) for problems that aredifferentiable

Our intention is to work with discretizethen optimizationalgorithms for smooth functions Thus we follow the DOapproach Galerkin finite element method incorporated withpolynomial chaos expansion is used for discretizing theSPDE By computing gradient and Hessian of the Lagrangianaugmented function preconditionedNewtonrsquos conjugate gra-dient method is used to compute the best right hand side vec-tor (control values at grid points) where the correspondingsolution of SPDE (state values) according to this vector staysas near as to the desired value approximation and has lowestcost in the objective function The organization of this paperis as follows in Section 2 problem formulation in additionto KLE PCE and stochastic Galerkin method is presentedIn Section 3 distributed control of random forced diffusionequation with some numerical examples is proposed

2 Problem Formulation

We consider optimal control problems of the form

min119906isin119880 119911isinZ

119869 (119906 119911)

st 119890 (119906 119911) = 0

(5)

where 119869 119867

1

0(119863)⨂119871

2(Ω) rarr R is the objective function

119890 119880 = 119867

1

0(119863)⨂119871

2(Ω) rarr Z = 119871

2(119863) is an operator which

stands as the constraint 119863 is the spatial domain Ω is theprobability space and⨂ is the tensor product

To provide a concrete setting for our discussion thefollowing problem is considered as the constraint operator

minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω

119906 (x 120596) = 0 on 120597119863 times Ω

(6)

where 119886(x 120596) is the correlated random fields Hence thesolutions 119906(x 120596) of the SPDE (6) are also random fields It is

assumed that the control space 119911(x) isin 119871

2(119863) The weak form

of the constraint function (6) can be defined for all V isin 119880 as

⟨119890 (119906 119911) V⟩119880lowast 119880

= int

Ω

120588 (120596)int

119863

(119886 (x 120596) nabla119906 (x 120596)

sdot nablaV (x 120596) 119889x119889120596 minus119911 (x) V (x 120596)) 119889x119889120596

(7)

It is assumed that the objective or cost functional is as

119869 (119906 119911) = 119864 [

1

2

119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

(8)

where 119864(sdot) denotes the expected value is a given desiredfunction and 120572 is the regularization parameter Obviouslycontrolling the solution of this problem leads to controllingthe statistics of the solution for example the expected valueof the solution given by

119864[119906 (x 120596) minus (x)]21198712(119863)

le 119864 [119906 (x 120596) minus (x)21198712(119863)

]

le 119906 (x 120596) minus (x)21198712(119863)otimes119871

2(Ω)

(9)

So the optimal control problem can be interpretedas follows given the random field 119886(x 120596) minimize theobjective function (8) over all 119911 isin Z subject to 119906 isin 119880 suchthat for the given random field 119886(x 120596) satisfies in the weakformulation (7) [1]

21 Basic Definitions In (5) 119869 and 119890 are assumed to becontinuously F-differentiable and that for each 119911 isin Z thestate equation 119890(119906 119911) = 0 possesses a unique correspondingsolution 119906(119911) isin 119880 Similar to the deterministic PDEsexistence and uniqueness of the solution for these kind ofSPDEs can be established using the Lax-Milgram theoremThus there is a solution operator 119911 isin Z rarr 119906(119911) isin 119880In addition it is assumed that 119890

119906(119906(119911) 119911) isin L(119880Z) is

continuously invertible Then existence and uniqueness ofthe solution for the above optimal control problems as well asthe continuously differentiability of 119906(119911) are ensured by theimplicit function theorem [8]

22 Karhunen-Loeve Expansion (KLE) Consider a randomfield 119886(x 120596) x isin 119863 with finite second order moment

int

Ω

119864 [119886

2(x 120596)] 119889119909 lt infin (10)

Assume that 119864[119886] = 119886(x) It is possible to expand 119886(x 120596)for a given orthonormal basis 120595

119896 in 119871

2(119863) as a generalized

Fourier series

119886 (x 120596) = 119886 (x) +infin

sum

119896=1

119886

119896 (120596) 120595119896 (

x) (11)

ISRN Applied Mathematics 3

where

119886

119896 (120596) = int

Ω

119886 (x 120596) 120595119896 (x) 119889x 119896 = 1 2 (12)

are random variables with zero means It is importantnow to find a special basis 120601

119896 that makes corresponding

119886

119896uncorrelated 119864[119886

119894119886

119895] = 0 for all 119894 = 119895 Denoting the

covariance function of 119886(x 120596) by 119877(x y) = 119864[119886(x 120596)119886(y 120596)]the basis functions 120601

119896 should satisfy

119864 [119886

119894119886

119895] = int

119863

120601

119894 (x) 119889xint

119863

119877 (x y) 120601119895(y) 119889y = 0 119894 = 119895

(13)

Completion and orthonormality of 120601119896 in 119871

2(119863) follow

that 120601119896(x) are eigenfunctions of 119877(x y)

int

119863

119877 (x y) 120601119895(y) 119889119910 = 120582

119895120601

119895 (x) 119895 = 1 2 (14)

where 120582

119895= 119864[119886

2

119895] gt 0 Indeed by choosing basis

functions 120601119896(119909) as the solutions of the eigenproblem (14) the

randomvariables 119886119896(120596)will be uncorrelated In (14) denoting

120579

119896= 119886

119896radic120582

119896 yields the following expansion

119886 (x 120596) = 119886 (x) +infin

sum

119896=1

radic

120582

119896120579

119896(120596) 120601119896 (

x) (15)

where 120579

119896satisfy 119864[120579

119896] = 0 and 119864[120579

119894120579

119895] = 120575

119894119895 In the case

that 119886(x 120596) is considered as a Gaussian process 119886119896(120596) 119896 =

1 2 will be independent Gaussian random variables Theexpansion (15) is known as the Karhunen-Loeve expansion(KLE) of the stochastic process 119886(x 120596)

Using the KLE (15) the stochastic process can be rep-resented as a series of uncorrelated random variables Sincethe basis functions 120601

119896(x) are deterministic the spatial depen-

dence of the random process can be resolved by them Theconvergence property of the KLE to the random process119886(x 120596) can be represented in the mean square sense

lim119873rarrinfin

int

119863

119864

1003816

1003816

1003816

1003816

119886 (x 120596) minus 119886

119873 (x 120596)

1003816

1003816

1003816

1003816

2119889x = 0 (16)

where

119886

119873= 119886 (x) +

119873

sum

119896=1

radic

120582

119896120579

119896120601

119896(17)

is a finite term KLE [4 5]

23 Polynomial Chaos Expansion (PCE) There are problemsas the solution of a PDE with random inputs that thecovariance function of a random process 119906(x 120596) x isin 119863 isnot knownThe solution of such problems can be representedusing a polynomial chaos expansion (PCE) given by

119906 (x 120596) =

infin

sum

119896=1

119906

119896 (x) Ψ119896 (120585) (18)

where the functions 119906

119896(x) are deterministic coefficients 120585

is a vector of orthonormal random variables and Ψ

119896(120585) are

multidimensional orthogonal polynomials that satisfy in thefollowing properties

⟨Ψ

1⟩ equiv 119864 [Ψ

1] = 1 ⟨Ψ

119896⟩ = 0 119896 gt 1

⟨Ψ

119894Ψ

119895⟩ = ℎ

119894120575

119894119895

(19)

The convergence property of PCE for a random quantityin 119871

2 is ensured by the Cameron-Martin theorem [6 9] thatis

⟨119906 (x 120596) minus

infin

sum

119896=1

119906

119896 (x) Ψ119896 (120585)⟩

1198712

997888rarr 0 (20)

Hence this convergence justifies a truncation of PCE to afinite number of terms

119906 (x 120596) =

119875

sum

119896=1

119906

119896 (x) Ψ119896 (120585) (21)

where the value of 119875 is determined by the highest degree ofpolynomial 119870 used to represent 119906 and the number 119873 ofrandom variablesmdashthe length of 120585mdashwith the formula 119875+1 =

(119873 + 119870)119873119870 Generally the value of 119873 is the same asthe number of uncorrelated random variables in the systemor equivalently the truncation length of the truncated KLETypically the value of119870 is chosen by some heuristic methodIndeed in the case of119870 = 1 and119873 randomvariables the KLEis a special case of the PCE [9 10]

24 Stochastic Galerkin Suppose 119882

119894sub 119871

2

120588119894

(Ω

119894) with dimen-

sion 120588

119894for 119894 = 1 119872 and 119881 sub 119867

1

0(119863) with dimension

119873 In addition let 120595

119894

119899

119901119894

119899=1for 119894 = 1 119872 be basis of

119882

119894and 120601

119894

119873

119894=1a basis of 119881 The finite dimensional tensor

product space 119882

1⨂sdot sdot sdot⨂119882

119872⨂119881 can be defined as the

space spanned by the functions 120595

1

1198991

120595

119872

119899119872

120601

119894 for all 119899

1isin

1 119901

1 119899

119872isin 1 119901

119872 and 119894 isin 1 119873

For simplification let n denote a multi-index whose 119896thcomponent 119899

119896isin 1 119901

119896 and I denote the set of such

multi-indices then the basis functions for the tensor productspace have the form

Vn119894 (x 120596) = 120601

119894(x) Ψ119899 (120596) (22)

where

Ψ

119899=

119872

prod

119896=1

120595

119896

119899119896

(120596

119896) (23)

Then the Galerkin method is looking for a solution isin 119882

1⨂sdot sdot sdot⨂119882

119872⨂119881 such that for x = (119909 119910) all n isinI

and 119894 = 1 119873

int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla119906

ℎ119875 (x 120596) sdot nablaVn119894 (x 120596) 119889x119889120596

= int

Ω

120588 (120596) int

119863

119911 (x 120596) Vn119894 (x 120596) 119889x119889120596

(24)

4 ISRN Applied Mathematics

where 120588(120596) is the integration weight Equation (24) is equiv-alent to the following equation which results by plugging inthe explicit formula for the basis functions

int

Ω

120588 (120596)Ψn (120596) int

119863

119886 (x 120596) nabla119906

ℎ119875 (x 120596) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψn (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(25)

for all n isinI and 119894 = 1 119873Now considering the approximate solution as

119906

ℎ119875 (x 120596) =

119873

sum

119895=1

sum

misinI119906

119895m120601

119895 (x) Ψm (120596) (26)

and substituting (26) into (25) yields119873

sum

119895=1

sum

misinI119906

119895m int

Ω

120588 (120596)Ψn (120596)Ψm (120596)

times int

119863

119886 (x 120596) nabla120601

119895 (x) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψn (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(27)

for allm isinI and 119894 = 1 119873 Let 119875 = |I| = prod

119872

119896=1119901

119896 then

a natural bijection between 1 119875 and Ican be definedThus (27) can be written as

119875

sum

119898=1

119873

sum

119895=1

119906

119898119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596)

times int

119863

119886 (x 120596) nabla120601

119895 (x) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψ119899 (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(28)

Now if 119886 and 119911 have the following KLE

119886 (x 120596) = 119886

0 (x) +

119872

sum

119894=1

120596

119894119886

119894 (x)

119911 (x 120596) = 119911

0 (x) +

119872

sum

119894=1

120596

119894119911

119894 (x)

(29)

then we can rewrite the Galerkin projection (28) as119875

sum

119898=1

119873

sum

119895=1

119906

119898119895((119870

0)

119894119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596

+

119872

sum

119896=1

(119870

119896)

119894119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 120596119896

119889120596)

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(30)

where

(119870

119896)

119894119895= int

119863

119886

119896 (x) nabla120601

119894 (x) sdot nabla120601

119895 (x) 119889x

cong sum

119903

119908

119903119886

119896(x119903) nabla120601

119894(x119903) sdot nabla120601

119895(x119903)

(119911

119896)

119894= int

119863

119911

119894 (x) 120601119894 (x) 119889x cong sum

119903

119908

119903119911

119894(x119903) 120601

119894(x119903)

(31)

for 119896 = 1 119872 [3] Equation (31) is approximated byquadrature rule in spatial domain Now assume that thereexist functions Ψ

119899 119899 = 1 119875 such that

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596 = 120575

119898119899

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 120596119896

119889120596 = 119862

119896119899120575

119899119898

(32)

where for 119898 = 119899 the above integral is approximated usingquadrature rule in random domain as follows

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119899 (

120596) 120596119896119889120596 cong sum

119904

119908

1015840

119904120588 (120596

119904) Ψ

2

119899(120596

119904) 120596

119896 (33)

Then (30) becomes

119875

sum

119898=1

119873

sum

119895=1

119906

119898119895((119870

0)

119894119895+

119872

sum

119896=1

119862

119896119899(119870

119896)

119894119895)120575

119899119898

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(34)

or equivalently

119873

sum

119895=1

119906m119895 ((119870

0)

119894119895+

119872

sum

119896=1

119862

119896119899(119870

119896)

119894119895)

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(35)

In (34) and (35) we have

int

Ω

120588 (120596)Ψ119899 (120596) 119889120596 = 0 for 119899 = 0

int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596 = int

Ω

120588 (120596)Ψ0 (120596)Ψ119899 (

120596) 120596119896119889120596

= 119862

119896119899120575

0119899= 119862

1198960

(36)

ISRN Applied Mathematics 5

Finally (35) can be considered as the following blockdiagonal system

(

(

(

119870

0+

119872

sum

119896=1

119862

1198961119870

119896sdot sdot sdot 0

d

0 sdot sdot sdot 119870

0+

119872

sum

119896=1

119862

119896119901119870

119896

)

)

)

(

1

119875

)

= (

119911

1

119911

119901

)

(37)

or briefly 119870 =

119911

3 Distributed Optimal Control Problem

Using the stochastic Galerkin method for general objectivefunctions computing the derivatives becomes very tediousand rather messy Consider the following optimal controlproblem

min119906isin119880 119911isinz

119869 (119906 119911) =

1

2

119864 [119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

st minus nabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω

119906 (x 120596) = 0 on 120597119863 times Ω

(38)

The solutions to linear elliptic SPDEs live in the space119880 =

119867

1

0(119863)⨂119871

2(Ω) and the control space Z = 119871

2(119863) Denoting

the constraint equation by 119890(119906 119911) = 0 usually it is possibleto invoke the implicit function theorem to find a solution119906(119911) to the constraint equation This leads to an implicitlydefined objective function

119869(119911) = 119869(119906(119911) 119911) Now we focuson computing derivatives of 119869 [11] For any direction 119904 isin Z

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= ⟨119869

119906 (119906 (119911) 119911) 119906119911 (

119911) 119904⟩

119880lowast119880

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(39)

where119880lowast andZlowast are the dual space of119880 andZ respectivelyNow computing the derivative of the constraint 119890with respectto 119911 in the direction 119904 and applying it to the implicit solutionto 119890(119906(119911) 119911) = 0 yield

119890

119906 (119906 (119911) 119911) 119906119911 (

119911) 119904 + 119890

119911 (119906 (119911) 119911) 119904 = 0 (40)

So 119906

119911(119911) is

119906

119911 (119911) 119904 = minus119890

119906(119906 (119911) 119911)

minus1119890

119911 (119906 (119911) 119911) 119904

(41)

Hence

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= minus ⟨119869

119906 (119906 (119911) 119911) 119890119906(

119906 (119911) 119911)

minus1

times 119890

119911 (119906 (119911) 119911) 119904⟩

119880lowast119880

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(42)

Applying the adjoint to the 119906

119911(119911) term

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= minus ⟨119890

119911(119906 (119911) 119911)

lowast119890

119906(119906 (119911) 119911)

minuslowast

times 119869

119906 (119906 (119911) 119911) 119904⟩

Zlowast Z

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

= minus ⟨119890

119911(119906 (119911) 119911)

lowast119890

119906(119906 (119911) 119911)

minuslowast119869

119906(119906 (119911) 119911)

+119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(43)

Considering the adjoint state 120583 = 120583(119911) isin Zlowast as the solutionto

119890

119906(119906 (119911) 119911)

lowast120583 = minus119869

119906 (119906 (119911) 119911) (44)

the derivative of 1198691015840 becomes

119869

1015840(119911) = 119890

119911(119906 (119911) 119911)

lowast120583 (119911) + 119869

119911 (119906 (119911) 119911)

(45)

Theweak formof the constraint function is defined as followsfor all V isin 119880

⟨119890 (119906 119911) V⟩119880lowast 119880 = int

Ω

120588 (120596)int

119863

(119886 (x 120596) nabla119906 (x 120596) sdot nablaV (x 120596)

minus119911 (x 120596) V (x 120596)) 119889x119889120596

(46)

Thus the constraint function 119890 119880 times Z rarr 119880

lowast and 119884 = 119880

lowastSince the constraint function 119890 acts linearly with respect to 119906

and 119911 thus the corresponding derivative of 119890 with respect to119911 in the direction 120575119911 isin Z for all V isin 119880 is given by

⟨119890

119911 (119906 119911) 120575119911 V⟩

119880lowast119880

= minusint

Ω

120588 (120596)int

119863

120575119911 (x 120596) V (x 120596) 119889x119889120596

(47)

Similarly the derivative of 119890 with respect to 119906 for anydirection 120575119906 isin 119880 and for all V isin 119880 is given by

⟨119890

119906 (119906 119911) 120575119906 V⟩

119880lowast119880

= int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla120575119906 (x 120596)

sdot nablaV (x 120596) 119889x119889120596

(48)

Both of these derivatives are symmetric so 119890

119906(119906 119911)

lowast=

119890

119906(119906 119911) and 119890

119911(119906 119911)

lowast= 119890

119911(119906 119911) From here the adjoint can

be computed as

119890

119906 (119906 (119911) 119911) 120583 = minus119869

119906 (119906 (119911) 119911) (49)

Using this to any direction V isin 119880

⟨119890

119906 (119906 (119911) 119911) 120583 V⟩

119880lowast119880

= int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla120583 (x 120596)

sdot nablaV (x 120596) 119889x119889120596

= minus119869

119906 (119906 (119911) 119911) V

(50)

6 ISRN Applied Mathematics

By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is

⟨119869

119906 (119906 119911) 120575119906⟩

119880lowast119880

= 119864 [119906 (x 120596) minus (x)21198712(119863)

times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)

]

(51)

Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin

Z direction is

⟨119869

119911 (119906 119911) 120575119911⟩

Zlowast Z= 120572⟨119911 120575119911⟩Z (52)

With these computations one can compute the firstderivative of

119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883

ℎsub 119867

1

0(119863) is a finite dimensional

subspace of dimension 119873 and 119884

ℎsub 119871

2(Ω) is a finite

dimensional subspace of dimension 119875 then 119883

ℎ⨂119884

ℎis a

finite dimensional subspace of the state space 119880 Similarlylet Z

ℎsub Z be a finite dimensional subspace of the control

space (with dimension) For the stochastic Galerkin methodit is assumed that 119884

ℎ= P119875minus1 the space of polynomials with

highest degree 119875minus1 and119883

ℎis any finite element space here

119883

ℎis the space of linear functions built on a given meshT

ℎ

We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884

ℎ that is 119884

ℎ= spanΨ

1(120596) Ψ

119875(120596)

where

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596 = 120575

119898119899 (53)

The discretized optimization problem in the stochasticGalerkin framework is

min119911ℎisinZℎ

119869

ℎ119875(119911

ℎ) =

1

2

119864

[

[

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119875

sum

119896=1

119873

sum

119895=1

(

119896(119911

ℎ))

119895Ψ

119896 (120596) 120601119895 (

x)

minus (x)1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(119863)

]

]

+

120572

2

1003817

1003817

1003817

1003817

119911

ℎ

1003817

1003817

1003817

1003817

2

1198712(119863)

(54)

where

119896(119911

ℎ) = (

1(119911

ℎ)

119879

119875(119911

ℎ)

119879)

119879

=

(55)

is the stochastic Galerkin solution to the state equation

(

119870

11sdot sdot sdot 119870

1119875

d

119870

1198751sdot sdot sdot 119870

119875119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119870

(

1

119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=

= (

119864 [Ψ

1]119872

119911

ℎ

119864 [Ψ

119875]119872

119911

ℎ

)

= (

120575

11119872

119911

ℎ

120575

1119875119872

119911

ℎ

)

(56)

since 119864[Ψ

119896] = 119864[1Ψ

119896] = 119864[Ψ

1Ψ

119896] = 120575

1119896 The blocks of the

above 119870matrix have the form

(119870

119896119897)

119894119895= int

Ω

120588 (120596)Ψ

119896(120596)Ψ119897 (

120596)

times int

119863

119886 (x 120596) nabla120601

119894 (x) sdot nabla120601

119895 (x) 119889x119889120596

(57)

First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations

minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))

in 119863 times Ω

120583 (x 120596) = 0 on 120597119863 times Ω

(58)

Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865

119879

1 119865

119879

119901)

119879 and 119865

119896is defined

as

(

119865

119896)

119894= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596

= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895120601

119895 (x) Ψ119897 (120596) 120601119894 (

x)

minus (x) 120601119894 (x))119889x119889120596

= minus

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

(x) 120601119894 (x) 119889x

=

119887

119894120575

1119896minus

119875

sum

119897=1

119873

sum

119895=1

119872

119894119895(

119897)

119895120575

119896119897

=

119887

119894120575

1119896minus

119873

sum

119895=1

119872

119894119895(

119897)

119895

(59)

in which

119887

119894= int 120601

119894 and119872

119894119895= int 120601

119894120601

119895

Thus

(

119865

119896)

119894=

119887 minus 119872119906

1if 119896 = 1

minus119872119906 if 119896 = 2 119875

(60)

where b= [

119887

1

119887

119873] and the matrix 119872 = (119872

119894119895) is of order

119873 times 119873

ISRN Applied Mathematics 7

Now the derivative of

119869(119911) in the direction 120601

119895(x) for all

119895 = 1 119873 with the computed adjoint state is

⟨

119869

1015840(119911) 120601119895 (

x)⟩Zlowast Z

= ⟨119890

119911(119906 (119911 sdot sdot) 119911)

lowast120583

+ 119869

119911 (119906 (119911 sdot sdot) 119911) 120601119895 (

x) ⟩Zlowast Z

= minusint

Ω

120588 (120596)int

119863

120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596

+ 120572int

119863

119911 (119909) 120601119895 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894int

Ω

120588 (120596)Ψ119896 (120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ 120572

119873

sum

119894=1

119911

119894int

119863

120601

119895 (x) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894119872

119894119895120575

1119896+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

= minus

119873

sum

119894=1

(

1)

119894119872

119894119895+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

(61)

Therefore the gradient can be computed as follows

119869

1015840(119911) = 119872(120572

119911

ℎminus

1)

(62)

Now for any vector V isin Z it is possible to write V as

V =

119873

sum

119896=1

V119896120601

119896 (x) (63)

In order to compute the Hessian times a vector that ismultiply the Hessian of the

119869 with a some vector V isin Z theequation 119890

119906(119906(119911) 119911)119908 = 119890

119911(119906(119911) 119911)V for 119908 must be solved

Similar to the adjoint computation119908 solves the linear system119870 = 119866 where

119866 = (

119866

119879

1

119866

119879

119875) and

119866

119896for 119896 = 1 119875 is

(

119866

119896)

119894= int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

119873

sum

119895=1

V119895120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895V119895120575

1119896

(64)

for all 119894 = 1 119873 Thus

119866

119896=

119872V if 119896 = 1

0 if 119896 = 2 119875

(65)

Now solving 119890

119906(119906(119911) 119911)119902 = 119869

119906119906(119906(119911) 119911)119908 for 119902 requires

the solution to the linear system 119870 119902 = 119867 where

(

119867

119896)

119894=

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895(

119896)

119895

(66)

or equivalently

119867

119896= 119872

119896for all 119896 = 1 119875 Hence in the

direction 120601

119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by

⟨

119869

10158401015840(119911) V 120601119894⟩Zlowast Z

asymp ⟨

119869

10158401015840

ℎ119901(119911

ℎ) V 120601

119894⟩

Zlowast Z

= minusint

Ω

120588 (120596)int

119863

119875

sum

119896=1

119873

sum

119895=1

( 119902

119896)

119895120601

119895 (x) Ψ119896 (120596)

times 120601

119894 (x) 119889x119889120596

+ 120572int

119863

V (119909) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119895=1

119872

119894119895( 119902

119896)

119895120575

1119896+ 120572

119873

sum

119895=1

119872

119894119895V119895

=

119873

sum

119895=1

119872

119894119895(120572V

119895minus ( 119902

1)

119895)

(67)

Therefore

119869

10158401015840(119911) V asymp 119872(120572V minus 119902

1)

(68)

Having (62) and (68) we can use the iterative solvers tothe Newton equation

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) (69)

Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572

119896and using this step size to generate

the iterate 119911

119896+1= 119911

119896+ 120572

119896119878

119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is

119869 (119911

119896+ 120572

119896119878

119896) le 119869 (119911

119896) + 119888120572

119896119869

1015840(119911

119896) 119878

119896 (70)

where 119888 isin (0 1) and typically is quite small for example 119888 =

10

minus4 [12 13]

8 ISRN Applied Mathematics

(1) Given 119911

0 and tol gt 0 set 119896 = 0

(2) Compute 119870119872 119860 and

119887

(3) Compute (

119865

119896)

119894=

119887 minus 119872119906 if 119896 = 1

minus119872119906 if 119896 = 2 119875

(4) Solve 119870 120583 = 119865

(5) Compute

119869

1015840(119911

119896) = 119872(120572119911

119896minus

1)

(6) Compute

119866

1= 119872V and

119866

119905= 0 for 119905 = 2 119875

(7) Solve 119870 = 119866 = (

119866

119879

1

119866

119879

119875)

119879

(8) Compute

119867

119896= 119872

119896for 119896 = 1 119875

(9) Solve 119870 119902 = 119867 = (

119867

119879

1

119867

119879

119875)

119879

(10) Compute

119869

10158401015840(119911)V asymp 119872(120572V minus 119902

1)

(11) If 1003817100381710038171003817

1003817

nabla

119869 (119911

119896)

1003817

1003817

1003817

1003817

1003817

lt tol stop(12) Solve

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) using PCG method with preconditioner

119875 = 119866⨂119870

0

(i) Set 1199090= 0

(ii) Set 1199030larr

119869

10158401015840(119911

119896) 119909

0+

119869

1015840(119911

119896)

(iii) Solve

119875119910

0= 119903

0for 119910

0

(iv) Set 1199010= minus119903

0 119905 larr 0

(v) While 1003817

1003817

1003817

1003817

119903

119905

1003817

1003817

1003817

1003817

gt tol

120572

119896larr

119903

119879

119905119910

119905

119901

119879

119905119860119901

119905

119909

119905+1larr 119909

119905+ 120572

119905119901

119905

119903

119905+1larr 119903

119905+ 120572

119905119860119901

119905

119875119910

119905+1larr 119903

119905+1

120573

119905+1larr

119903

119879

119905+1119910

119905+1

119903

119879

119905119910

119905

119901

119905+1larr minus119910

119905+1+ 120573

119905+1119901

119905

119905 larr 119905 + 1

End (while)(vi) 119904119896 = 119909

119905+1

(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911

119896+ 120572

119896119878

119896)

(ii) While 119869 (119911

119896+ 120572

119896119878

119896) gt 119869 (119911

119896) + 10

minus4120572

119896119878

119896nabla

119869 (119911

119896) do

(i) Set 120572119896 = 120572

1198962 and evaluate 119869 (119911

119896+ 120572

119896119878

119896)

(14) Set 119911119896+1 = 119911

119896+ 120572

119896119878

119896 119896 larr 119896 + 1 Go to (2)

Algorithm 1

31 Kronecker Product Preconditioners Let us to rewrite (37)in the form

119860 =

119911 where

119860 = (

119872

sum

119896=0

119866

119896⨂119870

119896) (71)

The strategy here is to introduce

119875 = 119866⨂119870

0as a

preconditioner such that

119866 = argmin 119867 isin R119875times119875

1003817

1003817

1003817

1003817

1003817

119860 minus 119867⨂119870

0

1003817

1003817

1003817

1003817

1003817119865 (72)

where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot

119865denotes the Frobe-

nius norm The closed form of the solution can be written asfollow [14]

119866 = 119868 + sum

119875

tr (119870119879

119896119870

0)

tr (119870119879

0119870

0)

119866

119896

(73)

Here tr(119870119879

119896119870

0) = sum

119873119902

119894=1[119870

119896]

119894119894[119870

0]

119894119894and hence the coeffi-

cients in the above equality can be computed straightforwardIn addition since

119860 and 119870

0are symmetric and positive

definite thus 119866 and

119875 = 119866⨂119870

0have also these properties

[15 16]

Example 1 Consider the following distributed optimal con-trol problem

min119906119911

119869 (119906 119911) = 119864 [

1

2

119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω

(74)

where 119863 = [minus1 1]

2 Ω sim 119880[minus1 1] the boundary conditionsare given as

119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0

(75)

ISRN Applied Mathematics 9

005

1

005

10

05

1y d

xy

(a) Desired values

005

1

005

10

05

1

uM

ean

xy

times10minus3

(b) Initial state

005

1

005

10

05

1

u opt

xy

(c) Approximated solution (state)

005

1

005

10

200

400

z opt

xy

(d) Control

005

1

0

05

10

05

1

times 10minus3

xy

120583

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

05

1

15

2

25

times 10minus3

(f) Expected value

20 40

10

20

30

40

50

05

1

15

2

25

times 10minus3

(g) Standard dev

0 5 10times 104

0

5

10

times 104 0

(h) Sparsity of global matrix119899119911 = 1815980

Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))

2+ (119910 minus (12))

2)]

119873 = 3 119870 = 5 and 120572 = 000005

and the desired function is given by

(119909 119910) = exp [minus64 ((119909 minus

1

2

)

2

+ (119910 minus

1

2

)

2

)] (76)

The random field 119886(x 120596) is characterized by its mean andcovariance function

119864 [119886] = 119886 = 10

119877 (119909

1 119909

2) = 119890

minus|1199091minus1199092| 119909

1 119909

2isin [minus1 1]

(77)

Following Section 22 the truncated KLE can be representedas

119886 (119909 119910 120596) = 10 +

119873

sum

119895=0

radic120582

119895120579

119895120601

119895 (119909) (78)

The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867

1

0(119863)⨂119871

2(Ω) and all 119911isin 119871

2(119863)

119869 (119906opt 119911opt) le 119869 (119906 119911) (79)

The eigenpairs 120582119895 120579

119895 in truncated KLE solve the integral

equation

int

1

minus1

119890

minus|1199091minus1199092|120601

119895(119909

2) 119889119909

2= 120582

119895120601

119895(119909

1)

(80)

For this special case of the covariance function we haveexplicit expressions for 120601

119895and 120582

119895 [6] Let120596

119895evenand120596

119895oddsolve

the equations

1 minus 120596

119895eventan (120596

119895even) = 0

120596

119895odd+ tan (120596

119895odd) = 0

(81)

Then the even and odd indexed eigenfunctions are given by

120601

119895even(119909) =

cos (120596119895even

119909)

radic1 + (sin (2120596

119895even) 2120596

119895even)

120601

119895odd(119909)

=

sin (120596

119895odd119909)

radic1 minus (sin (2120596

119895odd) 2120596

119895odd)

(82)

10 ISRN Applied Mathematics

005

1

005

10

05

1y ha

nd

xy

(a) Desired values

005

1

005

1minus05

05

0

u Mea

n

xy

(b) Mean values

005

1

005

1

0

Num

sol

xy

0

1

(c) Approximated solution (State)

005

1

005

10

05

1

xy

(d) Control

005

1

005

10

05

1

Lend

a

xy

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

0

005

01

015

02

(f) Expected value-SGS

20 40

10

20

30

40

50

0

005

01

015

02

(g) Standard dev-SGS

0 2 4 6 8times 104

0

2

4

6

8

times 104

(h) Sparsity of global matrix119899119911 = 3046160

Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001

and corresponding eigenvalues

120582

119895even(119909) =

2

120596

2

119895even+ 1

120582

119895odd(119909) =

2

120596

2

119895odd+ 1

(83)

We choose 120579 = (120579

1 120579

119873)

119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]

First of all we find numerical approximations of120596119895even

and120596

119895oddwith bisection method and then with the eigenpairs

evaluated by this 120596119895even

and 120596

119895odd by choosing dimension 119873 =

3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1

In Figure 1(a) the desired function (119909 119910) is plottedfor 119909

119894= 119910

119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial

state is computed and plotted in Figure 1(b) by solving (37)where

119911 = 0 Approximated 119906opt corresponding to the

optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial

119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911

119900119901119905must approximate the

initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane

In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use

Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary

ISRN Applied Mathematics 11

conditions is replaced by (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2

4 Conclusion

The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996

[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987

[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005

[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947

[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946

[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991

[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007

[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009

[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938

[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993

[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008

[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999

[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006

[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010

[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010

[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007

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Stochastic AnalysisInternational Journal of

2 ISRN Applied Mathematics

Karhunen-Loeve expansion of the permeability field 119886(x 120596)Then (1) is approximated by

nabla (119886

119872 (x 120596) nabla119906) = 119911 (4)

For approximation of the function 119906 with 119873randomvariables and the Gaussian random variables of the highestdegree119870 the number of PCE coefficients are(119873+119870)119873119870[6] Now weak formulation of (4) and then Galerkinfinite element method are used to solve the SPDE problemapproximately Generally in nature optimization with SPDE-constraint is infinite dimensional large and complex Thereare two numerical approaches for solving this problem [2 7]

(i) discretize then optimize (DO) for problems that canbe trivially discretized first

(ii) optimize then discretize (OD) for problems that aredifferentiable

Our intention is to work with discretizethen optimizationalgorithms for smooth functions Thus we follow the DOapproach Galerkin finite element method incorporated withpolynomial chaos expansion is used for discretizing theSPDE By computing gradient and Hessian of the Lagrangianaugmented function preconditionedNewtonrsquos conjugate gra-dient method is used to compute the best right hand side vec-tor (control values at grid points) where the correspondingsolution of SPDE (state values) according to this vector staysas near as to the desired value approximation and has lowestcost in the objective function The organization of this paperis as follows in Section 2 problem formulation in additionto KLE PCE and stochastic Galerkin method is presentedIn Section 3 distributed control of random forced diffusionequation with some numerical examples is proposed

2 Problem Formulation

We consider optimal control problems of the form

min119906isin119880 119911isinZ

119869 (119906 119911)

st 119890 (119906 119911) = 0

(5)

where 119869 119867

1

0(119863)⨂119871

2(Ω) rarr R is the objective function

119890 119880 = 119867

1

0(119863)⨂119871

2(Ω) rarr Z = 119871

2(119863) is an operator which

stands as the constraint 119863 is the spatial domain Ω is theprobability space and⨂ is the tensor product

To provide a concrete setting for our discussion thefollowing problem is considered as the constraint operator

minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω

119906 (x 120596) = 0 on 120597119863 times Ω

(6)

where 119886(x 120596) is the correlated random fields Hence thesolutions 119906(x 120596) of the SPDE (6) are also random fields It is

assumed that the control space 119911(x) isin 119871

2(119863) The weak form

of the constraint function (6) can be defined for all V isin 119880 as

⟨119890 (119906 119911) V⟩119880lowast 119880

= int

Ω

120588 (120596)int

119863

(119886 (x 120596) nabla119906 (x 120596)

sdot nablaV (x 120596) 119889x119889120596 minus119911 (x) V (x 120596)) 119889x119889120596

(7)

It is assumed that the objective or cost functional is as

119869 (119906 119911) = 119864 [

1

2

119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

(8)

where 119864(sdot) denotes the expected value is a given desiredfunction and 120572 is the regularization parameter Obviouslycontrolling the solution of this problem leads to controllingthe statistics of the solution for example the expected valueof the solution given by

119864[119906 (x 120596) minus (x)]21198712(119863)

le 119864 [119906 (x 120596) minus (x)21198712(119863)

]

le 119906 (x 120596) minus (x)21198712(119863)otimes119871

2(Ω)

(9)

So the optimal control problem can be interpretedas follows given the random field 119886(x 120596) minimize theobjective function (8) over all 119911 isin Z subject to 119906 isin 119880 suchthat for the given random field 119886(x 120596) satisfies in the weakformulation (7) [1]

21 Basic Definitions In (5) 119869 and 119890 are assumed to becontinuously F-differentiable and that for each 119911 isin Z thestate equation 119890(119906 119911) = 0 possesses a unique correspondingsolution 119906(119911) isin 119880 Similar to the deterministic PDEsexistence and uniqueness of the solution for these kind ofSPDEs can be established using the Lax-Milgram theoremThus there is a solution operator 119911 isin Z rarr 119906(119911) isin 119880In addition it is assumed that 119890

119906(119906(119911) 119911) isin L(119880Z) is

continuously invertible Then existence and uniqueness ofthe solution for the above optimal control problems as well asthe continuously differentiability of 119906(119911) are ensured by theimplicit function theorem [8]

22 Karhunen-Loeve Expansion (KLE) Consider a randomfield 119886(x 120596) x isin 119863 with finite second order moment

int

Ω

119864 [119886

2(x 120596)] 119889119909 lt infin (10)

Assume that 119864[119886] = 119886(x) It is possible to expand 119886(x 120596)for a given orthonormal basis 120595

119896 in 119871

2(119863) as a generalized

Fourier series

119886 (x 120596) = 119886 (x) +infin

sum

119896=1

119886

119896 (120596) 120595119896 (

x) (11)

ISRN Applied Mathematics 3

where

119886

119896 (120596) = int

Ω

119886 (x 120596) 120595119896 (x) 119889x 119896 = 1 2 (12)

are random variables with zero means It is importantnow to find a special basis 120601

119896 that makes corresponding

119886

119896uncorrelated 119864[119886

119894119886

119895] = 0 for all 119894 = 119895 Denoting the

covariance function of 119886(x 120596) by 119877(x y) = 119864[119886(x 120596)119886(y 120596)]the basis functions 120601

119896 should satisfy

119864 [119886

119894119886

119895] = int

119863

120601

119894 (x) 119889xint

119863

119877 (x y) 120601119895(y) 119889y = 0 119894 = 119895

(13)

Completion and orthonormality of 120601119896 in 119871

2(119863) follow

that 120601119896(x) are eigenfunctions of 119877(x y)

int

119863

119877 (x y) 120601119895(y) 119889119910 = 120582

119895120601

119895 (x) 119895 = 1 2 (14)

where 120582

119895= 119864[119886

2

119895] gt 0 Indeed by choosing basis

functions 120601119896(119909) as the solutions of the eigenproblem (14) the

randomvariables 119886119896(120596)will be uncorrelated In (14) denoting

120579

119896= 119886

119896radic120582

119896 yields the following expansion

119886 (x 120596) = 119886 (x) +infin

sum

119896=1

radic

120582

119896120579

119896(120596) 120601119896 (

x) (15)

where 120579

119896satisfy 119864[120579

119896] = 0 and 119864[120579

119894120579

119895] = 120575

119894119895 In the case

that 119886(x 120596) is considered as a Gaussian process 119886119896(120596) 119896 =

1 2 will be independent Gaussian random variables Theexpansion (15) is known as the Karhunen-Loeve expansion(KLE) of the stochastic process 119886(x 120596)

Using the KLE (15) the stochastic process can be rep-resented as a series of uncorrelated random variables Sincethe basis functions 120601

119896(x) are deterministic the spatial depen-

dence of the random process can be resolved by them Theconvergence property of the KLE to the random process119886(x 120596) can be represented in the mean square sense

lim119873rarrinfin

int

119863

119864

1003816

1003816

1003816

1003816

119886 (x 120596) minus 119886

119873 (x 120596)

1003816

1003816

1003816

1003816

2119889x = 0 (16)

where

119886

119873= 119886 (x) +

119873

sum

119896=1

radic

120582

119896120579

119896120601

119896(17)

is a finite term KLE [4 5]

23 Polynomial Chaos Expansion (PCE) There are problemsas the solution of a PDE with random inputs that thecovariance function of a random process 119906(x 120596) x isin 119863 isnot knownThe solution of such problems can be representedusing a polynomial chaos expansion (PCE) given by

119906 (x 120596) =

infin

sum

119896=1

119906

119896 (x) Ψ119896 (120585) (18)

where the functions 119906

119896(x) are deterministic coefficients 120585

is a vector of orthonormal random variables and Ψ

119896(120585) are

multidimensional orthogonal polynomials that satisfy in thefollowing properties

⟨Ψ

1⟩ equiv 119864 [Ψ

1] = 1 ⟨Ψ

119896⟩ = 0 119896 gt 1

⟨Ψ

119894Ψ

119895⟩ = ℎ

119894120575

119894119895

(19)

The convergence property of PCE for a random quantityin 119871

2 is ensured by the Cameron-Martin theorem [6 9] thatis

⟨119906 (x 120596) minus

infin

sum

119896=1

119906

119896 (x) Ψ119896 (120585)⟩

1198712

997888rarr 0 (20)

Hence this convergence justifies a truncation of PCE to afinite number of terms

119906 (x 120596) =

119875

sum

119896=1

119906

119896 (x) Ψ119896 (120585) (21)

where the value of 119875 is determined by the highest degree ofpolynomial 119870 used to represent 119906 and the number 119873 ofrandom variablesmdashthe length of 120585mdashwith the formula 119875+1 =

(119873 + 119870)119873119870 Generally the value of 119873 is the same asthe number of uncorrelated random variables in the systemor equivalently the truncation length of the truncated KLETypically the value of119870 is chosen by some heuristic methodIndeed in the case of119870 = 1 and119873 randomvariables the KLEis a special case of the PCE [9 10]

24 Stochastic Galerkin Suppose 119882

119894sub 119871

2

120588119894

(Ω

119894) with dimen-

sion 120588

119894for 119894 = 1 119872 and 119881 sub 119867

1

0(119863) with dimension

119873 In addition let 120595

119894

119899

119901119894

119899=1for 119894 = 1 119872 be basis of

119882

119894and 120601

119894

119873

119894=1a basis of 119881 The finite dimensional tensor

product space 119882

1⨂sdot sdot sdot⨂119882

119872⨂119881 can be defined as the

space spanned by the functions 120595

1

1198991

120595

119872

119899119872

120601

119894 for all 119899

1isin

1 119901

1 119899

119872isin 1 119901

119872 and 119894 isin 1 119873

For simplification let n denote a multi-index whose 119896thcomponent 119899

119896isin 1 119901

119896 and I denote the set of such

multi-indices then the basis functions for the tensor productspace have the form

Vn119894 (x 120596) = 120601

119894(x) Ψ119899 (120596) (22)

where

Ψ

119899=

119872

prod

119896=1

120595

119896

119899119896

(120596

119896) (23)

Then the Galerkin method is looking for a solution isin 119882

1⨂sdot sdot sdot⨂119882

119872⨂119881 such that for x = (119909 119910) all n isinI

and 119894 = 1 119873

int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla119906

ℎ119875 (x 120596) sdot nablaVn119894 (x 120596) 119889x119889120596

= int

Ω

120588 (120596) int

119863

119911 (x 120596) Vn119894 (x 120596) 119889x119889120596

(24)

4 ISRN Applied Mathematics

where 120588(120596) is the integration weight Equation (24) is equiv-alent to the following equation which results by plugging inthe explicit formula for the basis functions

int

Ω

120588 (120596)Ψn (120596) int

119863

119886 (x 120596) nabla119906

ℎ119875 (x 120596) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψn (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(25)

for all n isinI and 119894 = 1 119873Now considering the approximate solution as

119906

ℎ119875 (x 120596) =

119873

sum

119895=1

sum

misinI119906

119895m120601

119895 (x) Ψm (120596) (26)

and substituting (26) into (25) yields119873

sum

119895=1

sum

misinI119906

119895m int

Ω

120588 (120596)Ψn (120596)Ψm (120596)

times int

119863

119886 (x 120596) nabla120601

119895 (x) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψn (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(27)

for allm isinI and 119894 = 1 119873 Let 119875 = |I| = prod

119872

119896=1119901

119896 then

a natural bijection between 1 119875 and Ican be definedThus (27) can be written as

119875

sum

119898=1

119873

sum

119895=1

119906

119898119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596)

times int

119863

119886 (x 120596) nabla120601

119895 (x) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψ119899 (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(28)

Now if 119886 and 119911 have the following KLE

119886 (x 120596) = 119886

0 (x) +

119872

sum

119894=1

120596

119894119886

119894 (x)

119911 (x 120596) = 119911

0 (x) +

119872

sum

119894=1

120596

119894119911

119894 (x)

(29)

then we can rewrite the Galerkin projection (28) as119875

sum

119898=1

119873

sum

119895=1

119906

119898119895((119870

0)

119894119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596

+

119872

sum

119896=1

(119870

119896)

119894119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 120596119896

119889120596)

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(30)

where

(119870

119896)

119894119895= int

119863

119886

119896 (x) nabla120601

119894 (x) sdot nabla120601

119895 (x) 119889x

cong sum

119903

119908

119903119886

119896(x119903) nabla120601

119894(x119903) sdot nabla120601

119895(x119903)

(119911

119896)

119894= int

119863

119911

119894 (x) 120601119894 (x) 119889x cong sum

119903

119908

119903119911

119894(x119903) 120601

119894(x119903)

(31)

for 119896 = 1 119872 [3] Equation (31) is approximated byquadrature rule in spatial domain Now assume that thereexist functions Ψ

119899 119899 = 1 119875 such that

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596 = 120575

119898119899

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 120596119896

119889120596 = 119862

119896119899120575

119899119898

(32)

where for 119898 = 119899 the above integral is approximated usingquadrature rule in random domain as follows

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119899 (

120596) 120596119896119889120596 cong sum

119904

119908

1015840

119904120588 (120596

119904) Ψ

2

119899(120596

119904) 120596

119896 (33)

Then (30) becomes

119875

sum

119898=1

119873

sum

119895=1

119906

119898119895((119870

0)

119894119895+

119872

sum

119896=1

119862

119896119899(119870

119896)

119894119895)120575

119899119898

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(34)

or equivalently

119873

sum

119895=1

119906m119895 ((119870

0)

119894119895+

119872

sum

119896=1

119862

119896119899(119870

119896)

119894119895)

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(35)

In (34) and (35) we have

int

Ω

120588 (120596)Ψ119899 (120596) 119889120596 = 0 for 119899 = 0

int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596 = int

Ω

120588 (120596)Ψ0 (120596)Ψ119899 (

120596) 120596119896119889120596

= 119862

119896119899120575

0119899= 119862

1198960

(36)

ISRN Applied Mathematics 5

Finally (35) can be considered as the following blockdiagonal system

(

(

(

119870

0+

119872

sum

119896=1

119862

1198961119870

119896sdot sdot sdot 0

d

0 sdot sdot sdot 119870

0+

119872

sum

119896=1

119862

119896119901119870

119896

)

)

)

(

1

119875

)

= (

119911

1

119911

119901

)

(37)

or briefly 119870 =

119911

3 Distributed Optimal Control Problem

Using the stochastic Galerkin method for general objectivefunctions computing the derivatives becomes very tediousand rather messy Consider the following optimal controlproblem

min119906isin119880 119911isinz

119869 (119906 119911) =

1

2

119864 [119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

st minus nabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω

119906 (x 120596) = 0 on 120597119863 times Ω

(38)

The solutions to linear elliptic SPDEs live in the space119880 =

119867

1

0(119863)⨂119871

2(Ω) and the control space Z = 119871

2(119863) Denoting

the constraint equation by 119890(119906 119911) = 0 usually it is possibleto invoke the implicit function theorem to find a solution119906(119911) to the constraint equation This leads to an implicitlydefined objective function

119869(119911) = 119869(119906(119911) 119911) Now we focuson computing derivatives of 119869 [11] For any direction 119904 isin Z

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= ⟨119869

119906 (119906 (119911) 119911) 119906119911 (

119911) 119904⟩

119880lowast119880

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(39)

where119880lowast andZlowast are the dual space of119880 andZ respectivelyNow computing the derivative of the constraint 119890with respectto 119911 in the direction 119904 and applying it to the implicit solutionto 119890(119906(119911) 119911) = 0 yield

119890

119906 (119906 (119911) 119911) 119906119911 (

119911) 119904 + 119890

119911 (119906 (119911) 119911) 119904 = 0 (40)

So 119906

119911(119911) is

119906

119911 (119911) 119904 = minus119890

119906(119906 (119911) 119911)

minus1119890

119911 (119906 (119911) 119911) 119904

(41)

Hence

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= minus ⟨119869

119906 (119906 (119911) 119911) 119890119906(

119906 (119911) 119911)

minus1

times 119890

119911 (119906 (119911) 119911) 119904⟩

119880lowast119880

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(42)

Applying the adjoint to the 119906

119911(119911) term

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= minus ⟨119890

119911(119906 (119911) 119911)

lowast119890

119906(119906 (119911) 119911)

minuslowast

times 119869

119906 (119906 (119911) 119911) 119904⟩

Zlowast Z

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

= minus ⟨119890

119911(119906 (119911) 119911)

lowast119890

119906(119906 (119911) 119911)

minuslowast119869

119906(119906 (119911) 119911)

+119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(43)

Considering the adjoint state 120583 = 120583(119911) isin Zlowast as the solutionto

119890

119906(119906 (119911) 119911)

lowast120583 = minus119869

119906 (119906 (119911) 119911) (44)

the derivative of 1198691015840 becomes

119869

1015840(119911) = 119890

119911(119906 (119911) 119911)

lowast120583 (119911) + 119869

119911 (119906 (119911) 119911)

(45)

Theweak formof the constraint function is defined as followsfor all V isin 119880

⟨119890 (119906 119911) V⟩119880lowast 119880 = int

Ω

120588 (120596)int

119863

(119886 (x 120596) nabla119906 (x 120596) sdot nablaV (x 120596)

minus119911 (x 120596) V (x 120596)) 119889x119889120596

(46)

Thus the constraint function 119890 119880 times Z rarr 119880

lowast and 119884 = 119880

lowastSince the constraint function 119890 acts linearly with respect to 119906

and 119911 thus the corresponding derivative of 119890 with respect to119911 in the direction 120575119911 isin Z for all V isin 119880 is given by

⟨119890

119911 (119906 119911) 120575119911 V⟩

119880lowast119880

= minusint

Ω

120588 (120596)int

119863

120575119911 (x 120596) V (x 120596) 119889x119889120596

(47)

Similarly the derivative of 119890 with respect to 119906 for anydirection 120575119906 isin 119880 and for all V isin 119880 is given by

⟨119890

119906 (119906 119911) 120575119906 V⟩

119880lowast119880

= int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla120575119906 (x 120596)

sdot nablaV (x 120596) 119889x119889120596

(48)

Both of these derivatives are symmetric so 119890

119906(119906 119911)

lowast=

119890

119906(119906 119911) and 119890

119911(119906 119911)

lowast= 119890

119911(119906 119911) From here the adjoint can

be computed as

119890

119906 (119906 (119911) 119911) 120583 = minus119869

119906 (119906 (119911) 119911) (49)

Using this to any direction V isin 119880

⟨119890

119906 (119906 (119911) 119911) 120583 V⟩

119880lowast119880

= int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla120583 (x 120596)

sdot nablaV (x 120596) 119889x119889120596

= minus119869

119906 (119906 (119911) 119911) V

(50)

6 ISRN Applied Mathematics

By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is

⟨119869

119906 (119906 119911) 120575119906⟩

119880lowast119880

= 119864 [119906 (x 120596) minus (x)21198712(119863)

times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)

]

(51)

Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin

Z direction is

⟨119869

119911 (119906 119911) 120575119911⟩

Zlowast Z= 120572⟨119911 120575119911⟩Z (52)

With these computations one can compute the firstderivative of

119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883

ℎsub 119867

1

0(119863) is a finite dimensional

subspace of dimension 119873 and 119884

ℎsub 119871

2(Ω) is a finite

dimensional subspace of dimension 119875 then 119883

ℎ⨂119884

ℎis a

finite dimensional subspace of the state space 119880 Similarlylet Z

ℎsub Z be a finite dimensional subspace of the control

space (with dimension) For the stochastic Galerkin methodit is assumed that 119884

ℎ= P119875minus1 the space of polynomials with

highest degree 119875minus1 and119883

ℎis any finite element space here

119883

ℎis the space of linear functions built on a given meshT

ℎ

We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884

ℎ that is 119884

ℎ= spanΨ

1(120596) Ψ

119875(120596)

where

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596 = 120575

119898119899 (53)

The discretized optimization problem in the stochasticGalerkin framework is

min119911ℎisinZℎ

119869

ℎ119875(119911

ℎ) =

1

2

119864

[

[

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119875

sum

119896=1

119873

sum

119895=1

(

119896(119911

ℎ))

119895Ψ

119896 (120596) 120601119895 (

x)

minus (x)1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(119863)

]

]

+

120572

2

1003817

1003817

1003817

1003817

119911

ℎ

1003817

1003817

1003817

1003817

2

1198712(119863)

(54)

where

119896(119911

ℎ) = (

1(119911

ℎ)

119879

119875(119911

ℎ)

119879)

119879

=

(55)

is the stochastic Galerkin solution to the state equation

(

119870

11sdot sdot sdot 119870

1119875

d

119870

1198751sdot sdot sdot 119870

119875119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119870

(

1

119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=

= (

119864 [Ψ

1]119872

119911

ℎ

119864 [Ψ

119875]119872

119911

ℎ

)

= (

120575

11119872

119911

ℎ

120575

1119875119872

119911

ℎ

)

(56)

since 119864[Ψ

119896] = 119864[1Ψ

119896] = 119864[Ψ

1Ψ

119896] = 120575

1119896 The blocks of the

above 119870matrix have the form

(119870

119896119897)

119894119895= int

Ω

120588 (120596)Ψ

119896(120596)Ψ119897 (

120596)

times int

119863

119886 (x 120596) nabla120601

119894 (x) sdot nabla120601

119895 (x) 119889x119889120596

(57)

First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations

minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))

in 119863 times Ω

120583 (x 120596) = 0 on 120597119863 times Ω

(58)

Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865

119879

1 119865

119879

119901)

119879 and 119865

119896is defined

as

(

119865

119896)

119894= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596

= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895120601

119895 (x) Ψ119897 (120596) 120601119894 (

x)

minus (x) 120601119894 (x))119889x119889120596

= minus

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

(x) 120601119894 (x) 119889x

=

119887

119894120575

1119896minus

119875

sum

119897=1

119873

sum

119895=1

119872

119894119895(

119897)

119895120575

119896119897

=

119887

119894120575

1119896minus

119873

sum

119895=1

119872

119894119895(

119897)

119895

(59)

in which

119887

119894= int 120601

119894 and119872

119894119895= int 120601

119894120601

119895

Thus

(

119865

119896)

119894=

119887 minus 119872119906

1if 119896 = 1

minus119872119906 if 119896 = 2 119875

(60)

where b= [

119887

1

119887

119873] and the matrix 119872 = (119872

119894119895) is of order

119873 times 119873

ISRN Applied Mathematics 7

Now the derivative of

119869(119911) in the direction 120601

119895(x) for all

119895 = 1 119873 with the computed adjoint state is

⟨

119869

1015840(119911) 120601119895 (

x)⟩Zlowast Z

= ⟨119890

119911(119906 (119911 sdot sdot) 119911)

lowast120583

+ 119869

119911 (119906 (119911 sdot sdot) 119911) 120601119895 (

x) ⟩Zlowast Z

= minusint

Ω

120588 (120596)int

119863

120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596

+ 120572int

119863

119911 (119909) 120601119895 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894int

Ω

120588 (120596)Ψ119896 (120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ 120572

119873

sum

119894=1

119911

119894int

119863

120601

119895 (x) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894119872

119894119895120575

1119896+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

= minus

119873

sum

119894=1

(

1)

119894119872

119894119895+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

(61)

Therefore the gradient can be computed as follows

119869

1015840(119911) = 119872(120572

119911

ℎminus

1)

(62)

Now for any vector V isin Z it is possible to write V as

V =

119873

sum

119896=1

V119896120601

119896 (x) (63)

In order to compute the Hessian times a vector that ismultiply the Hessian of the

119869 with a some vector V isin Z theequation 119890

119906(119906(119911) 119911)119908 = 119890

119911(119906(119911) 119911)V for 119908 must be solved

Similar to the adjoint computation119908 solves the linear system119870 = 119866 where

119866 = (

119866

119879

1

119866

119879

119875) and

119866

119896for 119896 = 1 119875 is

(

119866

119896)

119894= int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

119873

sum

119895=1

V119895120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895V119895120575

1119896

(64)

for all 119894 = 1 119873 Thus

119866

119896=

119872V if 119896 = 1

0 if 119896 = 2 119875

(65)

Now solving 119890

119906(119906(119911) 119911)119902 = 119869

119906119906(119906(119911) 119911)119908 for 119902 requires

the solution to the linear system 119870 119902 = 119867 where

(

119867

119896)

119894=

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895(

119896)

119895

(66)

or equivalently

119867

119896= 119872

119896for all 119896 = 1 119875 Hence in the

direction 120601

119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by

⟨

119869

10158401015840(119911) V 120601119894⟩Zlowast Z

asymp ⟨

119869

10158401015840

ℎ119901(119911

ℎ) V 120601

119894⟩

Zlowast Z

= minusint

Ω

120588 (120596)int

119863

119875

sum

119896=1

119873

sum

119895=1

( 119902

119896)

119895120601

119895 (x) Ψ119896 (120596)

times 120601

119894 (x) 119889x119889120596

+ 120572int

119863

V (119909) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119895=1

119872

119894119895( 119902

119896)

119895120575

1119896+ 120572

119873

sum

119895=1

119872

119894119895V119895

=

119873

sum

119895=1

119872

119894119895(120572V

119895minus ( 119902

1)

119895)

(67)

Therefore

119869

10158401015840(119911) V asymp 119872(120572V minus 119902

1)

(68)

Having (62) and (68) we can use the iterative solvers tothe Newton equation

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) (69)

Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572

119896and using this step size to generate

the iterate 119911

119896+1= 119911

119896+ 120572

119896119878

119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is

119869 (119911

119896+ 120572

119896119878

119896) le 119869 (119911

119896) + 119888120572

119896119869

1015840(119911

119896) 119878

119896 (70)

where 119888 isin (0 1) and typically is quite small for example 119888 =

10

minus4 [12 13]

8 ISRN Applied Mathematics

(1) Given 119911

0 and tol gt 0 set 119896 = 0

(2) Compute 119870119872 119860 and

119887

(3) Compute (

119865

119896)

119894=

119887 minus 119872119906 if 119896 = 1

minus119872119906 if 119896 = 2 119875

(4) Solve 119870 120583 = 119865

(5) Compute

119869

1015840(119911

119896) = 119872(120572119911

119896minus

1)

(6) Compute

119866

1= 119872V and

119866

119905= 0 for 119905 = 2 119875

(7) Solve 119870 = 119866 = (

119866

119879

1

119866

119879

119875)

119879

(8) Compute

119867

119896= 119872

119896for 119896 = 1 119875

(9) Solve 119870 119902 = 119867 = (

119867

119879

1

119867

119879

119875)

119879

(10) Compute

119869

10158401015840(119911)V asymp 119872(120572V minus 119902

1)

(11) If 1003817100381710038171003817

1003817

nabla

119869 (119911

119896)

1003817

1003817

1003817

1003817

1003817

lt tol stop(12) Solve

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) using PCG method with preconditioner

119875 = 119866⨂119870

0

(i) Set 1199090= 0

(ii) Set 1199030larr

119869

10158401015840(119911

119896) 119909

0+

119869

1015840(119911

119896)

(iii) Solve

119875119910

0= 119903

0for 119910

0

(iv) Set 1199010= minus119903

0 119905 larr 0

(v) While 1003817

1003817

1003817

1003817

119903

119905

1003817

1003817

1003817

1003817

gt tol

120572

119896larr

119903

119879

119905119910

119905

119901

119879

119905119860119901

119905

119909

119905+1larr 119909

119905+ 120572

119905119901

119905

119903

119905+1larr 119903

119905+ 120572

119905119860119901

119905

119875119910

119905+1larr 119903

119905+1

120573

119905+1larr

119903

119879

119905+1119910

119905+1

119903

119879

119905119910

119905

119901

119905+1larr minus119910

119905+1+ 120573

119905+1119901

119905

119905 larr 119905 + 1

End (while)(vi) 119904119896 = 119909

119905+1

(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911

119896+ 120572

119896119878

119896)

(ii) While 119869 (119911

119896+ 120572

119896119878

119896) gt 119869 (119911

119896) + 10

minus4120572

119896119878

119896nabla

119869 (119911

119896) do

(i) Set 120572119896 = 120572

1198962 and evaluate 119869 (119911

119896+ 120572

119896119878

119896)

(14) Set 119911119896+1 = 119911

119896+ 120572

119896119878

119896 119896 larr 119896 + 1 Go to (2)

Algorithm 1

31 Kronecker Product Preconditioners Let us to rewrite (37)in the form

119860 =

119911 where

119860 = (

119872

sum

119896=0

119866

119896⨂119870

119896) (71)

The strategy here is to introduce

119875 = 119866⨂119870

0as a

preconditioner such that

119866 = argmin 119867 isin R119875times119875

1003817

1003817

1003817

1003817

1003817

119860 minus 119867⨂119870

0

1003817

1003817

1003817

1003817

1003817119865 (72)

where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot

119865denotes the Frobe-

nius norm The closed form of the solution can be written asfollow [14]

119866 = 119868 + sum

119875

tr (119870119879

119896119870

0)

tr (119870119879

0119870

0)

119866

119896

(73)

Here tr(119870119879

119896119870

0) = sum

119873119902

119894=1[119870

119896]

119894119894[119870

0]

119894119894and hence the coeffi-

cients in the above equality can be computed straightforwardIn addition since

119860 and 119870

0are symmetric and positive

definite thus 119866 and

119875 = 119866⨂119870

0have also these properties

[15 16]

Example 1 Consider the following distributed optimal con-trol problem

min119906119911

119869 (119906 119911) = 119864 [

1

2

119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω

(74)

where 119863 = [minus1 1]

2 Ω sim 119880[minus1 1] the boundary conditionsare given as

119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0

(75)

ISRN Applied Mathematics 9

005

1

005

10

05

1y d

xy

(a) Desired values

005

1

005

10

05

1

uM

ean

xy

times10minus3

(b) Initial state

005

1

005

10

05

1

u opt

xy

(c) Approximated solution (state)

005

1

005

10

200

400

z opt

xy

(d) Control

005

1

0

05

10

05

1

times 10minus3

xy

120583

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

05

1

15

2

25

times 10minus3

(f) Expected value

20 40

10

20

30

40

50

05

1

15

2

25

times 10minus3

(g) Standard dev

0 5 10times 104

0

5

10

times 104 0

(h) Sparsity of global matrix119899119911 = 1815980

Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))

2+ (119910 minus (12))

2)]

119873 = 3 119870 = 5 and 120572 = 000005

and the desired function is given by

(119909 119910) = exp [minus64 ((119909 minus

1

2

)

2

+ (119910 minus

1

2

)

2

)] (76)

The random field 119886(x 120596) is characterized by its mean andcovariance function

119864 [119886] = 119886 = 10

119877 (119909

1 119909

2) = 119890

minus|1199091minus1199092| 119909

1 119909

2isin [minus1 1]

(77)

Following Section 22 the truncated KLE can be representedas

119886 (119909 119910 120596) = 10 +

119873

sum

119895=0

radic120582

119895120579

119895120601

119895 (119909) (78)

The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867

1

0(119863)⨂119871

2(Ω) and all 119911isin 119871

2(119863)

119869 (119906opt 119911opt) le 119869 (119906 119911) (79)

The eigenpairs 120582119895 120579

119895 in truncated KLE solve the integral

equation

int

1

minus1

119890

minus|1199091minus1199092|120601

119895(119909

2) 119889119909

2= 120582

119895120601

119895(119909

1)

(80)

For this special case of the covariance function we haveexplicit expressions for 120601

119895and 120582

119895 [6] Let120596

119895evenand120596

119895oddsolve

the equations

1 minus 120596

119895eventan (120596

119895even) = 0

120596

119895odd+ tan (120596

119895odd) = 0

(81)

Then the even and odd indexed eigenfunctions are given by

120601

119895even(119909) =

cos (120596119895even

119909)

radic1 + (sin (2120596

119895even) 2120596

119895even)

120601

119895odd(119909)

=

sin (120596

119895odd119909)

radic1 minus (sin (2120596

119895odd) 2120596

119895odd)

(82)

10 ISRN Applied Mathematics

005

1

005

10

05

1y ha

nd

xy

(a) Desired values

005

1

005

1minus05

05

0

u Mea

n

xy

(b) Mean values

005

1

005

1

0

Num

sol

xy

0

1

(c) Approximated solution (State)

005

1

005

10

05

1

xy

(d) Control

005

1

005

10

05

1

Lend

a

xy

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

0

005

01

015

02

(f) Expected value-SGS

20 40

10

20

30

40

50

0

005

01

015

02

(g) Standard dev-SGS

0 2 4 6 8times 104

0

2

4

6

8

times 104

(h) Sparsity of global matrix119899119911 = 3046160

Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001

and corresponding eigenvalues

120582

119895even(119909) =

2

120596

2

119895even+ 1

120582

119895odd(119909) =

2

120596

2

119895odd+ 1

(83)

We choose 120579 = (120579

1 120579

119873)

119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]

First of all we find numerical approximations of120596119895even

and120596

119895oddwith bisection method and then with the eigenpairs

evaluated by this 120596119895even

and 120596

119895odd by choosing dimension 119873 =

3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1

In Figure 1(a) the desired function (119909 119910) is plottedfor 119909

119894= 119910

119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial

state is computed and plotted in Figure 1(b) by solving (37)where

119911 = 0 Approximated 119906opt corresponding to the

optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial

119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911

119900119901119905must approximate the

initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane

In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use

Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary

ISRN Applied Mathematics 11

conditions is replaced by (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2

4 Conclusion

The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996

[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987

[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005

[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947

[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946

[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991

[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007

[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009

[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938

[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993

[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008

[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999

[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006

[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010

[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010

[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Applied Mathematics 3

where

119886

119896 (120596) = int

Ω

119886 (x 120596) 120595119896 (x) 119889x 119896 = 1 2 (12)

are random variables with zero means It is importantnow to find a special basis 120601

119896 that makes corresponding

119886

119896uncorrelated 119864[119886

119894119886

119895] = 0 for all 119894 = 119895 Denoting the

covariance function of 119886(x 120596) by 119877(x y) = 119864[119886(x 120596)119886(y 120596)]the basis functions 120601

119896 should satisfy

119864 [119886

119894119886

119895] = int

119863

120601

119894 (x) 119889xint

119863

119877 (x y) 120601119895(y) 119889y = 0 119894 = 119895

(13)

Completion and orthonormality of 120601119896 in 119871

2(119863) follow

that 120601119896(x) are eigenfunctions of 119877(x y)

int

119863

119877 (x y) 120601119895(y) 119889119910 = 120582

119895120601

119895 (x) 119895 = 1 2 (14)

where 120582

119895= 119864[119886

2

119895] gt 0 Indeed by choosing basis

functions 120601119896(119909) as the solutions of the eigenproblem (14) the

randomvariables 119886119896(120596)will be uncorrelated In (14) denoting

120579

119896= 119886

119896radic120582

119896 yields the following expansion

119886 (x 120596) = 119886 (x) +infin

sum

119896=1

radic

120582

119896120579

119896(120596) 120601119896 (

x) (15)

where 120579

119896satisfy 119864[120579

119896] = 0 and 119864[120579

119894120579

119895] = 120575

119894119895 In the case

that 119886(x 120596) is considered as a Gaussian process 119886119896(120596) 119896 =

1 2 will be independent Gaussian random variables Theexpansion (15) is known as the Karhunen-Loeve expansion(KLE) of the stochastic process 119886(x 120596)

Using the KLE (15) the stochastic process can be rep-resented as a series of uncorrelated random variables Sincethe basis functions 120601

119896(x) are deterministic the spatial depen-

dence of the random process can be resolved by them Theconvergence property of the KLE to the random process119886(x 120596) can be represented in the mean square sense

lim119873rarrinfin

int

119863

119864

1003816

1003816

1003816

1003816

119886 (x 120596) minus 119886

119873 (x 120596)

1003816

1003816

1003816

1003816

2119889x = 0 (16)

where

119886

119873= 119886 (x) +

119873

sum

119896=1

radic

120582

119896120579

119896120601

119896(17)

is a finite term KLE [4 5]

23 Polynomial Chaos Expansion (PCE) There are problemsas the solution of a PDE with random inputs that thecovariance function of a random process 119906(x 120596) x isin 119863 isnot knownThe solution of such problems can be representedusing a polynomial chaos expansion (PCE) given by

119906 (x 120596) =

infin

sum

119896=1

119906

119896 (x) Ψ119896 (120585) (18)

where the functions 119906

119896(x) are deterministic coefficients 120585

is a vector of orthonormal random variables and Ψ

119896(120585) are

multidimensional orthogonal polynomials that satisfy in thefollowing properties

⟨Ψ

1⟩ equiv 119864 [Ψ

1] = 1 ⟨Ψ

119896⟩ = 0 119896 gt 1

⟨Ψ

119894Ψ

119895⟩ = ℎ

119894120575

119894119895

(19)

The convergence property of PCE for a random quantityin 119871

2 is ensured by the Cameron-Martin theorem [6 9] thatis

⟨119906 (x 120596) minus

infin

sum

119896=1

119906

119896 (x) Ψ119896 (120585)⟩

1198712

997888rarr 0 (20)

Hence this convergence justifies a truncation of PCE to afinite number of terms

119906 (x 120596) =

119875

sum

119896=1

119906

119896 (x) Ψ119896 (120585) (21)

where the value of 119875 is determined by the highest degree ofpolynomial 119870 used to represent 119906 and the number 119873 ofrandom variablesmdashthe length of 120585mdashwith the formula 119875+1 =

(119873 + 119870)119873119870 Generally the value of 119873 is the same asthe number of uncorrelated random variables in the systemor equivalently the truncation length of the truncated KLETypically the value of119870 is chosen by some heuristic methodIndeed in the case of119870 = 1 and119873 randomvariables the KLEis a special case of the PCE [9 10]

24 Stochastic Galerkin Suppose 119882

119894sub 119871

2

120588119894

(Ω

119894) with dimen-

sion 120588

119894for 119894 = 1 119872 and 119881 sub 119867

1

0(119863) with dimension

119873 In addition let 120595

119894

119899

119901119894

119899=1for 119894 = 1 119872 be basis of

119882

119894and 120601

119894

119873

119894=1a basis of 119881 The finite dimensional tensor

product space 119882

1⨂sdot sdot sdot⨂119882

119872⨂119881 can be defined as the

space spanned by the functions 120595

1

1198991

120595

119872

119899119872

120601

119894 for all 119899

1isin

1 119901

1 119899

119872isin 1 119901

119872 and 119894 isin 1 119873

For simplification let n denote a multi-index whose 119896thcomponent 119899

119896isin 1 119901

119896 and I denote the set of such

multi-indices then the basis functions for the tensor productspace have the form

Vn119894 (x 120596) = 120601

119894(x) Ψ119899 (120596) (22)

where

Ψ

119899=

119872

prod

119896=1

120595

119896

119899119896

(120596

119896) (23)

Then the Galerkin method is looking for a solution isin 119882

1⨂sdot sdot sdot⨂119882

119872⨂119881 such that for x = (119909 119910) all n isinI

and 119894 = 1 119873

int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla119906

ℎ119875 (x 120596) sdot nablaVn119894 (x 120596) 119889x119889120596

= int

Ω

120588 (120596) int

119863

119911 (x 120596) Vn119894 (x 120596) 119889x119889120596

(24)

4 ISRN Applied Mathematics

where 120588(120596) is the integration weight Equation (24) is equiv-alent to the following equation which results by plugging inthe explicit formula for the basis functions

int

Ω

120588 (120596)Ψn (120596) int

119863

119886 (x 120596) nabla119906

ℎ119875 (x 120596) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψn (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(25)

for all n isinI and 119894 = 1 119873Now considering the approximate solution as

119906

ℎ119875 (x 120596) =

119873

sum

119895=1

sum

misinI119906

119895m120601

119895 (x) Ψm (120596) (26)

and substituting (26) into (25) yields119873

sum

119895=1

sum

misinI119906

119895m int

Ω

120588 (120596)Ψn (120596)Ψm (120596)

times int

119863

119886 (x 120596) nabla120601

119895 (x) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψn (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(27)

for allm isinI and 119894 = 1 119873 Let 119875 = |I| = prod

119872

119896=1119901

119896 then

a natural bijection between 1 119875 and Ican be definedThus (27) can be written as

119875

sum

119898=1

119873

sum

119895=1

119906

119898119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596)

times int

119863

119886 (x 120596) nabla120601

119895 (x) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψ119899 (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(28)

Now if 119886 and 119911 have the following KLE

119886 (x 120596) = 119886

0 (x) +

119872

sum

119894=1

120596

119894119886

119894 (x)

119911 (x 120596) = 119911

0 (x) +

119872

sum

119894=1

120596

119894119911

119894 (x)

(29)

then we can rewrite the Galerkin projection (28) as119875

sum

119898=1

119873

sum

119895=1

119906

119898119895((119870

0)

119894119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596

+

119872

sum

119896=1

(119870

119896)

119894119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 120596119896

119889120596)

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(30)

where

(119870

119896)

119894119895= int

119863

119886

119896 (x) nabla120601

119894 (x) sdot nabla120601

119895 (x) 119889x

cong sum

119903

119908

119903119886

119896(x119903) nabla120601

119894(x119903) sdot nabla120601

119895(x119903)

(119911

119896)

119894= int

119863

119911

119894 (x) 120601119894 (x) 119889x cong sum

119903

119908

119903119911

119894(x119903) 120601

119894(x119903)

(31)

for 119896 = 1 119872 [3] Equation (31) is approximated byquadrature rule in spatial domain Now assume that thereexist functions Ψ

119899 119899 = 1 119875 such that

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596 = 120575

119898119899

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 120596119896

119889120596 = 119862

119896119899120575

119899119898

(32)

where for 119898 = 119899 the above integral is approximated usingquadrature rule in random domain as follows

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119899 (

120596) 120596119896119889120596 cong sum

119904

119908

1015840

119904120588 (120596

119904) Ψ

2

119899(120596

119904) 120596

119896 (33)

Then (30) becomes

119875

sum

119898=1

119873

sum

119895=1

119906

119898119895((119870

0)

119894119895+

119872

sum

119896=1

119862

119896119899(119870

119896)

119894119895)120575

119899119898

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(34)

or equivalently

119873

sum

119895=1

119906m119895 ((119870

0)

119894119895+

119872

sum

119896=1

119862

119896119899(119870

119896)

119894119895)

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(35)

In (34) and (35) we have

int

Ω

120588 (120596)Ψ119899 (120596) 119889120596 = 0 for 119899 = 0

int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596 = int

Ω

120588 (120596)Ψ0 (120596)Ψ119899 (

120596) 120596119896119889120596

= 119862

119896119899120575

0119899= 119862

1198960

(36)

ISRN Applied Mathematics 5

Finally (35) can be considered as the following blockdiagonal system

(

(

(

119870

0+

119872

sum

119896=1

119862

1198961119870

119896sdot sdot sdot 0

d

0 sdot sdot sdot 119870

0+

119872

sum

119896=1

119862

119896119901119870

119896

)

)

)

(

1

119875

)

= (

119911

1

119911

119901

)

(37)

or briefly 119870 =

119911

3 Distributed Optimal Control Problem

Using the stochastic Galerkin method for general objectivefunctions computing the derivatives becomes very tediousand rather messy Consider the following optimal controlproblem

min119906isin119880 119911isinz

119869 (119906 119911) =

1

2

119864 [119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

st minus nabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω

119906 (x 120596) = 0 on 120597119863 times Ω

(38)

The solutions to linear elliptic SPDEs live in the space119880 =

119867

1

0(119863)⨂119871

2(Ω) and the control space Z = 119871

2(119863) Denoting

the constraint equation by 119890(119906 119911) = 0 usually it is possibleto invoke the implicit function theorem to find a solution119906(119911) to the constraint equation This leads to an implicitlydefined objective function

119869(119911) = 119869(119906(119911) 119911) Now we focuson computing derivatives of 119869 [11] For any direction 119904 isin Z

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= ⟨119869

119906 (119906 (119911) 119911) 119906119911 (

119911) 119904⟩

119880lowast119880

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(39)

where119880lowast andZlowast are the dual space of119880 andZ respectivelyNow computing the derivative of the constraint 119890with respectto 119911 in the direction 119904 and applying it to the implicit solutionto 119890(119906(119911) 119911) = 0 yield

119890

119906 (119906 (119911) 119911) 119906119911 (

119911) 119904 + 119890

119911 (119906 (119911) 119911) 119904 = 0 (40)

So 119906

119911(119911) is

119906

119911 (119911) 119904 = minus119890

119906(119906 (119911) 119911)

minus1119890

119911 (119906 (119911) 119911) 119904

(41)

Hence

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= minus ⟨119869

119906 (119906 (119911) 119911) 119890119906(

119906 (119911) 119911)

minus1

times 119890

119911 (119906 (119911) 119911) 119904⟩

119880lowast119880

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(42)

Applying the adjoint to the 119906

119911(119911) term

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= minus ⟨119890

119911(119906 (119911) 119911)

lowast119890

119906(119906 (119911) 119911)

minuslowast

times 119869

119906 (119906 (119911) 119911) 119904⟩

Zlowast Z

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

= minus ⟨119890

119911(119906 (119911) 119911)

lowast119890

119906(119906 (119911) 119911)

minuslowast119869

119906(119906 (119911) 119911)

+119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(43)

Considering the adjoint state 120583 = 120583(119911) isin Zlowast as the solutionto

119890

119906(119906 (119911) 119911)

lowast120583 = minus119869

119906 (119906 (119911) 119911) (44)

the derivative of 1198691015840 becomes

119869

1015840(119911) = 119890

119911(119906 (119911) 119911)

lowast120583 (119911) + 119869

119911 (119906 (119911) 119911)

(45)

Theweak formof the constraint function is defined as followsfor all V isin 119880

⟨119890 (119906 119911) V⟩119880lowast 119880 = int

Ω

120588 (120596)int

119863

(119886 (x 120596) nabla119906 (x 120596) sdot nablaV (x 120596)

minus119911 (x 120596) V (x 120596)) 119889x119889120596

(46)

Thus the constraint function 119890 119880 times Z rarr 119880

lowast and 119884 = 119880

lowastSince the constraint function 119890 acts linearly with respect to 119906

and 119911 thus the corresponding derivative of 119890 with respect to119911 in the direction 120575119911 isin Z for all V isin 119880 is given by

⟨119890

119911 (119906 119911) 120575119911 V⟩

119880lowast119880

= minusint

Ω

120588 (120596)int

119863

120575119911 (x 120596) V (x 120596) 119889x119889120596

(47)

Similarly the derivative of 119890 with respect to 119906 for anydirection 120575119906 isin 119880 and for all V isin 119880 is given by

⟨119890

119906 (119906 119911) 120575119906 V⟩

119880lowast119880

= int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla120575119906 (x 120596)

sdot nablaV (x 120596) 119889x119889120596

(48)

Both of these derivatives are symmetric so 119890

119906(119906 119911)

lowast=

119890

119906(119906 119911) and 119890

119911(119906 119911)

lowast= 119890

119911(119906 119911) From here the adjoint can

be computed as

119890

119906 (119906 (119911) 119911) 120583 = minus119869

119906 (119906 (119911) 119911) (49)

Using this to any direction V isin 119880

⟨119890

119906 (119906 (119911) 119911) 120583 V⟩

119880lowast119880

= int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla120583 (x 120596)

sdot nablaV (x 120596) 119889x119889120596

= minus119869

119906 (119906 (119911) 119911) V

(50)

6 ISRN Applied Mathematics

By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is

⟨119869

119906 (119906 119911) 120575119906⟩

119880lowast119880

= 119864 [119906 (x 120596) minus (x)21198712(119863)

times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)

]

(51)

Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin

Z direction is

⟨119869

119911 (119906 119911) 120575119911⟩

Zlowast Z= 120572⟨119911 120575119911⟩Z (52)

With these computations one can compute the firstderivative of

119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883

ℎsub 119867

1

0(119863) is a finite dimensional

subspace of dimension 119873 and 119884

ℎsub 119871

2(Ω) is a finite

dimensional subspace of dimension 119875 then 119883

ℎ⨂119884

ℎis a

finite dimensional subspace of the state space 119880 Similarlylet Z

ℎsub Z be a finite dimensional subspace of the control

space (with dimension) For the stochastic Galerkin methodit is assumed that 119884

ℎ= P119875minus1 the space of polynomials with

highest degree 119875minus1 and119883

ℎis any finite element space here

119883

ℎis the space of linear functions built on a given meshT

ℎ

We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884

ℎ that is 119884

ℎ= spanΨ

1(120596) Ψ

119875(120596)

where

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596 = 120575

119898119899 (53)

The discretized optimization problem in the stochasticGalerkin framework is

min119911ℎisinZℎ

119869

ℎ119875(119911

ℎ) =

1

2

119864

[

[

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119875

sum

119896=1

119873

sum

119895=1

(

119896(119911

ℎ))

119895Ψ

119896 (120596) 120601119895 (

x)

minus (x)1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(119863)

]

]

+

120572

2

1003817

1003817

1003817

1003817

119911

ℎ

1003817

1003817

1003817

1003817

2

1198712(119863)

(54)

where

119896(119911

ℎ) = (

1(119911

ℎ)

119879

119875(119911

ℎ)

119879)

119879

=

(55)

is the stochastic Galerkin solution to the state equation

(

119870

11sdot sdot sdot 119870

1119875

d

119870

1198751sdot sdot sdot 119870

119875119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119870

(

1

119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=

= (

119864 [Ψ

1]119872

119911

ℎ

119864 [Ψ

119875]119872

119911

ℎ

)

= (

120575

11119872

119911

ℎ

120575

1119875119872

119911

ℎ

)

(56)

since 119864[Ψ

119896] = 119864[1Ψ

119896] = 119864[Ψ

1Ψ

119896] = 120575

1119896 The blocks of the

above 119870matrix have the form

(119870

119896119897)

119894119895= int

Ω

120588 (120596)Ψ

119896(120596)Ψ119897 (

120596)

times int

119863

119886 (x 120596) nabla120601

119894 (x) sdot nabla120601

119895 (x) 119889x119889120596

(57)

First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations

minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))

in 119863 times Ω

120583 (x 120596) = 0 on 120597119863 times Ω

(58)

Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865

119879

1 119865

119879

119901)

119879 and 119865

119896is defined

as

(

119865

119896)

119894= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596

= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895120601

119895 (x) Ψ119897 (120596) 120601119894 (

x)

minus (x) 120601119894 (x))119889x119889120596

= minus

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

(x) 120601119894 (x) 119889x

=

119887

119894120575

1119896minus

119875

sum

119897=1

119873

sum

119895=1

119872

119894119895(

119897)

119895120575

119896119897

=

119887

119894120575

1119896minus

119873

sum

119895=1

119872

119894119895(

119897)

119895

(59)

in which

119887

119894= int 120601

119894 and119872

119894119895= int 120601

119894120601

119895

Thus

(

119865

119896)

119894=

119887 minus 119872119906

1if 119896 = 1

minus119872119906 if 119896 = 2 119875

(60)

where b= [

119887

1

119887

119873] and the matrix 119872 = (119872

119894119895) is of order

119873 times 119873

ISRN Applied Mathematics 7

Now the derivative of

119869(119911) in the direction 120601

119895(x) for all

119895 = 1 119873 with the computed adjoint state is

⟨

119869

1015840(119911) 120601119895 (

x)⟩Zlowast Z

= ⟨119890

119911(119906 (119911 sdot sdot) 119911)

lowast120583

+ 119869

119911 (119906 (119911 sdot sdot) 119911) 120601119895 (

x) ⟩Zlowast Z

= minusint

Ω

120588 (120596)int

119863

120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596

+ 120572int

119863

119911 (119909) 120601119895 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894int

Ω

120588 (120596)Ψ119896 (120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ 120572

119873

sum

119894=1

119911

119894int

119863

120601

119895 (x) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894119872

119894119895120575

1119896+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

= minus

119873

sum

119894=1

(

1)

119894119872

119894119895+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

(61)

Therefore the gradient can be computed as follows

119869

1015840(119911) = 119872(120572

119911

ℎminus

1)

(62)

Now for any vector V isin Z it is possible to write V as

V =

119873

sum

119896=1

V119896120601

119896 (x) (63)

In order to compute the Hessian times a vector that ismultiply the Hessian of the

119869 with a some vector V isin Z theequation 119890

119906(119906(119911) 119911)119908 = 119890

119911(119906(119911) 119911)V for 119908 must be solved

Similar to the adjoint computation119908 solves the linear system119870 = 119866 where

119866 = (

119866

119879

1

119866

119879

119875) and

119866

119896for 119896 = 1 119875 is

(

119866

119896)

119894= int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

119873

sum

119895=1

V119895120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895V119895120575

1119896

(64)

for all 119894 = 1 119873 Thus

119866

119896=

119872V if 119896 = 1

0 if 119896 = 2 119875

(65)

Now solving 119890

119906(119906(119911) 119911)119902 = 119869

119906119906(119906(119911) 119911)119908 for 119902 requires

the solution to the linear system 119870 119902 = 119867 where

(

119867

119896)

119894=

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895(

119896)

119895

(66)

or equivalently

119867

119896= 119872

119896for all 119896 = 1 119875 Hence in the

direction 120601

119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by

⟨

119869

10158401015840(119911) V 120601119894⟩Zlowast Z

asymp ⟨

119869

10158401015840

ℎ119901(119911

ℎ) V 120601

119894⟩

Zlowast Z

= minusint

Ω

120588 (120596)int

119863

119875

sum

119896=1

119873

sum

119895=1

( 119902

119896)

119895120601

119895 (x) Ψ119896 (120596)

times 120601

119894 (x) 119889x119889120596

+ 120572int

119863

V (119909) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119895=1

119872

119894119895( 119902

119896)

119895120575

1119896+ 120572

119873

sum

119895=1

119872

119894119895V119895

=

119873

sum

119895=1

119872

119894119895(120572V

119895minus ( 119902

1)

119895)

(67)

Therefore

119869

10158401015840(119911) V asymp 119872(120572V minus 119902

1)

(68)

Having (62) and (68) we can use the iterative solvers tothe Newton equation

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) (69)

Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572

119896and using this step size to generate

the iterate 119911

119896+1= 119911

119896+ 120572

119896119878

119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is

119869 (119911

119896+ 120572

119896119878

119896) le 119869 (119911

119896) + 119888120572

119896119869

1015840(119911

119896) 119878

119896 (70)

where 119888 isin (0 1) and typically is quite small for example 119888 =

10

minus4 [12 13]

8 ISRN Applied Mathematics

(1) Given 119911

0 and tol gt 0 set 119896 = 0

(2) Compute 119870119872 119860 and

119887

(3) Compute (

119865

119896)

119894=

119887 minus 119872119906 if 119896 = 1

minus119872119906 if 119896 = 2 119875

(4) Solve 119870 120583 = 119865

(5) Compute

119869

1015840(119911

119896) = 119872(120572119911

119896minus

1)

(6) Compute

119866

1= 119872V and

119866

119905= 0 for 119905 = 2 119875

(7) Solve 119870 = 119866 = (

119866

119879

1

119866

119879

119875)

119879

(8) Compute

119867

119896= 119872

119896for 119896 = 1 119875

(9) Solve 119870 119902 = 119867 = (

119867

119879

1

119867

119879

119875)

119879

(10) Compute

119869

10158401015840(119911)V asymp 119872(120572V minus 119902

1)

(11) If 1003817100381710038171003817

1003817

nabla

119869 (119911

119896)

1003817

1003817

1003817

1003817

1003817

lt tol stop(12) Solve

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) using PCG method with preconditioner

119875 = 119866⨂119870

0

(i) Set 1199090= 0

(ii) Set 1199030larr

119869

10158401015840(119911

119896) 119909

0+

119869

1015840(119911

119896)

(iii) Solve

119875119910

0= 119903

0for 119910

0

(iv) Set 1199010= minus119903

0 119905 larr 0

(v) While 1003817

1003817

1003817

1003817

119903

119905

1003817

1003817

1003817

1003817

gt tol

120572

119896larr

119903

119879

119905119910

119905

119901

119879

119905119860119901

119905

119909

119905+1larr 119909

119905+ 120572

119905119901

119905

119903

119905+1larr 119903

119905+ 120572

119905119860119901

119905

119875119910

119905+1larr 119903

119905+1

120573

119905+1larr

119903

119879

119905+1119910

119905+1

119903

119879

119905119910

119905

119901

119905+1larr minus119910

119905+1+ 120573

119905+1119901

119905

119905 larr 119905 + 1

End (while)(vi) 119904119896 = 119909

119905+1

(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911

119896+ 120572

119896119878

119896)

(ii) While 119869 (119911

119896+ 120572

119896119878

119896) gt 119869 (119911

119896) + 10

minus4120572

119896119878

119896nabla

119869 (119911

119896) do

(i) Set 120572119896 = 120572

1198962 and evaluate 119869 (119911

119896+ 120572

119896119878

119896)

(14) Set 119911119896+1 = 119911

119896+ 120572

119896119878

119896 119896 larr 119896 + 1 Go to (2)

Algorithm 1

31 Kronecker Product Preconditioners Let us to rewrite (37)in the form

119860 =

119911 where

119860 = (

119872

sum

119896=0

119866

119896⨂119870

119896) (71)

The strategy here is to introduce

119875 = 119866⨂119870

0as a

preconditioner such that

119866 = argmin 119867 isin R119875times119875

1003817

1003817

1003817

1003817

1003817

119860 minus 119867⨂119870

0

1003817

1003817

1003817

1003817

1003817119865 (72)

where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot

119865denotes the Frobe-

nius norm The closed form of the solution can be written asfollow [14]

119866 = 119868 + sum

119875

tr (119870119879

119896119870

0)

tr (119870119879

0119870

0)

119866

119896

(73)

Here tr(119870119879

119896119870

0) = sum

119873119902

119894=1[119870

119896]

119894119894[119870

0]

119894119894and hence the coeffi-

cients in the above equality can be computed straightforwardIn addition since

119860 and 119870

0are symmetric and positive

definite thus 119866 and

119875 = 119866⨂119870

0have also these properties

[15 16]

Example 1 Consider the following distributed optimal con-trol problem

min119906119911

119869 (119906 119911) = 119864 [

1

2

119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω

(74)

where 119863 = [minus1 1]

2 Ω sim 119880[minus1 1] the boundary conditionsare given as

119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0

(75)

ISRN Applied Mathematics 9

005

1

005

10

05

1y d

xy

(a) Desired values

005

1

005

10

05

1

uM

ean

xy

times10minus3

(b) Initial state

005

1

005

10

05

1

u opt

xy

(c) Approximated solution (state)

005

1

005

10

200

400

z opt

xy

(d) Control

005

1

0

05

10

05

1

times 10minus3

xy

120583

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

05

1

15

2

25

times 10minus3

(f) Expected value

20 40

10

20

30

40

50

05

1

15

2

25

times 10minus3

(g) Standard dev

0 5 10times 104

0

5

10

times 104 0

(h) Sparsity of global matrix119899119911 = 1815980

Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))

2+ (119910 minus (12))

2)]

119873 = 3 119870 = 5 and 120572 = 000005

and the desired function is given by

(119909 119910) = exp [minus64 ((119909 minus

1

2

)

2

+ (119910 minus

1

2

)

2

)] (76)

The random field 119886(x 120596) is characterized by its mean andcovariance function

119864 [119886] = 119886 = 10

119877 (119909

1 119909

2) = 119890

minus|1199091minus1199092| 119909

1 119909

2isin [minus1 1]

(77)

Following Section 22 the truncated KLE can be representedas

119886 (119909 119910 120596) = 10 +

119873

sum

119895=0

radic120582

119895120579

119895120601

119895 (119909) (78)

The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867

1

0(119863)⨂119871

2(Ω) and all 119911isin 119871

2(119863)

119869 (119906opt 119911opt) le 119869 (119906 119911) (79)

The eigenpairs 120582119895 120579

119895 in truncated KLE solve the integral

equation

int

1

minus1

119890

minus|1199091minus1199092|120601

119895(119909

2) 119889119909

2= 120582

119895120601

119895(119909

1)

(80)

For this special case of the covariance function we haveexplicit expressions for 120601

119895and 120582

119895 [6] Let120596

119895evenand120596

119895oddsolve

the equations

1 minus 120596

119895eventan (120596

119895even) = 0

120596

119895odd+ tan (120596

119895odd) = 0

(81)

Then the even and odd indexed eigenfunctions are given by

120601

119895even(119909) =

cos (120596119895even

119909)

radic1 + (sin (2120596

119895even) 2120596

119895even)

120601

119895odd(119909)

=

sin (120596

119895odd119909)

radic1 minus (sin (2120596

119895odd) 2120596

119895odd)

(82)

10 ISRN Applied Mathematics

005

1

005

10

05

1y ha

nd

xy

(a) Desired values

005

1

005

1minus05

05

0

u Mea

n

xy

(b) Mean values

005

1

005

1

0

Num

sol

xy

0

1

(c) Approximated solution (State)

005

1

005

10

05

1

xy

(d) Control

005

1

005

10

05

1

Lend

a

xy

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

0

005

01

015

02

(f) Expected value-SGS

20 40

10

20

30

40

50

0

005

01

015

02

(g) Standard dev-SGS

0 2 4 6 8times 104

0

2

4

6

8

times 104

(h) Sparsity of global matrix119899119911 = 3046160

Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001

and corresponding eigenvalues

120582

119895even(119909) =

2

120596

2

119895even+ 1

120582

119895odd(119909) =

2

120596

2

119895odd+ 1

(83)

We choose 120579 = (120579

1 120579

119873)

119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]

First of all we find numerical approximations of120596119895even

and120596

119895oddwith bisection method and then with the eigenpairs

evaluated by this 120596119895even

and 120596

119895odd by choosing dimension 119873 =

3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1

In Figure 1(a) the desired function (119909 119910) is plottedfor 119909

119894= 119910

119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial

state is computed and plotted in Figure 1(b) by solving (37)where

119911 = 0 Approximated 119906opt corresponding to the

optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial

119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911

119900119901119905must approximate the

initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane

In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use

Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary

ISRN Applied Mathematics 11

conditions is replaced by (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2

4 Conclusion

The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996

[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987

[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005

[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947

[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946

[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991

[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007

[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009

[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938

[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993

[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008

[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999

[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006

[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010

[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010

[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Complex AnalysisJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 ISRN Applied Mathematics

where 120588(120596) is the integration weight Equation (24) is equiv-alent to the following equation which results by plugging inthe explicit formula for the basis functions

int

Ω

120588 (120596)Ψn (120596) int

119863

119886 (x 120596) nabla119906

ℎ119875 (x 120596) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψn (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(25)

for all n isinI and 119894 = 1 119873Now considering the approximate solution as

119906

ℎ119875 (x 120596) =

119873

sum

119895=1

sum

misinI119906

119895m120601

119895 (x) Ψm (120596) (26)

and substituting (26) into (25) yields119873

sum

119895=1

sum

misinI119906

119895m int

Ω

120588 (120596)Ψn (120596)Ψm (120596)

times int

119863

119886 (x 120596) nabla120601

119895 (x) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψn (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(27)

for allm isinI and 119894 = 1 119873 Let 119875 = |I| = prod

119872

119896=1119901

119896 then

a natural bijection between 1 119875 and Ican be definedThus (27) can be written as

119875

sum

119898=1

119873

sum

119895=1

119906

119898119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596)

times int

119863

119886 (x 120596) nabla120601

119895 (x) sdot nabla120601

119894 (x) 119889x119889120596

= int

Ω

120588 (120596)Ψ119899 (120596) int

119863

119911 (x 120596) 120601119894 (x) 119889x119889120596

(28)

Now if 119886 and 119911 have the following KLE

119886 (x 120596) = 119886

0 (x) +

119872

sum

119894=1

120596

119894119886

119894 (x)

119911 (x 120596) = 119911

0 (x) +

119872

sum

119894=1

120596

119894119911

119894 (x)

(29)

then we can rewrite the Galerkin projection (28) as119875

sum

119898=1

119873

sum

119895=1

119906

119898119895((119870

0)

119894119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596

+

119872

sum

119896=1

(119870

119896)

119894119895int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 120596119896

119889120596)

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(30)

where

(119870

119896)

119894119895= int

119863

119886

119896 (x) nabla120601

119894 (x) sdot nabla120601

119895 (x) 119889x

cong sum

119903

119908

119903119886

119896(x119903) nabla120601

119894(x119903) sdot nabla120601

119895(x119903)

(119911

119896)

119894= int

119863

119911

119894 (x) 120601119894 (x) 119889x cong sum

119903

119908

119903119911

119894(x119903) 120601

119894(x119903)

(31)

for 119896 = 1 119872 [3] Equation (31) is approximated byquadrature rule in spatial domain Now assume that thereexist functions Ψ

119899 119899 = 1 119875 such that

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596 = 120575

119898119899

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 120596119896

119889120596 = 119862

119896119899120575

119899119898

(32)

where for 119898 = 119899 the above integral is approximated usingquadrature rule in random domain as follows

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119899 (

120596) 120596119896119889120596 cong sum

119904

119908

1015840

119904120588 (120596

119904) Ψ

2

119899(120596

119904) 120596

119896 (33)

Then (30) becomes

119875

sum

119898=1

119873

sum

119895=1

119906

119898119895((119870

0)

119894119895+

119872

sum

119896=1

119862

119896119899(119870

119896)

119894119895)120575

119899119898

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(34)

or equivalently

119873

sum

119895=1

119906m119895 ((119870

0)

119894119895+

119872

sum

119896=1

119862

119896119899(119870

119896)

119894119895)

= (119911

0)

119894int

Ω

120588 (120596)Ψ119899 (120596) 119889120596

+

119872

sum

119896=1

(119911

119896)

119894int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596

(35)

In (34) and (35) we have

int

Ω

120588 (120596)Ψ119899 (120596) 119889120596 = 0 for 119899 = 0

int

Ω

120588 (120596)Ψ119899 (120596) 120596119896

119889120596 = int

Ω

120588 (120596)Ψ0 (120596)Ψ119899 (

120596) 120596119896119889120596

= 119862

119896119899120575

0119899= 119862

1198960

(36)

ISRN Applied Mathematics 5

Finally (35) can be considered as the following blockdiagonal system

(

(

(

119870

0+

119872

sum

119896=1

119862

1198961119870

119896sdot sdot sdot 0

d

0 sdot sdot sdot 119870

0+

119872

sum

119896=1

119862

119896119901119870

119896

)

)

)

(

1

119875

)

= (

119911

1

119911

119901

)

(37)

or briefly 119870 =

119911

3 Distributed Optimal Control Problem

Using the stochastic Galerkin method for general objectivefunctions computing the derivatives becomes very tediousand rather messy Consider the following optimal controlproblem

min119906isin119880 119911isinz

119869 (119906 119911) =

1

2

119864 [119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

st minus nabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω

119906 (x 120596) = 0 on 120597119863 times Ω

(38)

The solutions to linear elliptic SPDEs live in the space119880 =

119867

1

0(119863)⨂119871

2(Ω) and the control space Z = 119871

2(119863) Denoting

the constraint equation by 119890(119906 119911) = 0 usually it is possibleto invoke the implicit function theorem to find a solution119906(119911) to the constraint equation This leads to an implicitlydefined objective function

119869(119911) = 119869(119906(119911) 119911) Now we focuson computing derivatives of 119869 [11] For any direction 119904 isin Z

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= ⟨119869

119906 (119906 (119911) 119911) 119906119911 (

119911) 119904⟩

119880lowast119880

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(39)

where119880lowast andZlowast are the dual space of119880 andZ respectivelyNow computing the derivative of the constraint 119890with respectto 119911 in the direction 119904 and applying it to the implicit solutionto 119890(119906(119911) 119911) = 0 yield

119890

119906 (119906 (119911) 119911) 119906119911 (

119911) 119904 + 119890

119911 (119906 (119911) 119911) 119904 = 0 (40)

So 119906

119911(119911) is

119906

119911 (119911) 119904 = minus119890

119906(119906 (119911) 119911)

minus1119890

119911 (119906 (119911) 119911) 119904

(41)

Hence

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= minus ⟨119869

119906 (119906 (119911) 119911) 119890119906(

119906 (119911) 119911)

minus1

times 119890

119911 (119906 (119911) 119911) 119904⟩

119880lowast119880

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(42)

Applying the adjoint to the 119906

119911(119911) term

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= minus ⟨119890

119911(119906 (119911) 119911)

lowast119890

119906(119906 (119911) 119911)

minuslowast

times 119869

119906 (119906 (119911) 119911) 119904⟩

Zlowast Z

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

= minus ⟨119890

119911(119906 (119911) 119911)

lowast119890

119906(119906 (119911) 119911)

minuslowast119869

119906(119906 (119911) 119911)

+119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(43)

Considering the adjoint state 120583 = 120583(119911) isin Zlowast as the solutionto

119890

119906(119906 (119911) 119911)

lowast120583 = minus119869

119906 (119906 (119911) 119911) (44)

the derivative of 1198691015840 becomes

119869

1015840(119911) = 119890

119911(119906 (119911) 119911)

lowast120583 (119911) + 119869

119911 (119906 (119911) 119911)

(45)

Theweak formof the constraint function is defined as followsfor all V isin 119880

⟨119890 (119906 119911) V⟩119880lowast 119880 = int

Ω

120588 (120596)int

119863

(119886 (x 120596) nabla119906 (x 120596) sdot nablaV (x 120596)

minus119911 (x 120596) V (x 120596)) 119889x119889120596

(46)

Thus the constraint function 119890 119880 times Z rarr 119880

lowast and 119884 = 119880

lowastSince the constraint function 119890 acts linearly with respect to 119906

and 119911 thus the corresponding derivative of 119890 with respect to119911 in the direction 120575119911 isin Z for all V isin 119880 is given by

⟨119890

119911 (119906 119911) 120575119911 V⟩

119880lowast119880

= minusint

Ω

120588 (120596)int

119863

120575119911 (x 120596) V (x 120596) 119889x119889120596

(47)

Similarly the derivative of 119890 with respect to 119906 for anydirection 120575119906 isin 119880 and for all V isin 119880 is given by

⟨119890

119906 (119906 119911) 120575119906 V⟩

119880lowast119880

= int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla120575119906 (x 120596)

sdot nablaV (x 120596) 119889x119889120596

(48)

Both of these derivatives are symmetric so 119890

119906(119906 119911)

lowast=

119890

119906(119906 119911) and 119890

119911(119906 119911)

lowast= 119890

119911(119906 119911) From here the adjoint can

be computed as

119890

119906 (119906 (119911) 119911) 120583 = minus119869

119906 (119906 (119911) 119911) (49)

Using this to any direction V isin 119880

⟨119890

119906 (119906 (119911) 119911) 120583 V⟩

119880lowast119880

= int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla120583 (x 120596)

sdot nablaV (x 120596) 119889x119889120596

= minus119869

119906 (119906 (119911) 119911) V

(50)

6 ISRN Applied Mathematics

By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is

⟨119869

119906 (119906 119911) 120575119906⟩

119880lowast119880

= 119864 [119906 (x 120596) minus (x)21198712(119863)

times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)

]

(51)

Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin

Z direction is

⟨119869

119911 (119906 119911) 120575119911⟩

Zlowast Z= 120572⟨119911 120575119911⟩Z (52)

With these computations one can compute the firstderivative of

119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883

ℎsub 119867

1

0(119863) is a finite dimensional

subspace of dimension 119873 and 119884

ℎsub 119871

2(Ω) is a finite

dimensional subspace of dimension 119875 then 119883

ℎ⨂119884

ℎis a

finite dimensional subspace of the state space 119880 Similarlylet Z

ℎsub Z be a finite dimensional subspace of the control

space (with dimension) For the stochastic Galerkin methodit is assumed that 119884

ℎ= P119875minus1 the space of polynomials with

highest degree 119875minus1 and119883

ℎis any finite element space here

119883

ℎis the space of linear functions built on a given meshT

ℎ

We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884

ℎ that is 119884

ℎ= spanΨ

1(120596) Ψ

119875(120596)

where

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596 = 120575

119898119899 (53)

The discretized optimization problem in the stochasticGalerkin framework is

min119911ℎisinZℎ

119869

ℎ119875(119911

ℎ) =

1

2

119864

[

[

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119875

sum

119896=1

119873

sum

119895=1

(

119896(119911

ℎ))

119895Ψ

119896 (120596) 120601119895 (

x)

minus (x)1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(119863)

]

]

+

120572

2

1003817

1003817

1003817

1003817

119911

ℎ

1003817

1003817

1003817

1003817

2

1198712(119863)

(54)

where

119896(119911

ℎ) = (

1(119911

ℎ)

119879

119875(119911

ℎ)

119879)

119879

=

(55)

is the stochastic Galerkin solution to the state equation

(

119870

11sdot sdot sdot 119870

1119875

d

119870

1198751sdot sdot sdot 119870

119875119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119870

(

1

119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=

= (

119864 [Ψ

1]119872

119911

ℎ

119864 [Ψ

119875]119872

119911

ℎ

)

= (

120575

11119872

119911

ℎ

120575

1119875119872

119911

ℎ

)

(56)

since 119864[Ψ

119896] = 119864[1Ψ

119896] = 119864[Ψ

1Ψ

119896] = 120575

1119896 The blocks of the

above 119870matrix have the form

(119870

119896119897)

119894119895= int

Ω

120588 (120596)Ψ

119896(120596)Ψ119897 (

120596)

times int

119863

119886 (x 120596) nabla120601

119894 (x) sdot nabla120601

119895 (x) 119889x119889120596

(57)

First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations

minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))

in 119863 times Ω

120583 (x 120596) = 0 on 120597119863 times Ω

(58)

Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865

119879

1 119865

119879

119901)

119879 and 119865

119896is defined

as

(

119865

119896)

119894= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596

= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895120601

119895 (x) Ψ119897 (120596) 120601119894 (

x)

minus (x) 120601119894 (x))119889x119889120596

= minus

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

(x) 120601119894 (x) 119889x

=

119887

119894120575

1119896minus

119875

sum

119897=1

119873

sum

119895=1

119872

119894119895(

119897)

119895120575

119896119897

=

119887

119894120575

1119896minus

119873

sum

119895=1

119872

119894119895(

119897)

119895

(59)

in which

119887

119894= int 120601

119894 and119872

119894119895= int 120601

119894120601

119895

Thus

(

119865

119896)

119894=

119887 minus 119872119906

1if 119896 = 1

minus119872119906 if 119896 = 2 119875

(60)

where b= [

119887

1

119887

119873] and the matrix 119872 = (119872

119894119895) is of order

119873 times 119873

ISRN Applied Mathematics 7

Now the derivative of

119869(119911) in the direction 120601

119895(x) for all

119895 = 1 119873 with the computed adjoint state is

⟨

119869

1015840(119911) 120601119895 (

x)⟩Zlowast Z

= ⟨119890

119911(119906 (119911 sdot sdot) 119911)

lowast120583

+ 119869

119911 (119906 (119911 sdot sdot) 119911) 120601119895 (

x) ⟩Zlowast Z

= minusint

Ω

120588 (120596)int

119863

120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596

+ 120572int

119863

119911 (119909) 120601119895 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894int

Ω

120588 (120596)Ψ119896 (120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ 120572

119873

sum

119894=1

119911

119894int

119863

120601

119895 (x) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894119872

119894119895120575

1119896+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

= minus

119873

sum

119894=1

(

1)

119894119872

119894119895+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

(61)

Therefore the gradient can be computed as follows

119869

1015840(119911) = 119872(120572

119911

ℎminus

1)

(62)

Now for any vector V isin Z it is possible to write V as

V =

119873

sum

119896=1

V119896120601

119896 (x) (63)

In order to compute the Hessian times a vector that ismultiply the Hessian of the

119869 with a some vector V isin Z theequation 119890

119906(119906(119911) 119911)119908 = 119890

119911(119906(119911) 119911)V for 119908 must be solved

Similar to the adjoint computation119908 solves the linear system119870 = 119866 where

119866 = (

119866

119879

1

119866

119879

119875) and

119866

119896for 119896 = 1 119875 is

(

119866

119896)

119894= int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

119873

sum

119895=1

V119895120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895V119895120575

1119896

(64)

for all 119894 = 1 119873 Thus

119866

119896=

119872V if 119896 = 1

0 if 119896 = 2 119875

(65)

Now solving 119890

119906(119906(119911) 119911)119902 = 119869

119906119906(119906(119911) 119911)119908 for 119902 requires

the solution to the linear system 119870 119902 = 119867 where

(

119867

119896)

119894=

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895(

119896)

119895

(66)

or equivalently

119867

119896= 119872

119896for all 119896 = 1 119875 Hence in the

direction 120601

119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by

⟨

119869

10158401015840(119911) V 120601119894⟩Zlowast Z

asymp ⟨

119869

10158401015840

ℎ119901(119911

ℎ) V 120601

119894⟩

Zlowast Z

= minusint

Ω

120588 (120596)int

119863

119875

sum

119896=1

119873

sum

119895=1

( 119902

119896)

119895120601

119895 (x) Ψ119896 (120596)

times 120601

119894 (x) 119889x119889120596

+ 120572int

119863

V (119909) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119895=1

119872

119894119895( 119902

119896)

119895120575

1119896+ 120572

119873

sum

119895=1

119872

119894119895V119895

=

119873

sum

119895=1

119872

119894119895(120572V

119895minus ( 119902

1)

119895)

(67)

Therefore

119869

10158401015840(119911) V asymp 119872(120572V minus 119902

1)

(68)

Having (62) and (68) we can use the iterative solvers tothe Newton equation

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) (69)

Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572

119896and using this step size to generate

the iterate 119911

119896+1= 119911

119896+ 120572

119896119878

119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is

119869 (119911

119896+ 120572

119896119878

119896) le 119869 (119911

119896) + 119888120572

119896119869

1015840(119911

119896) 119878

119896 (70)

where 119888 isin (0 1) and typically is quite small for example 119888 =

10

minus4 [12 13]

8 ISRN Applied Mathematics

(1) Given 119911

0 and tol gt 0 set 119896 = 0

(2) Compute 119870119872 119860 and

119887

(3) Compute (

119865

119896)

119894=

119887 minus 119872119906 if 119896 = 1

minus119872119906 if 119896 = 2 119875

(4) Solve 119870 120583 = 119865

(5) Compute

119869

1015840(119911

119896) = 119872(120572119911

119896minus

1)

(6) Compute

119866

1= 119872V and

119866

119905= 0 for 119905 = 2 119875

(7) Solve 119870 = 119866 = (

119866

119879

1

119866

119879

119875)

119879

(8) Compute

119867

119896= 119872

119896for 119896 = 1 119875

(9) Solve 119870 119902 = 119867 = (

119867

119879

1

119867

119879

119875)

119879

(10) Compute

119869

10158401015840(119911)V asymp 119872(120572V minus 119902

1)

(11) If 1003817100381710038171003817

1003817

nabla

119869 (119911

119896)

1003817

1003817

1003817

1003817

1003817

lt tol stop(12) Solve

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) using PCG method with preconditioner

119875 = 119866⨂119870

0

(i) Set 1199090= 0

(ii) Set 1199030larr

119869

10158401015840(119911

119896) 119909

0+

119869

1015840(119911

119896)

(iii) Solve

119875119910

0= 119903

0for 119910

0

(iv) Set 1199010= minus119903

0 119905 larr 0

(v) While 1003817

1003817

1003817

1003817

119903

119905

1003817

1003817

1003817

1003817

gt tol

120572

119896larr

119903

119879

119905119910

119905

119901

119879

119905119860119901

119905

119909

119905+1larr 119909

119905+ 120572

119905119901

119905

119903

119905+1larr 119903

119905+ 120572

119905119860119901

119905

119875119910

119905+1larr 119903

119905+1

120573

119905+1larr

119903

119879

119905+1119910

119905+1

119903

119879

119905119910

119905

119901

119905+1larr minus119910

119905+1+ 120573

119905+1119901

119905

119905 larr 119905 + 1

End (while)(vi) 119904119896 = 119909

119905+1

(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911

119896+ 120572

119896119878

119896)

(ii) While 119869 (119911

119896+ 120572

119896119878

119896) gt 119869 (119911

119896) + 10

minus4120572

119896119878

119896nabla

119869 (119911

119896) do

(i) Set 120572119896 = 120572

1198962 and evaluate 119869 (119911

119896+ 120572

119896119878

119896)

(14) Set 119911119896+1 = 119911

119896+ 120572

119896119878

119896 119896 larr 119896 + 1 Go to (2)

Algorithm 1

31 Kronecker Product Preconditioners Let us to rewrite (37)in the form

119860 =

119911 where

119860 = (

119872

sum

119896=0

119866

119896⨂119870

119896) (71)

The strategy here is to introduce

119875 = 119866⨂119870

0as a

preconditioner such that

119866 = argmin 119867 isin R119875times119875

1003817

1003817

1003817

1003817

1003817

119860 minus 119867⨂119870

0

1003817

1003817

1003817

1003817

1003817119865 (72)

where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot

119865denotes the Frobe-

nius norm The closed form of the solution can be written asfollow [14]

119866 = 119868 + sum

119875

tr (119870119879

119896119870

0)

tr (119870119879

0119870

0)

119866

119896

(73)

Here tr(119870119879

119896119870

0) = sum

119873119902

119894=1[119870

119896]

119894119894[119870

0]

119894119894and hence the coeffi-

cients in the above equality can be computed straightforwardIn addition since

119860 and 119870

0are symmetric and positive

definite thus 119866 and

119875 = 119866⨂119870

0have also these properties

[15 16]

Example 1 Consider the following distributed optimal con-trol problem

min119906119911

119869 (119906 119911) = 119864 [

1

2

119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω

(74)

where 119863 = [minus1 1]

2 Ω sim 119880[minus1 1] the boundary conditionsare given as

119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0

(75)

ISRN Applied Mathematics 9

005

1

005

10

05

1y d

xy

(a) Desired values

005

1

005

10

05

1

uM

ean

xy

times10minus3

(b) Initial state

005

1

005

10

05

1

u opt

xy

(c) Approximated solution (state)

005

1

005

10

200

400

z opt

xy

(d) Control

005

1

0

05

10

05

1

times 10minus3

xy

120583

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

05

1

15

2

25

times 10minus3

(f) Expected value

20 40

10

20

30

40

50

05

1

15

2

25

times 10minus3

(g) Standard dev

0 5 10times 104

0

5

10

times 104 0

(h) Sparsity of global matrix119899119911 = 1815980

Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))

2+ (119910 minus (12))

2)]

119873 = 3 119870 = 5 and 120572 = 000005

and the desired function is given by

(119909 119910) = exp [minus64 ((119909 minus

1

2

)

2

+ (119910 minus

1

2

)

2

)] (76)

The random field 119886(x 120596) is characterized by its mean andcovariance function

119864 [119886] = 119886 = 10

119877 (119909

1 119909

2) = 119890

minus|1199091minus1199092| 119909

1 119909

2isin [minus1 1]

(77)

Following Section 22 the truncated KLE can be representedas

119886 (119909 119910 120596) = 10 +

119873

sum

119895=0

radic120582

119895120579

119895120601

119895 (119909) (78)

The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867

1

0(119863)⨂119871

2(Ω) and all 119911isin 119871

2(119863)

119869 (119906opt 119911opt) le 119869 (119906 119911) (79)

The eigenpairs 120582119895 120579

119895 in truncated KLE solve the integral

equation

int

1

minus1

119890

minus|1199091minus1199092|120601

119895(119909

2) 119889119909

2= 120582

119895120601

119895(119909

1)

(80)

For this special case of the covariance function we haveexplicit expressions for 120601

119895and 120582

119895 [6] Let120596

119895evenand120596

119895oddsolve

the equations

1 minus 120596

119895eventan (120596

119895even) = 0

120596

119895odd+ tan (120596

119895odd) = 0

(81)

Then the even and odd indexed eigenfunctions are given by

120601

119895even(119909) =

cos (120596119895even

119909)

radic1 + (sin (2120596

119895even) 2120596

119895even)

120601

119895odd(119909)

=

sin (120596

119895odd119909)

radic1 minus (sin (2120596

119895odd) 2120596

119895odd)

(82)

10 ISRN Applied Mathematics

005

1

005

10

05

1y ha

nd

xy

(a) Desired values

005

1

005

1minus05

05

0

u Mea

n

xy

(b) Mean values

005

1

005

1

0

Num

sol

xy

0

1

(c) Approximated solution (State)

005

1

005

10

05

1

xy

(d) Control

005

1

005

10

05

1

Lend

a

xy

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

0

005

01

015

02

(f) Expected value-SGS

20 40

10

20

30

40

50

0

005

01

015

02

(g) Standard dev-SGS

0 2 4 6 8times 104

0

2

4

6

8

times 104

(h) Sparsity of global matrix119899119911 = 3046160

Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001

and corresponding eigenvalues

120582

119895even(119909) =

2

120596

2

119895even+ 1

120582

119895odd(119909) =

2

120596

2

119895odd+ 1

(83)

We choose 120579 = (120579

1 120579

119873)

119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]

First of all we find numerical approximations of120596119895even

and120596

119895oddwith bisection method and then with the eigenpairs

evaluated by this 120596119895even

and 120596

119895odd by choosing dimension 119873 =

3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1

In Figure 1(a) the desired function (119909 119910) is plottedfor 119909

119894= 119910

119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial

state is computed and plotted in Figure 1(b) by solving (37)where

119911 = 0 Approximated 119906opt corresponding to the

optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial

119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911

119900119901119905must approximate the

initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane

In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use

Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary

ISRN Applied Mathematics 11

conditions is replaced by (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2

4 Conclusion

The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996

[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987

[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005

[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947

[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946

[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991

[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007

[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009

[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938

[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993

[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008

[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999

[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006

[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010

[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010

[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Applied Mathematics 5

Finally (35) can be considered as the following blockdiagonal system

(

(

(

119870

0+

119872

sum

119896=1

119862

1198961119870

119896sdot sdot sdot 0

d

0 sdot sdot sdot 119870

0+

119872

sum

119896=1

119862

119896119901119870

119896

)

)

)

(

1

119875

)

= (

119911

1

119911

119901

)

(37)

or briefly 119870 =

119911

3 Distributed Optimal Control Problem

Using the stochastic Galerkin method for general objectivefunctions computing the derivatives becomes very tediousand rather messy Consider the following optimal controlproblem

min119906isin119880 119911isinz

119869 (119906 119911) =

1

2

119864 [119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

st minus nabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω

119906 (x 120596) = 0 on 120597119863 times Ω

(38)

The solutions to linear elliptic SPDEs live in the space119880 =

119867

1

0(119863)⨂119871

2(Ω) and the control space Z = 119871

2(119863) Denoting

the constraint equation by 119890(119906 119911) = 0 usually it is possibleto invoke the implicit function theorem to find a solution119906(119911) to the constraint equation This leads to an implicitlydefined objective function

119869(119911) = 119869(119906(119911) 119911) Now we focuson computing derivatives of 119869 [11] For any direction 119904 isin Z

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= ⟨119869

119906 (119906 (119911) 119911) 119906119911 (

119911) 119904⟩

119880lowast119880

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(39)

where119880lowast andZlowast are the dual space of119880 andZ respectivelyNow computing the derivative of the constraint 119890with respectto 119911 in the direction 119904 and applying it to the implicit solutionto 119890(119906(119911) 119911) = 0 yield

119890

119906 (119906 (119911) 119911) 119906119911 (

119911) 119904 + 119890

119911 (119906 (119911) 119911) 119904 = 0 (40)

So 119906

119911(119911) is

119906

119911 (119911) 119904 = minus119890

119906(119906 (119911) 119911)

minus1119890

119911 (119906 (119911) 119911) 119904

(41)

Hence

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= minus ⟨119869

119906 (119906 (119911) 119911) 119890119906(

119906 (119911) 119911)

minus1

times 119890

119911 (119906 (119911) 119911) 119904⟩

119880lowast119880

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(42)

Applying the adjoint to the 119906

119911(119911) term

⟨

119869

1015840(119911) 119904⟩

Zlowast Z= minus ⟨119890

119911(119906 (119911) 119911)

lowast119890

119906(119906 (119911) 119911)

minuslowast

times 119869

119906 (119906 (119911) 119911) 119904⟩

Zlowast Z

+ ⟨119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

= minus ⟨119890

119911(119906 (119911) 119911)

lowast119890

119906(119906 (119911) 119911)

minuslowast119869

119906(119906 (119911) 119911)

+119869

119911 (119906 (119911) 119911) 119904⟩

Zlowast Z

(43)

Considering the adjoint state 120583 = 120583(119911) isin Zlowast as the solutionto

119890

119906(119906 (119911) 119911)

lowast120583 = minus119869

119906 (119906 (119911) 119911) (44)

the derivative of 1198691015840 becomes

119869

1015840(119911) = 119890

119911(119906 (119911) 119911)

lowast120583 (119911) + 119869

119911 (119906 (119911) 119911)

(45)

Theweak formof the constraint function is defined as followsfor all V isin 119880

⟨119890 (119906 119911) V⟩119880lowast 119880 = int

Ω

120588 (120596)int

119863

(119886 (x 120596) nabla119906 (x 120596) sdot nablaV (x 120596)

minus119911 (x 120596) V (x 120596)) 119889x119889120596

(46)

Thus the constraint function 119890 119880 times Z rarr 119880

lowast and 119884 = 119880

lowastSince the constraint function 119890 acts linearly with respect to 119906

and 119911 thus the corresponding derivative of 119890 with respect to119911 in the direction 120575119911 isin Z for all V isin 119880 is given by

⟨119890

119911 (119906 119911) 120575119911 V⟩

119880lowast119880

= minusint

Ω

120588 (120596)int

119863

120575119911 (x 120596) V (x 120596) 119889x119889120596

(47)

Similarly the derivative of 119890 with respect to 119906 for anydirection 120575119906 isin 119880 and for all V isin 119880 is given by

⟨119890

119906 (119906 119911) 120575119906 V⟩

119880lowast119880

= int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla120575119906 (x 120596)

sdot nablaV (x 120596) 119889x119889120596

(48)

Both of these derivatives are symmetric so 119890

119906(119906 119911)

lowast=

119890

119906(119906 119911) and 119890

119911(119906 119911)

lowast= 119890

119911(119906 119911) From here the adjoint can

be computed as

119890

119906 (119906 (119911) 119911) 120583 = minus119869

119906 (119906 (119911) 119911) (49)

Using this to any direction V isin 119880

⟨119890

119906 (119906 (119911) 119911) 120583 V⟩

119880lowast119880

= int

Ω

120588 (120596)int

119863

119886 (x 120596) nabla120583 (x 120596)

sdot nablaV (x 120596) 119889x119889120596

= minus119869

119906 (119906 (119911) 119911) V

(50)

6 ISRN Applied Mathematics

By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is

⟨119869

119906 (119906 119911) 120575119906⟩

119880lowast119880

= 119864 [119906 (x 120596) minus (x)21198712(119863)

times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)

]

(51)

Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin

Z direction is

⟨119869

119911 (119906 119911) 120575119911⟩

Zlowast Z= 120572⟨119911 120575119911⟩Z (52)

With these computations one can compute the firstderivative of

119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883

ℎsub 119867

1

0(119863) is a finite dimensional

subspace of dimension 119873 and 119884

ℎsub 119871

2(Ω) is a finite

dimensional subspace of dimension 119875 then 119883

ℎ⨂119884

ℎis a

finite dimensional subspace of the state space 119880 Similarlylet Z

ℎsub Z be a finite dimensional subspace of the control

space (with dimension) For the stochastic Galerkin methodit is assumed that 119884

ℎ= P119875minus1 the space of polynomials with

highest degree 119875minus1 and119883

ℎis any finite element space here

119883

ℎis the space of linear functions built on a given meshT

ℎ

We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884

ℎ that is 119884

ℎ= spanΨ

1(120596) Ψ

119875(120596)

where

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596 = 120575

119898119899 (53)

The discretized optimization problem in the stochasticGalerkin framework is

min119911ℎisinZℎ

119869

ℎ119875(119911

ℎ) =

1

2

119864

[

[

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119875

sum

119896=1

119873

sum

119895=1

(

119896(119911

ℎ))

119895Ψ

119896 (120596) 120601119895 (

x)

minus (x)1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(119863)

]

]

+

120572

2

1003817

1003817

1003817

1003817

119911

ℎ

1003817

1003817

1003817

1003817

2

1198712(119863)

(54)

where

119896(119911

ℎ) = (

1(119911

ℎ)

119879

119875(119911

ℎ)

119879)

119879

=

(55)

is the stochastic Galerkin solution to the state equation

(

119870

11sdot sdot sdot 119870

1119875

d

119870

1198751sdot sdot sdot 119870

119875119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119870

(

1

119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=

= (

119864 [Ψ

1]119872

119911

ℎ

119864 [Ψ

119875]119872

119911

ℎ

)

= (

120575

11119872

119911

ℎ

120575

1119875119872

119911

ℎ

)

(56)

since 119864[Ψ

119896] = 119864[1Ψ

119896] = 119864[Ψ

1Ψ

119896] = 120575

1119896 The blocks of the

above 119870matrix have the form

(119870

119896119897)

119894119895= int

Ω

120588 (120596)Ψ

119896(120596)Ψ119897 (

120596)

times int

119863

119886 (x 120596) nabla120601

119894 (x) sdot nabla120601

119895 (x) 119889x119889120596

(57)

First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations

minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))

in 119863 times Ω

120583 (x 120596) = 0 on 120597119863 times Ω

(58)

Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865

119879

1 119865

119879

119901)

119879 and 119865

119896is defined

as

(

119865

119896)

119894= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596

= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895120601

119895 (x) Ψ119897 (120596) 120601119894 (

x)

minus (x) 120601119894 (x))119889x119889120596

= minus

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

(x) 120601119894 (x) 119889x

=

119887

119894120575

1119896minus

119875

sum

119897=1

119873

sum

119895=1

119872

119894119895(

119897)

119895120575

119896119897

=

119887

119894120575

1119896minus

119873

sum

119895=1

119872

119894119895(

119897)

119895

(59)

in which

119887

119894= int 120601

119894 and119872

119894119895= int 120601

119894120601

119895

Thus

(

119865

119896)

119894=

119887 minus 119872119906

1if 119896 = 1

minus119872119906 if 119896 = 2 119875

(60)

where b= [

119887

1

119887

119873] and the matrix 119872 = (119872

119894119895) is of order

119873 times 119873

ISRN Applied Mathematics 7

Now the derivative of

119869(119911) in the direction 120601

119895(x) for all

119895 = 1 119873 with the computed adjoint state is

⟨

119869

1015840(119911) 120601119895 (

x)⟩Zlowast Z

= ⟨119890

119911(119906 (119911 sdot sdot) 119911)

lowast120583

+ 119869

119911 (119906 (119911 sdot sdot) 119911) 120601119895 (

x) ⟩Zlowast Z

= minusint

Ω

120588 (120596)int

119863

120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596

+ 120572int

119863

119911 (119909) 120601119895 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894int

Ω

120588 (120596)Ψ119896 (120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ 120572

119873

sum

119894=1

119911

119894int

119863

120601

119895 (x) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894119872

119894119895120575

1119896+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

= minus

119873

sum

119894=1

(

1)

119894119872

119894119895+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

(61)

Therefore the gradient can be computed as follows

119869

1015840(119911) = 119872(120572

119911

ℎminus

1)

(62)

Now for any vector V isin Z it is possible to write V as

V =

119873

sum

119896=1

V119896120601

119896 (x) (63)

In order to compute the Hessian times a vector that ismultiply the Hessian of the

119869 with a some vector V isin Z theequation 119890

119906(119906(119911) 119911)119908 = 119890

119911(119906(119911) 119911)V for 119908 must be solved

Similar to the adjoint computation119908 solves the linear system119870 = 119866 where

119866 = (

119866

119879

1

119866

119879

119875) and

119866

119896for 119896 = 1 119875 is

(

119866

119896)

119894= int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

119873

sum

119895=1

V119895120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895V119895120575

1119896

(64)

for all 119894 = 1 119873 Thus

119866

119896=

119872V if 119896 = 1

0 if 119896 = 2 119875

(65)

Now solving 119890

119906(119906(119911) 119911)119902 = 119869

119906119906(119906(119911) 119911)119908 for 119902 requires

the solution to the linear system 119870 119902 = 119867 where

(

119867

119896)

119894=

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895(

119896)

119895

(66)

or equivalently

119867

119896= 119872

119896for all 119896 = 1 119875 Hence in the

direction 120601

119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by

⟨

119869

10158401015840(119911) V 120601119894⟩Zlowast Z

asymp ⟨

119869

10158401015840

ℎ119901(119911

ℎ) V 120601

119894⟩

Zlowast Z

= minusint

Ω

120588 (120596)int

119863

119875

sum

119896=1

119873

sum

119895=1

( 119902

119896)

119895120601

119895 (x) Ψ119896 (120596)

times 120601

119894 (x) 119889x119889120596

+ 120572int

119863

V (119909) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119895=1

119872

119894119895( 119902

119896)

119895120575

1119896+ 120572

119873

sum

119895=1

119872

119894119895V119895

=

119873

sum

119895=1

119872

119894119895(120572V

119895minus ( 119902

1)

119895)

(67)

Therefore

119869

10158401015840(119911) V asymp 119872(120572V minus 119902

1)

(68)

Having (62) and (68) we can use the iterative solvers tothe Newton equation

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) (69)

Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572

119896and using this step size to generate

the iterate 119911

119896+1= 119911

119896+ 120572

119896119878

119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is

119869 (119911

119896+ 120572

119896119878

119896) le 119869 (119911

119896) + 119888120572

119896119869

1015840(119911

119896) 119878

119896 (70)

where 119888 isin (0 1) and typically is quite small for example 119888 =

10

minus4 [12 13]

8 ISRN Applied Mathematics

(1) Given 119911

0 and tol gt 0 set 119896 = 0

(2) Compute 119870119872 119860 and

119887

(3) Compute (

119865

119896)

119894=

119887 minus 119872119906 if 119896 = 1

minus119872119906 if 119896 = 2 119875

(4) Solve 119870 120583 = 119865

(5) Compute

119869

1015840(119911

119896) = 119872(120572119911

119896minus

1)

(6) Compute

119866

1= 119872V and

119866

119905= 0 for 119905 = 2 119875

(7) Solve 119870 = 119866 = (

119866

119879

1

119866

119879

119875)

119879

(8) Compute

119867

119896= 119872

119896for 119896 = 1 119875

(9) Solve 119870 119902 = 119867 = (

119867

119879

1

119867

119879

119875)

119879

(10) Compute

119869

10158401015840(119911)V asymp 119872(120572V minus 119902

1)

(11) If 1003817100381710038171003817

1003817

nabla

119869 (119911

119896)

1003817

1003817

1003817

1003817

1003817

lt tol stop(12) Solve

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) using PCG method with preconditioner

119875 = 119866⨂119870

0

(i) Set 1199090= 0

(ii) Set 1199030larr

119869

10158401015840(119911

119896) 119909

0+

119869

1015840(119911

119896)

(iii) Solve

119875119910

0= 119903

0for 119910

0

(iv) Set 1199010= minus119903

0 119905 larr 0

(v) While 1003817

1003817

1003817

1003817

119903

119905

1003817

1003817

1003817

1003817

gt tol

120572

119896larr

119903

119879

119905119910

119905

119901

119879

119905119860119901

119905

119909

119905+1larr 119909

119905+ 120572

119905119901

119905

119903

119905+1larr 119903

119905+ 120572

119905119860119901

119905

119875119910

119905+1larr 119903

119905+1

120573

119905+1larr

119903

119879

119905+1119910

119905+1

119903

119879

119905119910

119905

119901

119905+1larr minus119910

119905+1+ 120573

119905+1119901

119905

119905 larr 119905 + 1

End (while)(vi) 119904119896 = 119909

119905+1

(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911

119896+ 120572

119896119878

119896)

(ii) While 119869 (119911

119896+ 120572

119896119878

119896) gt 119869 (119911

119896) + 10

minus4120572

119896119878

119896nabla

119869 (119911

119896) do

(i) Set 120572119896 = 120572

1198962 and evaluate 119869 (119911

119896+ 120572

119896119878

119896)

(14) Set 119911119896+1 = 119911

119896+ 120572

119896119878

119896 119896 larr 119896 + 1 Go to (2)

Algorithm 1

31 Kronecker Product Preconditioners Let us to rewrite (37)in the form

119860 =

119911 where

119860 = (

119872

sum

119896=0

119866

119896⨂119870

119896) (71)

The strategy here is to introduce

119875 = 119866⨂119870

0as a

preconditioner such that

119866 = argmin 119867 isin R119875times119875

1003817

1003817

1003817

1003817

1003817

119860 minus 119867⨂119870

0

1003817

1003817

1003817

1003817

1003817119865 (72)

where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot

119865denotes the Frobe-

nius norm The closed form of the solution can be written asfollow [14]

119866 = 119868 + sum

119875

tr (119870119879

119896119870

0)

tr (119870119879

0119870

0)

119866

119896

(73)

Here tr(119870119879

119896119870

0) = sum

119873119902

119894=1[119870

119896]

119894119894[119870

0]

119894119894and hence the coeffi-

cients in the above equality can be computed straightforwardIn addition since

119860 and 119870

0are symmetric and positive

definite thus 119866 and

119875 = 119866⨂119870

0have also these properties

[15 16]

Example 1 Consider the following distributed optimal con-trol problem

min119906119911

119869 (119906 119911) = 119864 [

1

2

119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω

(74)

where 119863 = [minus1 1]

2 Ω sim 119880[minus1 1] the boundary conditionsare given as

119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0

(75)

ISRN Applied Mathematics 9

005

1

005

10

05

1y d

xy

(a) Desired values

005

1

005

10

05

1

uM

ean

xy

times10minus3

(b) Initial state

005

1

005

10

05

1

u opt

xy

(c) Approximated solution (state)

005

1

005

10

200

400

z opt

xy

(d) Control

005

1

0

05

10

05

1

times 10minus3

xy

120583

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

05

1

15

2

25

times 10minus3

(f) Expected value

20 40

10

20

30

40

50

05

1

15

2

25

times 10minus3

(g) Standard dev

0 5 10times 104

0

5

10

times 104 0

(h) Sparsity of global matrix119899119911 = 1815980

Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))

2+ (119910 minus (12))

2)]

119873 = 3 119870 = 5 and 120572 = 000005

and the desired function is given by

(119909 119910) = exp [minus64 ((119909 minus

1

2

)

2

+ (119910 minus

1

2

)

2

)] (76)

The random field 119886(x 120596) is characterized by its mean andcovariance function

119864 [119886] = 119886 = 10

119877 (119909

1 119909

2) = 119890

minus|1199091minus1199092| 119909

1 119909

2isin [minus1 1]

(77)

Following Section 22 the truncated KLE can be representedas

119886 (119909 119910 120596) = 10 +

119873

sum

119895=0

radic120582

119895120579

119895120601

119895 (119909) (78)

The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867

1

0(119863)⨂119871

2(Ω) and all 119911isin 119871

2(119863)

119869 (119906opt 119911opt) le 119869 (119906 119911) (79)

The eigenpairs 120582119895 120579

119895 in truncated KLE solve the integral

equation

int

1

minus1

119890

minus|1199091minus1199092|120601

119895(119909

2) 119889119909

2= 120582

119895120601

119895(119909

1)

(80)

For this special case of the covariance function we haveexplicit expressions for 120601

119895and 120582

119895 [6] Let120596

119895evenand120596

119895oddsolve

the equations

1 minus 120596

119895eventan (120596

119895even) = 0

120596

119895odd+ tan (120596

119895odd) = 0

(81)

Then the even and odd indexed eigenfunctions are given by

120601

119895even(119909) =

cos (120596119895even

119909)

radic1 + (sin (2120596

119895even) 2120596

119895even)

120601

119895odd(119909)

=

sin (120596

119895odd119909)

radic1 minus (sin (2120596

119895odd) 2120596

119895odd)

(82)

10 ISRN Applied Mathematics

005

1

005

10

05

1y ha

nd

xy

(a) Desired values

005

1

005

1minus05

05

0

u Mea

n

xy

(b) Mean values

005

1

005

1

0

Num

sol

xy

0

1

(c) Approximated solution (State)

005

1

005

10

05

1

xy

(d) Control

005

1

005

10

05

1

Lend

a

xy

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

0

005

01

015

02

(f) Expected value-SGS

20 40

10

20

30

40

50

0

005

01

015

02

(g) Standard dev-SGS

0 2 4 6 8times 104

0

2

4

6

8

times 104

(h) Sparsity of global matrix119899119911 = 3046160

Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001

and corresponding eigenvalues

120582

119895even(119909) =

2

120596

2

119895even+ 1

120582

119895odd(119909) =

2

120596

2

119895odd+ 1

(83)

We choose 120579 = (120579

1 120579

119873)

119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]

First of all we find numerical approximations of120596119895even

and120596

119895oddwith bisection method and then with the eigenpairs

evaluated by this 120596119895even

and 120596

119895odd by choosing dimension 119873 =

3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1

In Figure 1(a) the desired function (119909 119910) is plottedfor 119909

119894= 119910

119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial

state is computed and plotted in Figure 1(b) by solving (37)where

119911 = 0 Approximated 119906opt corresponding to the

optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial

119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911

119900119901119905must approximate the

initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane

In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use

Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary

ISRN Applied Mathematics 11

conditions is replaced by (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2

4 Conclusion

The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996

[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987

[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005

[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947

[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946

[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991

[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007

[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009

[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938

[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993

[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008

[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999

[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006

[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010

[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010

[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 ISRN Applied Mathematics

By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is

⟨119869

119906 (119906 119911) 120575119906⟩

119880lowast119880

= 119864 [119906 (x 120596) minus (x)21198712(119863)

times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)

]

(51)

Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin

Z direction is

⟨119869

119911 (119906 119911) 120575119911⟩

Zlowast Z= 120572⟨119911 120575119911⟩Z (52)

With these computations one can compute the firstderivative of

119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883

ℎsub 119867

1

0(119863) is a finite dimensional

subspace of dimension 119873 and 119884

ℎsub 119871

2(Ω) is a finite

dimensional subspace of dimension 119875 then 119883

ℎ⨂119884

ℎis a

finite dimensional subspace of the state space 119880 Similarlylet Z

ℎsub Z be a finite dimensional subspace of the control

space (with dimension) For the stochastic Galerkin methodit is assumed that 119884

ℎ= P119875minus1 the space of polynomials with

highest degree 119875minus1 and119883

ℎis any finite element space here

119883

ℎis the space of linear functions built on a given meshT

ℎ

We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884

ℎ that is 119884

ℎ= spanΨ

1(120596) Ψ

119875(120596)

where

int

Ω

120588 (120596)Ψ

119899(120596)Ψ119898 (120596) 119889120596 = 120575

119898119899 (53)

The discretized optimization problem in the stochasticGalerkin framework is

min119911ℎisinZℎ

119869

ℎ119875(119911

ℎ) =

1

2

119864

[

[

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119875

sum

119896=1

119873

sum

119895=1

(

119896(119911

ℎ))

119895Ψ

119896 (120596) 120601119895 (

x)

minus (x)1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(119863)

]

]

+

120572

2

1003817

1003817

1003817

1003817

119911

ℎ

1003817

1003817

1003817

1003817

2

1198712(119863)

(54)

where

119896(119911

ℎ) = (

1(119911

ℎ)

119879

119875(119911

ℎ)

119879)

119879

=

(55)

is the stochastic Galerkin solution to the state equation

(

119870

11sdot sdot sdot 119870

1119875

d

119870

1198751sdot sdot sdot 119870

119875119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119870

(

1

119875

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=

= (

119864 [Ψ

1]119872

119911

ℎ

119864 [Ψ

119875]119872

119911

ℎ

)

= (

120575

11119872

119911

ℎ

120575

1119875119872

119911

ℎ

)

(56)

since 119864[Ψ

119896] = 119864[1Ψ

119896] = 119864[Ψ

1Ψ

119896] = 120575

1119896 The blocks of the

above 119870matrix have the form

(119870

119896119897)

119894119895= int

Ω

120588 (120596)Ψ

119896(120596)Ψ119897 (

120596)

times int

119863

119886 (x 120596) nabla120601

119894 (x) sdot nabla120601

119895 (x) 119889x119889120596

(57)

First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations

minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))

in 119863 times Ω

120583 (x 120596) = 0 on 120597119863 times Ω

(58)

Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865

119879

1 119865

119879

119901)

119879 and 119865

119896is defined

as

(

119865

119896)

119894= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596

= minusint

Ω

120588 (120596)Ψ119896 (120596) int

119863

(

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895120601

119895 (x) Ψ119897 (120596) 120601119894 (

x)

minus (x) 120601119894 (x))119889x119889120596

= minus

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

(x) 120601119894 (x) 119889x

=

119887

119894120575

1119896minus

119875

sum

119897=1

119873

sum

119895=1

119872

119894119895(

119897)

119895120575

119896119897

=

119887

119894120575

1119896minus

119873

sum

119895=1

119872

119894119895(

119897)

119895

(59)

in which

119887

119894= int 120601

119894 and119872

119894119895= int 120601

119894120601

119895

Thus

(

119865

119896)

119894=

119887 minus 119872119906

1if 119896 = 1

minus119872119906 if 119896 = 2 119875

(60)

where b= [

119887

1

119887

119873] and the matrix 119872 = (119872

119894119895) is of order

119873 times 119873

ISRN Applied Mathematics 7

Now the derivative of

119869(119911) in the direction 120601

119895(x) for all

119895 = 1 119873 with the computed adjoint state is

⟨

119869

1015840(119911) 120601119895 (

x)⟩Zlowast Z

= ⟨119890

119911(119906 (119911 sdot sdot) 119911)

lowast120583

+ 119869

119911 (119906 (119911 sdot sdot) 119911) 120601119895 (

x) ⟩Zlowast Z

= minusint

Ω

120588 (120596)int

119863

120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596

+ 120572int

119863

119911 (119909) 120601119895 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894int

Ω

120588 (120596)Ψ119896 (120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ 120572

119873

sum

119894=1

119911

119894int

119863

120601

119895 (x) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894119872

119894119895120575

1119896+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

= minus

119873

sum

119894=1

(

1)

119894119872

119894119895+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

(61)

Therefore the gradient can be computed as follows

119869

1015840(119911) = 119872(120572

119911

ℎminus

1)

(62)

Now for any vector V isin Z it is possible to write V as

V =

119873

sum

119896=1

V119896120601

119896 (x) (63)

In order to compute the Hessian times a vector that ismultiply the Hessian of the

119869 with a some vector V isin Z theequation 119890

119906(119906(119911) 119911)119908 = 119890

119911(119906(119911) 119911)V for 119908 must be solved

Similar to the adjoint computation119908 solves the linear system119870 = 119866 where

119866 = (

119866

119879

1

119866

119879

119875) and

119866

119896for 119896 = 1 119875 is

(

119866

119896)

119894= int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

119873

sum

119895=1

V119895120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895V119895120575

1119896

(64)

for all 119894 = 1 119873 Thus

119866

119896=

119872V if 119896 = 1

0 if 119896 = 2 119875

(65)

Now solving 119890

119906(119906(119911) 119911)119902 = 119869

119906119906(119906(119911) 119911)119908 for 119902 requires

the solution to the linear system 119870 119902 = 119867 where

(

119867

119896)

119894=

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895(

119896)

119895

(66)

or equivalently

119867

119896= 119872

119896for all 119896 = 1 119875 Hence in the

direction 120601

119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by

⟨

119869

10158401015840(119911) V 120601119894⟩Zlowast Z

asymp ⟨

119869

10158401015840

ℎ119901(119911

ℎ) V 120601

119894⟩

Zlowast Z

= minusint

Ω

120588 (120596)int

119863

119875

sum

119896=1

119873

sum

119895=1

( 119902

119896)

119895120601

119895 (x) Ψ119896 (120596)

times 120601

119894 (x) 119889x119889120596

+ 120572int

119863

V (119909) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119895=1

119872

119894119895( 119902

119896)

119895120575

1119896+ 120572

119873

sum

119895=1

119872

119894119895V119895

=

119873

sum

119895=1

119872

119894119895(120572V

119895minus ( 119902

1)

119895)

(67)

Therefore

119869

10158401015840(119911) V asymp 119872(120572V minus 119902

1)

(68)

Having (62) and (68) we can use the iterative solvers tothe Newton equation

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) (69)

Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572

119896and using this step size to generate

the iterate 119911

119896+1= 119911

119896+ 120572

119896119878

119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is

119869 (119911

119896+ 120572

119896119878

119896) le 119869 (119911

119896) + 119888120572

119896119869

1015840(119911

119896) 119878

119896 (70)

where 119888 isin (0 1) and typically is quite small for example 119888 =

10

minus4 [12 13]

8 ISRN Applied Mathematics

(1) Given 119911

0 and tol gt 0 set 119896 = 0

(2) Compute 119870119872 119860 and

119887

(3) Compute (

119865

119896)

119894=

119887 minus 119872119906 if 119896 = 1

minus119872119906 if 119896 = 2 119875

(4) Solve 119870 120583 = 119865

(5) Compute

119869

1015840(119911

119896) = 119872(120572119911

119896minus

1)

(6) Compute

119866

1= 119872V and

119866

119905= 0 for 119905 = 2 119875

(7) Solve 119870 = 119866 = (

119866

119879

1

119866

119879

119875)

119879

(8) Compute

119867

119896= 119872

119896for 119896 = 1 119875

(9) Solve 119870 119902 = 119867 = (

119867

119879

1

119867

119879

119875)

119879

(10) Compute

119869

10158401015840(119911)V asymp 119872(120572V minus 119902

1)

(11) If 1003817100381710038171003817

1003817

nabla

119869 (119911

119896)

1003817

1003817

1003817

1003817

1003817

lt tol stop(12) Solve

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) using PCG method with preconditioner

119875 = 119866⨂119870

0

(i) Set 1199090= 0

(ii) Set 1199030larr

119869

10158401015840(119911

119896) 119909

0+

119869

1015840(119911

119896)

(iii) Solve

119875119910

0= 119903

0for 119910

0

(iv) Set 1199010= minus119903

0 119905 larr 0

(v) While 1003817

1003817

1003817

1003817

119903

119905

1003817

1003817

1003817

1003817

gt tol

120572

119896larr

119903

119879

119905119910

119905

119901

119879

119905119860119901

119905

119909

119905+1larr 119909

119905+ 120572

119905119901

119905

119903

119905+1larr 119903

119905+ 120572

119905119860119901

119905

119875119910

119905+1larr 119903

119905+1

120573

119905+1larr

119903

119879

119905+1119910

119905+1

119903

119879

119905119910

119905

119901

119905+1larr minus119910

119905+1+ 120573

119905+1119901

119905

119905 larr 119905 + 1

End (while)(vi) 119904119896 = 119909

119905+1

(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911

119896+ 120572

119896119878

119896)

(ii) While 119869 (119911

119896+ 120572

119896119878

119896) gt 119869 (119911

119896) + 10

minus4120572

119896119878

119896nabla

119869 (119911

119896) do

(i) Set 120572119896 = 120572

1198962 and evaluate 119869 (119911

119896+ 120572

119896119878

119896)

(14) Set 119911119896+1 = 119911

119896+ 120572

119896119878

119896 119896 larr 119896 + 1 Go to (2)

Algorithm 1

31 Kronecker Product Preconditioners Let us to rewrite (37)in the form

119860 =

119911 where

119860 = (

119872

sum

119896=0

119866

119896⨂119870

119896) (71)

The strategy here is to introduce

119875 = 119866⨂119870

0as a

preconditioner such that

119866 = argmin 119867 isin R119875times119875

1003817

1003817

1003817

1003817

1003817

119860 minus 119867⨂119870

0

1003817

1003817

1003817

1003817

1003817119865 (72)

where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot

119865denotes the Frobe-

nius norm The closed form of the solution can be written asfollow [14]

119866 = 119868 + sum

119875

tr (119870119879

119896119870

0)

tr (119870119879

0119870

0)

119866

119896

(73)

Here tr(119870119879

119896119870

0) = sum

119873119902

119894=1[119870

119896]

119894119894[119870

0]

119894119894and hence the coeffi-

cients in the above equality can be computed straightforwardIn addition since

119860 and 119870

0are symmetric and positive

definite thus 119866 and

119875 = 119866⨂119870

0have also these properties

[15 16]

Example 1 Consider the following distributed optimal con-trol problem

min119906119911

119869 (119906 119911) = 119864 [

1

2

119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω

(74)

where 119863 = [minus1 1]

2 Ω sim 119880[minus1 1] the boundary conditionsare given as

119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0

(75)

ISRN Applied Mathematics 9

005

1

005

10

05

1y d

xy

(a) Desired values

005

1

005

10

05

1

uM

ean

xy

times10minus3

(b) Initial state

005

1

005

10

05

1

u opt

xy

(c) Approximated solution (state)

005

1

005

10

200

400

z opt

xy

(d) Control

005

1

0

05

10

05

1

times 10minus3

xy

120583

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

05

1

15

2

25

times 10minus3

(f) Expected value

20 40

10

20

30

40

50

05

1

15

2

25

times 10minus3

(g) Standard dev

0 5 10times 104

0

5

10

times 104 0

(h) Sparsity of global matrix119899119911 = 1815980

Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))

2+ (119910 minus (12))

2)]

119873 = 3 119870 = 5 and 120572 = 000005

and the desired function is given by

(119909 119910) = exp [minus64 ((119909 minus

1

2

)

2

+ (119910 minus

1

2

)

2

)] (76)

The random field 119886(x 120596) is characterized by its mean andcovariance function

119864 [119886] = 119886 = 10

119877 (119909

1 119909

2) = 119890

minus|1199091minus1199092| 119909

1 119909

2isin [minus1 1]

(77)

Following Section 22 the truncated KLE can be representedas

119886 (119909 119910 120596) = 10 +

119873

sum

119895=0

radic120582

119895120579

119895120601

119895 (119909) (78)

The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867

1

0(119863)⨂119871

2(Ω) and all 119911isin 119871

2(119863)

119869 (119906opt 119911opt) le 119869 (119906 119911) (79)

The eigenpairs 120582119895 120579

119895 in truncated KLE solve the integral

equation

int

1

minus1

119890

minus|1199091minus1199092|120601

119895(119909

2) 119889119909

2= 120582

119895120601

119895(119909

1)

(80)

For this special case of the covariance function we haveexplicit expressions for 120601

119895and 120582

119895 [6] Let120596

119895evenand120596

119895oddsolve

the equations

1 minus 120596

119895eventan (120596

119895even) = 0

120596

119895odd+ tan (120596

119895odd) = 0

(81)

Then the even and odd indexed eigenfunctions are given by

120601

119895even(119909) =

cos (120596119895even

119909)

radic1 + (sin (2120596

119895even) 2120596

119895even)

120601

119895odd(119909)

=

sin (120596

119895odd119909)

radic1 minus (sin (2120596

119895odd) 2120596

119895odd)

(82)

10 ISRN Applied Mathematics

005

1

005

10

05

1y ha

nd

xy

(a) Desired values

005

1

005

1minus05

05

0

u Mea

n

xy

(b) Mean values

005

1

005

1

0

Num

sol

xy

0

1

(c) Approximated solution (State)

005

1

005

10

05

1

xy

(d) Control

005

1

005

10

05

1

Lend

a

xy

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

0

005

01

015

02

(f) Expected value-SGS

20 40

10

20

30

40

50

0

005

01

015

02

(g) Standard dev-SGS

0 2 4 6 8times 104

0

2

4

6

8

times 104

(h) Sparsity of global matrix119899119911 = 3046160

Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001

and corresponding eigenvalues

120582

119895even(119909) =

2

120596

2

119895even+ 1

120582

119895odd(119909) =

2

120596

2

119895odd+ 1

(83)

We choose 120579 = (120579

1 120579

119873)

119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]

First of all we find numerical approximations of120596119895even

and120596

119895oddwith bisection method and then with the eigenpairs

evaluated by this 120596119895even

and 120596

119895odd by choosing dimension 119873 =

3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1

In Figure 1(a) the desired function (119909 119910) is plottedfor 119909

119894= 119910

119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial

state is computed and plotted in Figure 1(b) by solving (37)where

119911 = 0 Approximated 119906opt corresponding to the

optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial

119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911

119900119901119905must approximate the

initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane

In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use

Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary

ISRN Applied Mathematics 11

conditions is replaced by (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2

4 Conclusion

The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996

[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987

[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005

[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947

[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946

[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991

[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007

[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009

[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938

[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993

[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008

[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999

[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006

[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010

[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010

[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007

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Stochastic AnalysisInternational Journal of

ISRN Applied Mathematics 7

Now the derivative of

119869(119911) in the direction 120601

119895(x) for all

119895 = 1 119873 with the computed adjoint state is

⟨

119869

1015840(119911) 120601119895 (

x)⟩Zlowast Z

= ⟨119890

119911(119906 (119911 sdot sdot) 119911)

lowast120583

+ 119869

119911 (119906 (119911 sdot sdot) 119911) 120601119895 (

x) ⟩Zlowast Z

= minusint

Ω

120588 (120596)int

119863

120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596

+ 120572int

119863

119911 (119909) 120601119895 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894int

Ω

120588 (120596)Ψ119896 (120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

+ 120572

119873

sum

119894=1

119911

119894int

119863

120601

119895 (x) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119894=1

(

119896)

119894119872

119894119895120575

1119896+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

= minus

119873

sum

119894=1

(

1)

119894119872

119894119895+ 120572

119873

sum

119894=1

119872

119894119895

119911

119894

(61)

Therefore the gradient can be computed as follows

119869

1015840(119911) = 119872(120572

119911

ℎminus

1)

(62)

Now for any vector V isin Z it is possible to write V as

V =

119873

sum

119896=1

V119896120601

119896 (x) (63)

In order to compute the Hessian times a vector that ismultiply the Hessian of the

119869 with a some vector V isin Z theequation 119890

119906(119906(119911) 119911)119908 = 119890

119911(119906(119911) 119911)V for 119908 must be solved

Similar to the adjoint computation119908 solves the linear system119870 = 119866 where

119866 = (

119866

119879

1

119866

119879

119875) and

119866

119896for 119896 = 1 119875 is

(

119866

119896)

119894= int

Ω

120588 (120596)Ψ119896 (120596) 119889120596int

119863

119873

sum

119895=1

V119895120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895V119895120575

1119896

(64)

for all 119894 = 1 119873 Thus

119866

119896=

119872V if 119896 = 1

0 if 119896 = 2 119875

(65)

Now solving 119890

119906(119906(119911) 119911)119902 = 119869

119906119906(119906(119911) 119911)119908 for 119902 requires

the solution to the linear system 119870 119902 = 119867 where

(

119867

119896)

119894=

119875

sum

119897=1

119873

sum

119895=1

(

119897)

119895int

Ω

120588 (120596)Ψ119896 (120596)Ψ119897 (

120596) 119889120596

times int

119863

120601

119895 (x) 120601119894 (x) 119889x

=

119873

sum

119895=1

119872

119894119895(

119896)

119895

(66)

or equivalently

119867

119896= 119872

119896for all 119896 = 1 119875 Hence in the

direction 120601

119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by

⟨

119869

10158401015840(119911) V 120601119894⟩Zlowast Z

asymp ⟨

119869

10158401015840

ℎ119901(119911

ℎ) V 120601

119894⟩

Zlowast Z

= minusint

Ω

120588 (120596)int

119863

119875

sum

119896=1

119873

sum

119895=1

( 119902

119896)

119895120601

119895 (x) Ψ119896 (120596)

times 120601

119894 (x) 119889x119889120596

+ 120572int

119863

V (119909) 120601119894 (x) 119889x

= minus

119875

sum

119896=1

119873

sum

119895=1

119872

119894119895( 119902

119896)

119895120575

1119896+ 120572

119873

sum

119895=1

119872

119894119895V119895

=

119873

sum

119895=1

119872

119894119895(120572V

119895minus ( 119902

1)

119895)

(67)

Therefore

119869

10158401015840(119911) V asymp 119872(120572V minus 119902

1)

(68)

Having (62) and (68) we can use the iterative solvers tothe Newton equation

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) (69)

Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572

119896and using this step size to generate

the iterate 119911

119896+1= 119911

119896+ 120572

119896119878

119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is

119869 (119911

119896+ 120572

119896119878

119896) le 119869 (119911

119896) + 119888120572

119896119869

1015840(119911

119896) 119878

119896 (70)

where 119888 isin (0 1) and typically is quite small for example 119888 =

10

minus4 [12 13]

8 ISRN Applied Mathematics

(1) Given 119911

0 and tol gt 0 set 119896 = 0

(2) Compute 119870119872 119860 and

119887

(3) Compute (

119865

119896)

119894=

119887 minus 119872119906 if 119896 = 1

minus119872119906 if 119896 = 2 119875

(4) Solve 119870 120583 = 119865

(5) Compute

119869

1015840(119911

119896) = 119872(120572119911

119896minus

1)

(6) Compute

119866

1= 119872V and

119866

119905= 0 for 119905 = 2 119875

(7) Solve 119870 = 119866 = (

119866

119879

1

119866

119879

119875)

119879

(8) Compute

119867

119896= 119872

119896for 119896 = 1 119875

(9) Solve 119870 119902 = 119867 = (

119867

119879

1

119867

119879

119875)

119879

(10) Compute

119869

10158401015840(119911)V asymp 119872(120572V minus 119902

1)

(11) If 1003817100381710038171003817

1003817

nabla

119869 (119911

119896)

1003817

1003817

1003817

1003817

1003817

lt tol stop(12) Solve

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) using PCG method with preconditioner

119875 = 119866⨂119870

0

(i) Set 1199090= 0

(ii) Set 1199030larr

119869

10158401015840(119911

119896) 119909

0+

119869

1015840(119911

119896)

(iii) Solve

119875119910

0= 119903

0for 119910

0

(iv) Set 1199010= minus119903

0 119905 larr 0

(v) While 1003817

1003817

1003817

1003817

119903

119905

1003817

1003817

1003817

1003817

gt tol

120572

119896larr

119903

119879

119905119910

119905

119901

119879

119905119860119901

119905

119909

119905+1larr 119909

119905+ 120572

119905119901

119905

119903

119905+1larr 119903

119905+ 120572

119905119860119901

119905

119875119910

119905+1larr 119903

119905+1

120573

119905+1larr

119903

119879

119905+1119910

119905+1

119903

119879

119905119910

119905

119901

119905+1larr minus119910

119905+1+ 120573

119905+1119901

119905

119905 larr 119905 + 1

End (while)(vi) 119904119896 = 119909

119905+1

(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911

119896+ 120572

119896119878

119896)

(ii) While 119869 (119911

119896+ 120572

119896119878

119896) gt 119869 (119911

119896) + 10

minus4120572

119896119878

119896nabla

119869 (119911

119896) do

(i) Set 120572119896 = 120572

1198962 and evaluate 119869 (119911

119896+ 120572

119896119878

119896)

(14) Set 119911119896+1 = 119911

119896+ 120572

119896119878

119896 119896 larr 119896 + 1 Go to (2)

Algorithm 1

31 Kronecker Product Preconditioners Let us to rewrite (37)in the form

119860 =

119911 where

119860 = (

119872

sum

119896=0

119866

119896⨂119870

119896) (71)

The strategy here is to introduce

119875 = 119866⨂119870

0as a

preconditioner such that

119866 = argmin 119867 isin R119875times119875

1003817

1003817

1003817

1003817

1003817

119860 minus 119867⨂119870

0

1003817

1003817

1003817

1003817

1003817119865 (72)

where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot

119865denotes the Frobe-

nius norm The closed form of the solution can be written asfollow [14]

119866 = 119868 + sum

119875

tr (119870119879

119896119870

0)

tr (119870119879

0119870

0)

119866

119896

(73)

Here tr(119870119879

119896119870

0) = sum

119873119902

119894=1[119870

119896]

119894119894[119870

0]

119894119894and hence the coeffi-

cients in the above equality can be computed straightforwardIn addition since

119860 and 119870

0are symmetric and positive

definite thus 119866 and

119875 = 119866⨂119870

0have also these properties

[15 16]

Example 1 Consider the following distributed optimal con-trol problem

min119906119911

119869 (119906 119911) = 119864 [

1

2

119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω

(74)

where 119863 = [minus1 1]

2 Ω sim 119880[minus1 1] the boundary conditionsare given as

119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0

(75)

ISRN Applied Mathematics 9

005

1

005

10

05

1y d

xy

(a) Desired values

005

1

005

10

05

1

uM

ean

xy

times10minus3

(b) Initial state

005

1

005

10

05

1

u opt

xy

(c) Approximated solution (state)

005

1

005

10

200

400

z opt

xy

(d) Control

005

1

0

05

10

05

1

times 10minus3

xy

120583

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

05

1

15

2

25

times 10minus3

(f) Expected value

20 40

10

20

30

40

50

05

1

15

2

25

times 10minus3

(g) Standard dev

0 5 10times 104

0

5

10

times 104 0

(h) Sparsity of global matrix119899119911 = 1815980

Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))

2+ (119910 minus (12))

2)]

119873 = 3 119870 = 5 and 120572 = 000005

and the desired function is given by

(119909 119910) = exp [minus64 ((119909 minus

1

2

)

2

+ (119910 minus

1

2

)

2

)] (76)

The random field 119886(x 120596) is characterized by its mean andcovariance function

119864 [119886] = 119886 = 10

119877 (119909

1 119909

2) = 119890

minus|1199091minus1199092| 119909

1 119909

2isin [minus1 1]

(77)

Following Section 22 the truncated KLE can be representedas

119886 (119909 119910 120596) = 10 +

119873

sum

119895=0

radic120582

119895120579

119895120601

119895 (119909) (78)

The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867

1

0(119863)⨂119871

2(Ω) and all 119911isin 119871

2(119863)

119869 (119906opt 119911opt) le 119869 (119906 119911) (79)

The eigenpairs 120582119895 120579

119895 in truncated KLE solve the integral

equation

int

1

minus1

119890

minus|1199091minus1199092|120601

119895(119909

2) 119889119909

2= 120582

119895120601

119895(119909

1)

(80)

For this special case of the covariance function we haveexplicit expressions for 120601

119895and 120582

119895 [6] Let120596

119895evenand120596

119895oddsolve

the equations

1 minus 120596

119895eventan (120596

119895even) = 0

120596

119895odd+ tan (120596

119895odd) = 0

(81)

Then the even and odd indexed eigenfunctions are given by

120601

119895even(119909) =

cos (120596119895even

119909)

radic1 + (sin (2120596

119895even) 2120596

119895even)

120601

119895odd(119909)

=

sin (120596

119895odd119909)

radic1 minus (sin (2120596

119895odd) 2120596

119895odd)

(82)

10 ISRN Applied Mathematics

005

1

005

10

05

1y ha

nd

xy

(a) Desired values

005

1

005

1minus05

05

0

u Mea

n

xy

(b) Mean values

005

1

005

1

0

Num

sol

xy

0

1

(c) Approximated solution (State)

005

1

005

10

05

1

xy

(d) Control

005

1

005

10

05

1

Lend

a

xy

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

0

005

01

015

02

(f) Expected value-SGS

20 40

10

20

30

40

50

0

005

01

015

02

(g) Standard dev-SGS

0 2 4 6 8times 104

0

2

4

6

8

times 104

(h) Sparsity of global matrix119899119911 = 3046160

Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001

and corresponding eigenvalues

120582

119895even(119909) =

2

120596

2

119895even+ 1

120582

119895odd(119909) =

2

120596

2

119895odd+ 1

(83)

We choose 120579 = (120579

1 120579

119873)

119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]

First of all we find numerical approximations of120596119895even

and120596

119895oddwith bisection method and then with the eigenpairs

evaluated by this 120596119895even

and 120596

119895odd by choosing dimension 119873 =

3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1

In Figure 1(a) the desired function (119909 119910) is plottedfor 119909

119894= 119910

119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial

state is computed and plotted in Figure 1(b) by solving (37)where

119911 = 0 Approximated 119906opt corresponding to the

optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial

119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911

119900119901119905must approximate the

initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane

In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use

Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary

ISRN Applied Mathematics 11

conditions is replaced by (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2

4 Conclusion

The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996

[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987

[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005

[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947

[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946

[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991

[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007

[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009

[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938

[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993

[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008

[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999

[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006

[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010

[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010

[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007

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Stochastic AnalysisInternational Journal of

8 ISRN Applied Mathematics

(1) Given 119911

0 and tol gt 0 set 119896 = 0

(2) Compute 119870119872 119860 and

119887

(3) Compute (

119865

119896)

119894=

119887 minus 119872119906 if 119896 = 1

minus119872119906 if 119896 = 2 119875

(4) Solve 119870 120583 = 119865

(5) Compute

119869

1015840(119911

119896) = 119872(120572119911

119896minus

1)

(6) Compute

119866

1= 119872V and

119866

119905= 0 for 119905 = 2 119875

(7) Solve 119870 = 119866 = (

119866

119879

1

119866

119879

119875)

119879

(8) Compute

119867

119896= 119872

119896for 119896 = 1 119875

(9) Solve 119870 119902 = 119867 = (

119867

119879

1

119867

119879

119875)

119879

(10) Compute

119869

10158401015840(119911)V asymp 119872(120572V minus 119902

1)

(11) If 1003817100381710038171003817

1003817

nabla

119869 (119911

119896)

1003817

1003817

1003817

1003817

1003817

lt tol stop(12) Solve

119869

10158401015840(119911

119896) 119904

119896= minus

119869

1015840(119911

119896) using PCG method with preconditioner

119875 = 119866⨂119870

0

(i) Set 1199090= 0

(ii) Set 1199030larr

119869

10158401015840(119911

119896) 119909

0+

119869

1015840(119911

119896)

(iii) Solve

119875119910

0= 119903

0for 119910

0

(iv) Set 1199010= minus119903

0 119905 larr 0

(v) While 1003817

1003817

1003817

1003817

119903

119905

1003817

1003817

1003817

1003817

gt tol

120572

119896larr

119903

119879

119905119910

119905

119901

119879

119905119860119901

119905

119909

119905+1larr 119909

119905+ 120572

119905119901

119905

119903

119905+1larr 119903

119905+ 120572

119905119860119901

119905

119875119910

119905+1larr 119903

119905+1

120573

119905+1larr

119903

119879

119905+1119910

119905+1

119903

119879

119905119910

119905

119901

119905+1larr minus119910

119905+1+ 120573

119905+1119901

119905

119905 larr 119905 + 1

End (while)(vi) 119904119896 = 119909

119905+1

(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911

119896+ 120572

119896119878

119896)

(ii) While 119869 (119911

119896+ 120572

119896119878

119896) gt 119869 (119911

119896) + 10

minus4120572

119896119878

119896nabla

119869 (119911

119896) do

(i) Set 120572119896 = 120572

1198962 and evaluate 119869 (119911

119896+ 120572

119896119878

119896)

(14) Set 119911119896+1 = 119911

119896+ 120572

119896119878

119896 119896 larr 119896 + 1 Go to (2)

Algorithm 1

31 Kronecker Product Preconditioners Let us to rewrite (37)in the form

119860 =

119911 where

119860 = (

119872

sum

119896=0

119866

119896⨂119870

119896) (71)

The strategy here is to introduce

119875 = 119866⨂119870

0as a

preconditioner such that

119866 = argmin 119867 isin R119875times119875

1003817

1003817

1003817

1003817

1003817

119860 minus 119867⨂119870

0

1003817

1003817

1003817

1003817

1003817119865 (72)

where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot

119865denotes the Frobe-

nius norm The closed form of the solution can be written asfollow [14]

119866 = 119868 + sum

119875

tr (119870119879

119896119870

0)

tr (119870119879

0119870

0)

119866

119896

(73)

Here tr(119870119879

119896119870

0) = sum

119873119902

119894=1[119870

119896]

119894119894[119870

0]

119894119894and hence the coeffi-

cients in the above equality can be computed straightforwardIn addition since

119860 and 119870

0are symmetric and positive

definite thus 119866 and

119875 = 119866⨂119870

0have also these properties

[15 16]

Example 1 Consider the following distributed optimal con-trol problem

min119906119911

119869 (119906 119911) = 119864 [

1

2

119906 (x 120596) minus (x)21198712(119863)

]

+

120572

2

119911 (x)21198712(119863)

minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω

(74)

where 119863 = [minus1 1]

2 Ω sim 119880[minus1 1] the boundary conditionsare given as

119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0

(75)

ISRN Applied Mathematics 9

005

1

005

10

05

1y d

xy

(a) Desired values

005

1

005

10

05

1

uM

ean

xy

times10minus3

(b) Initial state

005

1

005

10

05

1

u opt

xy

(c) Approximated solution (state)

005

1

005

10

200

400

z opt

xy

(d) Control

005

1

0

05

10

05

1

times 10minus3

xy

120583

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

05

1

15

2

25

times 10minus3

(f) Expected value

20 40

10

20

30

40

50

05

1

15

2

25

times 10minus3

(g) Standard dev

0 5 10times 104

0

5

10

times 104 0

(h) Sparsity of global matrix119899119911 = 1815980

Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))

2+ (119910 minus (12))

2)]

119873 = 3 119870 = 5 and 120572 = 000005

and the desired function is given by

(119909 119910) = exp [minus64 ((119909 minus

1

2

)

2

+ (119910 minus

1

2

)

2

)] (76)

The random field 119886(x 120596) is characterized by its mean andcovariance function

119864 [119886] = 119886 = 10

119877 (119909

1 119909

2) = 119890

minus|1199091minus1199092| 119909

1 119909

2isin [minus1 1]

(77)

Following Section 22 the truncated KLE can be representedas

119886 (119909 119910 120596) = 10 +

119873

sum

119895=0

radic120582

119895120579

119895120601

119895 (119909) (78)

The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867

1

0(119863)⨂119871

2(Ω) and all 119911isin 119871

2(119863)

119869 (119906opt 119911opt) le 119869 (119906 119911) (79)

The eigenpairs 120582119895 120579

119895 in truncated KLE solve the integral

equation

int

1

minus1

119890

minus|1199091minus1199092|120601

119895(119909

2) 119889119909

2= 120582

119895120601

119895(119909

1)

(80)

For this special case of the covariance function we haveexplicit expressions for 120601

119895and 120582

119895 [6] Let120596

119895evenand120596

119895oddsolve

the equations

1 minus 120596

119895eventan (120596

119895even) = 0

120596

119895odd+ tan (120596

119895odd) = 0

(81)

Then the even and odd indexed eigenfunctions are given by

120601

119895even(119909) =

cos (120596119895even

119909)

radic1 + (sin (2120596

119895even) 2120596

119895even)

120601

119895odd(119909)

=

sin (120596

119895odd119909)

radic1 minus (sin (2120596

119895odd) 2120596

119895odd)

(82)

10 ISRN Applied Mathematics

005

1

005

10

05

1y ha

nd

xy

(a) Desired values

005

1

005

1minus05

05

0

u Mea

n

xy

(b) Mean values

005

1

005

1

0

Num

sol

xy

0

1

(c) Approximated solution (State)

005

1

005

10

05

1

xy

(d) Control

005

1

005

10

05

1

Lend

a

xy

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

0

005

01

015

02

(f) Expected value-SGS

20 40

10

20

30

40

50

0

005

01

015

02

(g) Standard dev-SGS

0 2 4 6 8times 104

0

2

4

6

8

times 104

(h) Sparsity of global matrix119899119911 = 3046160

Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001

and corresponding eigenvalues

120582

119895even(119909) =

2

120596

2

119895even+ 1

120582

119895odd(119909) =

2

120596

2

119895odd+ 1

(83)

We choose 120579 = (120579

1 120579

119873)

119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]

First of all we find numerical approximations of120596119895even

and120596

119895oddwith bisection method and then with the eigenpairs

evaluated by this 120596119895even

and 120596

119895odd by choosing dimension 119873 =

3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1

In Figure 1(a) the desired function (119909 119910) is plottedfor 119909

119894= 119910

119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial

state is computed and plotted in Figure 1(b) by solving (37)where

119911 = 0 Approximated 119906opt corresponding to the

optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial

119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911

119900119901119905must approximate the

initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane

In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use

Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary

ISRN Applied Mathematics 11

conditions is replaced by (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2

4 Conclusion

The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996

[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987

[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005

[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947

[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946

[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991

[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007

[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009

[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938

[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993

[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008

[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999

[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006

[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010

[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010

[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Applied Mathematics 9

005

1

005

10

05

1y d

xy

(a) Desired values

005

1

005

10

05

1

uM

ean

xy

times10minus3

(b) Initial state

005

1

005

10

05

1

u opt

xy

(c) Approximated solution (state)

005

1

005

10

200

400

z opt

xy

(d) Control

005

1

0

05

10

05

1

times 10minus3

xy

120583

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

05

1

15

2

25

times 10minus3

(f) Expected value

20 40

10

20

30

40

50

05

1

15

2

25

times 10minus3

(g) Standard dev

0 5 10times 104

0

5

10

times 104 0

(h) Sparsity of global matrix119899119911 = 1815980

Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))

2+ (119910 minus (12))

2)]

119873 = 3 119870 = 5 and 120572 = 000005

and the desired function is given by

(119909 119910) = exp [minus64 ((119909 minus

1

2

)

2

+ (119910 minus

1

2

)

2

)] (76)

The random field 119886(x 120596) is characterized by its mean andcovariance function

119864 [119886] = 119886 = 10

119877 (119909

1 119909

2) = 119890

minus|1199091minus1199092| 119909

1 119909

2isin [minus1 1]

(77)

Following Section 22 the truncated KLE can be representedas

119886 (119909 119910 120596) = 10 +

119873

sum

119895=0

radic120582

119895120579

119895120601

119895 (119909) (78)

The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867

1

0(119863)⨂119871

2(Ω) and all 119911isin 119871

2(119863)

119869 (119906opt 119911opt) le 119869 (119906 119911) (79)

The eigenpairs 120582119895 120579

119895 in truncated KLE solve the integral

equation

int

1

minus1

119890

minus|1199091minus1199092|120601

119895(119909

2) 119889119909

2= 120582

119895120601

119895(119909

1)

(80)

For this special case of the covariance function we haveexplicit expressions for 120601

119895and 120582

119895 [6] Let120596

119895evenand120596

119895oddsolve

the equations

1 minus 120596

119895eventan (120596

119895even) = 0

120596

119895odd+ tan (120596

119895odd) = 0

(81)

Then the even and odd indexed eigenfunctions are given by

120601

119895even(119909) =

cos (120596119895even

119909)

radic1 + (sin (2120596

119895even) 2120596

119895even)

120601

119895odd(119909)

=

sin (120596

119895odd119909)

radic1 minus (sin (2120596

119895odd) 2120596

119895odd)

(82)

10 ISRN Applied Mathematics

005

1

005

10

05

1y ha

nd

xy

(a) Desired values

005

1

005

1minus05

05

0

u Mea

n

xy

(b) Mean values

005

1

005

1

0

Num

sol

xy

0

1

(c) Approximated solution (State)

005

1

005

10

05

1

xy

(d) Control

005

1

005

10

05

1

Lend

a

xy

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

0

005

01

015

02

(f) Expected value-SGS

20 40

10

20

30

40

50

0

005

01

015

02

(g) Standard dev-SGS

0 2 4 6 8times 104

0

2

4

6

8

times 104

(h) Sparsity of global matrix119899119911 = 3046160

Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001

and corresponding eigenvalues

120582

119895even(119909) =

2

120596

2

119895even+ 1

120582

119895odd(119909) =

2

120596

2

119895odd+ 1

(83)

We choose 120579 = (120579

1 120579

119873)

119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]

First of all we find numerical approximations of120596119895even

and120596

119895oddwith bisection method and then with the eigenpairs

evaluated by this 120596119895even

and 120596

119895odd by choosing dimension 119873 =

3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1

In Figure 1(a) the desired function (119909 119910) is plottedfor 119909

119894= 119910

119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial

state is computed and plotted in Figure 1(b) by solving (37)where

119911 = 0 Approximated 119906opt corresponding to the

optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial

119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911

119900119901119905must approximate the

initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane

In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use

Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary

ISRN Applied Mathematics 11

conditions is replaced by (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2

4 Conclusion

The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996

[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987

[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005

[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947

[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946

[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991

[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007

[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009

[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938

[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993

[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008

[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999

[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006

[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010

[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010

[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 ISRN Applied Mathematics

005

1

005

10

05

1y ha

nd

xy

(a) Desired values

005

1

005

1minus05

05

0

u Mea

n

xy

(b) Mean values

005

1

005

1

0

Num

sol

xy

0

1

(c) Approximated solution (State)

005

1

005

10

05

1

xy

(d) Control

005

1

005

10

05

1

Lend

a

xy

(e) Adjoint

minus1 0 1minus1

minus05

0

05

1

0

005

01

015

02

(f) Expected value-SGS

20 40

10

20

30

40

50

0

005

01

015

02

(g) Standard dev-SGS

0 2 4 6 8times 104

0

2

4

6

8

times 104

(h) Sparsity of global matrix119899119911 = 3046160

Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001

and corresponding eigenvalues

120582

119895even(119909) =

2

120596

2

119895even+ 1

120582

119895odd(119909) =

2

120596

2

119895odd+ 1

(83)

We choose 120579 = (120579

1 120579

119873)

119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]

First of all we find numerical approximations of120596119895even

and120596

119895oddwith bisection method and then with the eigenpairs

evaluated by this 120596119895even

and 120596

119895odd by choosing dimension 119873 =

3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1

In Figure 1(a) the desired function (119909 119910) is plottedfor 119909

119894= 119910

119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial

state is computed and plotted in Figure 1(b) by solving (37)where

119911 = 0 Approximated 119906opt corresponding to the

optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial

119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911

119900119901119905must approximate the

initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane

In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use

Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary

ISRN Applied Mathematics 11

conditions is replaced by (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2

4 Conclusion

The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996

[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987

[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005

[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947

[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946

[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991

[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007

[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009

[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938

[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993

[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008

[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999

[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006

[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010

[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010

[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Applied Mathematics 11

conditions is replaced by (119909 119910) = 2120587

2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2

4 Conclusion

The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996

[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987

[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005

[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947

[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946

[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991

[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007

[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009

[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938

[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993

[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008

[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999

[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006

[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010

[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010

[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of