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Model Reduction inPDE-Constrained Optimization

Matthias Heinkenschloss

Department of Computational and Applied MathematicsRice University, Houston, Texas

[email protected]

June 14, 2016

Funded in part by AFOSR and NSF

2016 EU Regional SchoolAachen Institute for

Advanced Study in Computational Engineering Science (AICES)RWTH Aachen University

Matthias Heinkenschloss June 14, 2016 1

Outline

Overview

Example Optimization Problems

Optimization Problem

Projection Based Model Reduction

Back to Optimization

Error Estimates

Linear-Quadratic Problems

Shape Optimization with Local Parameter Dependence

Semilinear Parabolic Problems

Trust-Region Framework


OverviewI PDE constrained optimization arises in many science and

engineering applicationsI Numerical solution is iterative and requires the solution of many

PDEs.I Can we systematically replace the underlying PDE by a (projection

based) reduced order model to reduce the computational cost?

Flow control

Reservoir optimization

Control of semillinear reactionadvection diffusion

Shape optimization of biochips


Reduced order modeling has a long history in optimization

I Newton’s method uses a ’reduced order’ (reduced nonlinearity)model.

I Surrogate optimization.

I Multilevel optimization.

I ....

This course

I Focusses on projection based reduced order models.

I There is a close connection with surrogate optimization andespecially with multilevel optimization.


I Original problem

min J(y,u)

s.t. c(y,u) = 0, (PDE, size n)

u ∈ Uad, (control constraints)

where y ∈ Rn, n large, are the states and u ∈ Rm are the controls.

I Reduced order problemConstruct W,V ∈ Rn×r, r � n. rank(V) = rank(W) = r.Reduced order problem:

min J(Vŷ,u)

s.t. WT c(Vŷ,u) = 0, (ROM PDE, size r)

u ∈ Uad.

Optimization variables: states ŷ ∈ Rr, r � n, and controls u ∈ Rm.Reduced state equation WT c(Vŷ,u) = 0 ∈ Rr.


Rich literature on projection based ROMs in optimization, incl.:I (Strongly convex) Linear-Quadratic Optimal Control Problems

[Antil et al., 2010], [Chen and Quarteroni, 2014], [Gubisch and Volkwein, 2013],[Kammann et al., 2013], [Kärcher and Grepl, 2014b],[Kärcher and Grepl, 2014a], [Kärcher et al., 2014], [Negri et al., 2013][Tröltzsch and Volkwein, 2009], [Volkwein, 2011], ...

I Shape/Design Optimization [Amsallem et al., 2015], [Antil et al., 2011],[Choi et al., 2015], [Rozza and Manzoni, 2010], [Zahr and Farhat, 2015], ...

I Flow control [Borggaard and Gugercin, 2015], [Kunisch and Volkwein, 1999],[Ravindran, 2000], [Rowley et al., 2004], ...

I Newton-Kantorovich type estimates [Dihlmann and Haasdonk, 2015],[Gohlke, 2013].

I Optimality system based [Kunisch and Volkwein, 2008], [Grimm et al., 2015],[Kunisch and Müller, 2015], ...

I Balanced Truncation: [Antil et al., 2010], [Antil et al., 2011],[Antil et al., 2012], [Sun et al., 2008].

I Trust-region based approaches [Afanasiev and Hinze, 2001], [Arian et al., 2000],[Fahl and Sachs, 2003], [Gohlke, 2013], [Yue and Meerbergen, 2013], ...

I Model Predictive control [Ghiglieri and Ulbrich, 2014],[Alla and Volkwein, 2014], ...

I Feedback control [Kunisch et al., 2004], [Kunisch and Xie, 2005],[Alla and Falcone, 2013], ...


Outline

Overview





Error Estimates






Overview

Some examples of optimization problems governed by partial differentialequations (PDEs)

I Linear Quadratic Elliptic Problem

I Linear Quadratic Parabolic Problem

I Shape Optimization with Local Parameter Dependence

I Oil Reservoir Waterflooding Optimization


Linear Quadratic Elliptic Problem

minimize1

2

∫Ωs

(y(x)− ŷ(x))2dx+ α2

∫∂Ωc

u2(x)dx

subject to

−κf∆y(x) + a(x) · ∇y(x) = 0, x ∈ Ωf ,−κs∆y(x) = f(x), x ∈ Ωs,

κ∂

∂ny(x) = 0, x ∈ ∂Ω \ ∂Ωc,

y(x) = d(x) + u(x), x ∈ ∂Ωc.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.75

1

x1

x 2

Solid / Fluid Model

SOLID

Velocity field a(x) and control

boundary ∂Ωc (bold vertical line)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x1

x 2

Uncontrolled Temperature

7580

85 9095

100

105

110

115

120

125

130

130

125

120115

110

105

100

9590

Uncontrolled temperature.


Finite element approximation

min1

2yTQy + yT c +

1

2uTRu,

s.t. Ay(t) + Bu(t) = f .

Strongly convex problem.


Linear Quadratic Parabolic Problem([Antil et al., 2010], modeled after [Dedé and Quarteroni, 2005])

Minimize1

2

∫ T0

∫D

(y(x, t)− d(x, t))2dx dt+ 10−4

2

∫ T0

∫U1∪U2

u2(x, t)dx dt,

subject to

∂

∂ty(x, t)−∇(κ∇y(x, t)) + a(x) · ∇y(x, t)

= u(x, t)χU1(x) + u(x, t)χU2(x) in Ω× (0, T ),

with boundary conditions y(x, t) = 0 on ΓD × (0, T ), κ ∂∂ny(x, t) = 0 onΓN × (0, T ) and initial conditions y(x, 0) = 0 in Ω.1032 L. DEDE’ AND A. QUARTERONI

0 0.2 0.4 0.6 0.8 1.0 1.2−0.4

−0.2

0

0.2

0.4

Ω

U1

U2

D

ΓD

ΓN

ΓN

ΓN

ΓN

ΓN

ΓN

(a)

0 0.2 0.4 0.6 0.8 1.0 1.2−0.4

−0.2

0

0.2

0.4

Ω Γ

Din

ΓD

ΓD

ΓD

ΓD

ΓN

ΓN

(b)

Figure 4. Test 1. Reference domain for the control problem. We report the boundary condi-tions for the advection–diffusion Equation (10) (a) and for the Stokes problem (64) (b).

where ρwK , ρpK and ρ

uK are defined in Equations (55) and (61) (for the sake of simplicity, we have dropped the

apex (j) on the error indicators). Results are compared with those obtained on fine grids, that we consider anaccurate guess of the exact solution.

4.1. Test 1: water pollution

Let us consider a first test case that is inspired to a problem of a water pollution. The optimal control problemconsists in regulating the emission rates of pollutants (rising e.g. from refusals of industrial or agricultural plants)to keep the concentration of such substances below a desired threshold in a branch of a river.

We refer to the domain reported in Figure 4a, that could represent a river that bifurcates into two branchespast a hole, which stands for, e.g., an island. Referring to Equation (10), we obtain the velocity field V as thesolution of the following Stokes problem:

−µ∆V + ∇p = 0, in Ω,V = (1 − ( y0.2 )

2, 0)T , on ΓinD ,V = 0, on ΓD,µ∇V · n− pn = 0, on ΓN ,

(64)

where p stands for the pressure, while ΓinD , ΓD and ΓN are indicated in Figure 4b. Adimensional quantitiesare used. Here the Stokes problem serves the only purpose to provide an appropriate velocity field for theadvection–diffusion problem; since the latter governs our control problem, the analysis provided in Section 1and Section 2 applies. Moreover, for the sake of simplicity, we adopt the method and the a posteriori errorestimate (54) proposed in Section 3. In fact, this approach is not fully coherent, being the velocity field Vcomputed numerically by means of the same grid adopted to solve the control problem, i.e. we consider Vhinstead of V.

For the Stokes problem we assume µ = 0.1 , for which the Reynolds number reads Re ≈ 10; we solve theproblem by means of linear finite elements with stabilization (see [16]), computed with respect to the same gridof the control problem. In Figure 5 we report the velocity field and its intensity as obtained by solving theStokes problem.

For our control problem we assume ν = 0.015, u = 50 in both the emission areas U1 and U2 and zd = 0.1 inthe observation area D. The initial value of the control function, u = 50, can be interpreted as the maximumrate of emission of pollutants (divided by the emission area), while the state variable w stands for the pollutant

Ω with boundary conditions for theadvection diffusion equation

0 0.2 0.4 0.6 0.8 1 1.2−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4Velocity

ξ1

ξ 2

the velocity field a (obtained bysolving steady Stokes equation


Finite element discretization in space

min j(u) ≡ 12

∫ T0

‖Cy(t)− d(t)‖2 + 12u(t)TDu(t)dt,

where y(t) = y(u; t) is the solution of

My′(t) = Ay(t) + Bu(t), t ∈ (0, T ),y(0) = y0.

Here y(t) ∈ Rn, M ∈ Rn×n invert., A ∈ Rn×n, B ∈ Rn×m, n large.Strongly convex problem.


Shape Optimization with Local Parameter Dependence([Antil et al., 2011, Antil et al., 2012])

Geometry motivated by biochip

Problems where the shape param. θ only influences a (small) subdomain:

Ω̄(θ) := Ω̄1 ∪ Ω̄2(θ), Ω1 ∩ Ω2(θ) = ∅, Γ = Ω̄1 ∩ Ω̄2(θ).Here Ω2(θ) is the top left yellow, square domain.


minθmin≤θ≤θmax

J(θ) =

T∫0

∫Ωobs

12 |∇×v(x, t; θ)|

2dx+

∫Ω2(θ)

12 |v(x, t; θ)−v

d(x, t)|2dxdt

where v(θ) and p(θ) solve the Stokes equations

vt(x, t)− µ∆v(x, t) +∇p(x, t) = f(x, t), in Ω(θ)× (0, T ),∇ · v(x, t) = 0, in Ω(θ)× (0, T ),

v(x, t) = vin(x, t) on Γin × (0, T ),v(x, t) = 0 on Γlat × (0, T ),

−(µ∇v(x, t)− p(x, t)I)n = 0 on Γout × (0, T ),v(x, 0) = 0 in Ω(θ).

Here Ω(θ) = Ω1 ∪ Ω2(θ) and Ω2(θ) is the top left yellow, square domain.Observation region Ωobs is part of the two reservoirs.


The semi-discretized minimization problem is

minθ∈Θad

J(θ) :=

∫ T0

1

2

∫ T0

‖Cv(t, θ) + Fp(t, θ) + Du(t)− d‖2 dt

where v(·, θ), p(·, θ) solves the semi-discretized Stokes equations

M(θ)d

dtv(t) + A(θ)v(t) + B(θ)p(t) = K(θ)u(t) t ∈ [0, T ] ,

BT (θ)v(t) = L(θ)u(t) t ∈ [0, T ] ,M(θ)v(0) = M(θ)v0,

θ ∈ Θad


Oil Reservoir Waterflooding Optimization([Deng, 2017, Magruder, 2017, Wiegand, 2010])

From : http://plant-engineering.tistory.com/267


Reservoir Model: Two-phase immiscible incompressible flow with capillarypressure (see, e.g., [Peaceman, 1977], [Chen et al., 2006]).States: saturations sw, pressure p, velocity v.

φ(x)∂

∂tsw(x, t)

+∇(fw(sw(x, t))

[v(x, t) + d(sw(x, t))

])= qw(x, t), x ∈ Ω, t > 0,

v(x, t) +K(x)λ(sw(x, t))∇p(x, t) = 0, x ∈ Ω, t > 0,

∇ · v(x, t) = q(x, t), x ∈ Ω, t > 0,

v(x, t) · n = 0, x ∈ ∂Ω, t > 0,vw(x, t) · n = 0, x ∈ ∂Ω, t > 0,sw (x, 0) = swinit(x), x ∈ Ω.

I Porosity φ(x), diagonal permeability K(x) from SPE 10 dataset.

I Phase mobility λα = krα/µα; total mobility λ = λo + λw;water fractional flow function fw = λw/λ.

I Capillary pressure, Brooks-Corey formula pc = Pd(

sw−swc1−sor−swc

)− 12.


Optimization Model Problem (Well Rate Optimization)

I Maximize Net Present Value (NPV)

∫ T0

(1 + rdis)−t[ water injection cost︷︸︸︷−rinj

∑i∈Iinj

γq(xi, t)−

water treatment cost︷︸︸︷roper

∑i∈Iprod

γ|q(xi, t)|fw(sw(xi, t))

+

oil revenue︷︸︸︷roil

∑i∈Iprod

γ|q(xi, t)|fo(sw(xi, t))]dt

I subject toI two-phase immiscible incompressible flow,I well rates sum up to zero (reservoir is closed)∑

i∈Iinj∪Iprod

q(xi, t) = 0, t ∈ (0, T ),

I well rate bounds on each well

qi,low ≤ q(xi, t) ≤ qi,upp, i ∈ Iinj ∪ Iprod, t ∈ (0, T ).

I Data: Daily discount rate rdis = 2× 10−4, oil price roil = 80,injection cost rinj = 5, production cost rpro = 5.


Example Result

I 500 days;

I 1000 time steps;

I 1200×600×10 ft.3;I 60× 60× 5 grid;

I 10 + 10 = 20 wells;

I 25-days const. rate.


Outline

Overview





Error Estimates






Overview

I Formulate abstract optimization problem to introduce notation.

I Introduce adjoint equation method to compute gradient and Hessianinformation.

I Illustrate adjoint equation method on example problems.


Abstract Optimization ProblemI Original problem

min J(y, u)

s.t. c(y, u) = 0, (governing PDE, state eqn.)

u ∈ Uad, (control constraints)where y: states, u: controls,

I (Y, ‖ · ‖Y), (C, ‖ · ‖C) Banach spaces, (U , ‖ · ‖U ) Hilbert spaceI Uad ⊂ U nonempty, closed convex set,I J : Y × U → R, c : Y × U → C are smooth mappings.I Can think (Y, ‖ · ‖Y) = (Rn, ‖ · ‖2), (C, ‖ · ‖C) = (Rn, ‖ · ‖2),

(U , ‖ · ‖U ) = (Rm, ‖ · ‖2).I Reduced order problem

min Ĵ(ŷ, u)

s.t. ĉ(ŷ, u) = 0,

u ∈ Uad,where ŷ: ROM states, u: controls,

I (Ŷ, ‖ · ‖Ŷ), (Ĉ, ‖ · ‖Ĉ) Banach spaces,I Ĵ : Ŷ × U → R, c : Ŷ × U → Ĉ are smooth mappings.


Problem Formulation

Original problem

min J(y, u)

s.t. c(y, u) = 0,u ∈ Uad.

⇓y(u) unique sol. of c(y, u) = 0

⇓

min j(u)

s.t. u ∈ Uad

where j(u)def= J(y(u), u).

Reduced order problem

min Ĵ(ŷ, u)

s.t. ĉ(ŷ, u) = 0,u ∈ Uad.

⇓ŷ(u) unique sol. of ĉ(ŷ, u) = 0

⇓

min ĵ(u)

s.t. u ∈ Uad

where ĵ(u)def= Ĵ(ŷ(u), u).


Derivative Computation

Compute gradient and Hessian information for

j(u) = J(y(u), u), where y(u) solves c(y, u) = 0.

Assumption

I J and c are twice continuously differentiable,

I cy(y, u) is continuously invertible.

Consider problems with large number of controls u.Use adjoint equation approach for derivative computaton.See, e.g., chapter 1 in [Hinze et al., 2009] or [Heinkenschloss, 2008].


Gradient Computation

I Derivative

〈Dj(u), v〉U∗,U = 〈DyJ(y(u), u), Dy(u)v〉Y∗,Y+〈DuJ(y(u), u), v〉U∗,U

I Implicit function theorem applied to c(y(u), u) = 0 gives

cy(y, u)(Dy(u)v)+cu(y, u)v = 0 =⇒ Dy(u)v = −cy(y, u)−1cu(y, u).

I Derivative (y = y(u))

〈Dj(u), v〉U∗,U= 〈DyJ(y, u), −cy(y, u)−1cu(y, u)v〉Y∗,Y + 〈DuJ(y, u), v〉U∗,U= 〈−cy(y, u)−∗DyJ(y, u)︸︷︷︸

=p

, cu(y, u)v〉C∗,C + 〈DuJ(y, u), v〉U∗,U

= 〈cu(y, u)∗p+DuJ(y, u), v〉U∗,U .


I Connection with Lagrangian L(y, u, p) = J(y, u) + 〈p, c(y, u)〉C∗,C :I The adjoint variable p solves cy(y, u)

∗p = −DyJ(y, u),which is equivalent to DyL(y, u, p) = 0.

I Derivative

〈Dj(u), v〉U∗,U = 〈cu(y, u)∗p+DuJ(y, u), v〉U∗,U= 〈DuL(y, u, p), v〉U∗,U .

I The gradient ∇j(u) is the vector in U such that

〈∇j(u), v〉U = 〈Dj(u), v〉U∗,U= 〈cu(y, u)∗p+DuJ(y, u), v〉U∗,U ∀v ∈ U

(Riesz representation)

Gradient Computation Using Adjoints

1. Given u, solve c(y, u) = 0 for y (if not done already).

2. Solve the adjoint equation cy(y(u), u)∗ p = −DyJ(y(u), u) for p.

Denote the solution by p(u).

3. Compute Dj(u) = DuJ(y(u), u) + cu(y(u), u)∗p(u).

Two PDE solves (possibly nonlinear PDE in step 1, linear PDE in step 2)


I Connection with Lagrangian L(y, u, p) = J(y, u) + 〈p, c(y, u)〉C∗,C :I The adjoint variable p solves cy(y, u)

∗p = −DyJ(y, u),which is equivalent to DyL(y, u, p) = 0.

I Derivative

〈Dj(u), v〉U∗,U = 〈cu(y, u)∗p+DuJ(y, u), v〉U∗,U= 〈DuL(y, u, p), v〉U∗,U .

I The gradient ∇j(u) is the vector in U such that


(Riesz representation)

Gradient Computation Using Adjoints

1. Given u, solve c(y, u) = 0 for y (if not done already).

2. Solve the adjoint equation cy(y(u), u)∗ p = −DyJ(y(u), u) for p.

Denote the solution by p(u).


Two PDE solves (possibly nonlinear PDE in step 1, linear PDE in step 2)Matthias Heinkenschloss June 14, 2016 26

Hessian ComputationI Apply implicit differentiation to

Dj(u) = DuJ(y(u), u) + cu(y(u), u)∗p(u)

to compute Hessian information.

I Hessian–Times–Vector Computation1. Given u, solve c(y, u) = 0 for y (if not done already).2. Solve adjoint eqn. cy(y, u)

∗ p = −DyJ(y, u) for p (if not donealready).

3. Solve cy(y, u)w = −cu(y, u)v.4. Solve cy(y, u)

∗ q = −DyyL(y, u, p)w −DyuL(y, u, p)v.5. Compute

D2j(u)v = cu(y, u)∗q +DuyL(y, u, p)w +DuuL(y, u, p)v.

Two linear PDE solves in steps 3+4 per direction v.

I Vector su solves Newton equation ∇2j(u) su = −∇j(u)if and only if (sy, su) solves the quadratic program

min 〈[DyJ(.)DuJ(.)

],

[sysu

]〉+ 12 〈

[sysu

],

[DyyL(..) DyuL(..)DuyL(..) DuuL(..)

] [sysu

]〉,

s.t. cy(.)sy + cu(.)su = 0,

where (.) = (y(u), u) and (..) = (y(u), u, p(u)).


Application to Example Problems


Example: Elliptic Optimal Control ProblemI Problem:

Minimize j(u) =1

2

∫D

(y(x)− d(x))2dx+ α2

∫Γc

|∇u(x)|2dσ

where y solves

−∇(κ∇y(x)) + a·∇y(x) = f(x) in Ω,y(x) = u(x) on Γc,

y(x) = 0 on ΓD,

κ∇y(x)n = 0, on ΓN .

I Control space H10 (Γc). State spaceV =

{v ∈ H1(Ω) : v = 0 on Γc ∪ ΓD

}.

I Handle Dirichlet boundary condition using the inverse trace theorem.(In finite element approximation: via interpolation):For every u ∈ H10 (Γc) there exists y(u; ·) ∈ H1(Ω) such thaty(u;x) = u(x) on Γc.Moreover H10 (Γc) 3 u 7→ y(u; ·) ∈ H1(Ω) is bounded and linear.

I Write y = y0 + y(u; ·), where y0 ∈ V.


Example: Elliptic Optimal Control ProblemI Problem:

Minimize j(u) =1

2

∫D

(y(x)− d(x))2dx+ α2

∫Γc

|∇u(x)|2dσ

where y solves

−∇(κ∇y(x)) + a·∇y(x) = f(x) in Ω,y(x) = u(x) on Γc,

y(x) = 0 on ΓD,

κ∇y(x)n = 0, on ΓN .

I Control space H10 (Γc). State spaceV =

{v ∈ H1(Ω) : v = 0 on Γc ∪ ΓD

}.

I Handle Dirichlet boundary condition using the inverse trace theorem.(In finite element approximation: via interpolation):For every u ∈ H10 (Γc) there exists y(u; ·) ∈ H1(Ω) such thaty(u;x) = u(x) on Γc.Moreover H10 (Γc) 3 u 7→ y(u; ·) ∈ H1(Ω) is bounded and linear.

I Write y = y0 + y(u; ·), where y0 ∈ V.Matthias Heinkenschloss June 14, 2016 29

I Lagrangian:

L(y, u, p) =1

2

∫D

(y0(x) + y(u;x)− yd(x))2dx+α

2

∫Γc

|∇u(x)|2dσ

+

∫Ω

κ∇y0∇p+ a·∇y0p dx

+

∫Ω

κ∇y(u; ·)∇p+ a·y(u; ·)∇p− fp dx

I Adjoint equation:

−∇(κ∇p(x))− a·∇p(x) = −(y0(x) + y(u;x)− yd(x))|D, in Ω,p(x) = 0, on Γc ∪ ΓD,

κ∇p(x) · n+ a · n p(x) = 0, on ΓN .I Derivative

〈Dj(u), v〉H10 (Γc)∗,H1(Γc)

=

∫∂Ω

α∇u(x)∇v(x) dσ +∫

Ω

κ∇y(v; ·)∇p+ a·y(v; ·)∇p dx

+

∫D

(y0(x) + y(u;x)− yd(x))y(v;x)dx

What’s the gradient?


I Lagrangian:

L(y, u, p) =1

2

∫D

(y0(x) + y(u;x)− yd(x))2dx+α

2

∫Γc

|∇u(x)|2dσ

+

∫Ω


+

∫Ω

κ∇y(u; ·)∇p+ a·y(u; ·)∇p− fp dx

I Adjoint equation:


κ∇p(x) · n+ a · n p(x) = 0, on ΓN .

I Derivative

〈Dj(u), v〉H10 (Γc)∗,H1(Γc)

=

∫∂Ω


Ω

κ∇y(v; ·)∇p+ a·y(v; ·)∇p dx

+

∫D


What’s the gradient?


I Lagrangian:

L(y, u, p) =1

2

∫D

(y0(x) + y(u;x)− yd(x))2dx+α

2

∫Γc

|∇u(x)|2dσ

+

∫Ω


+

∫Ω

κ∇y(u; ·)∇p+ a·y(u; ·)∇p− fp dx

I Adjoint equation:


κ∇p(x) · n+ a · n p(x) = 0, on ΓN .I Derivative

〈Dj(u), v〉H10 (Γc)∗,H1(Γc)

=

∫∂Ω


Ω

κ∇y(v; ·)∇p+ a·y(v; ·)∇p dx

+

∫D


What’s the gradient?Matthias Heinkenschloss June 14, 2016 30

I The gradient ∇j(u) = g ∈ H10 (Γc) is a function that satisfies

〈Dj(u), v〉H10 (Γc)∗,H1(Γc)

=

∫∂Ω


Ω

κ∇y(v; ·)∇p+ a·y(v; ·)∇p dx

+

∫D


=

∫∂Ω

g(x)v(x) +∇g(x)∇v(x)ds = 〈g, v〉H10 (Γc) ∀v ∈ H10 (Γc).

I Solve Laplace equation on the boundary,∫Γc

∇g̃(x)∇v(x)dσ =∫

Ω

κ∇y(v; ·)∇p+ a · y(v; ·)∇p dx

+

∫D

(y0(x) + y(u;x)− yd(x))y(v;x)dx ∀v ∈ H10 (Γc)

and then∇j(u) = αu+ g̃.

I Note: Other ways to incorporate Dirichlet boundary controls(Lagrange multipliers, very weak form of Laplace equation) may leadto different weak forms and to different control and state spaces.


Example: Parabolic Optimal Control ProblemI Problem:

Minimize1

2

∫ T0

∫D

(y(x, t)− d(x, t))2dx dt+ α2

∫ T0

∫U

u2(x, t)dx dt,

where y solves

∂

∂ty(x, t)−∇(κ∇y(x, t)) + a·∇y(x, t) = u(x, t)χU (x) in Ω× (0, T ),

y(x, t) = 0, on ΓD × (0, T ), κ∇y(x, t)n = 0, on ΓN × (0, T ),y(x, 0) = 0, in Ω.

I Lagrangian (formally)

L(y, u, p) =1

2

∫ T0

∫D

(y(x, t)− d(x, t))2dx dt+ α2

∫ T0

∫U

u2(x, t)dx dt

+

∫ T0

∫Ω

∂

∂ty(x, t)p(x, t) + κ∇y(x, t)∇p(x, t) + a·∇y(x, t)p(x, t)dxdt

−∫ T

0

∫U

u(x, t)p(x, t)dxdt

(For details chapter 1 in [Hinze et al., 2009] or [Tröltzsch, 2010a].)


I Adjoint equation

− ∂∂tp(x, t)−∇(κ∇p(x, t))

−a·∇p(x, t) = −(y(x, t)− d(x, t))χD(x) in Ω× (0, T ),p(x, t) = 0, on ΓD × (0, T ),

(κ∇p(x, t) + a p(x, t))n = 0, on ΓN × (0, T ),p(x, T ) = 0, in Ω.

I Gradient

∇j(u) =αu(x, t)− p(x, t) x ∈ U, t ∈ (0, T ).


Outline

Overview





Error Estimates






Reduced-Order Dynamical Systems

ẏ(t) = Ay(t) + Bu(t) + f(t)

s(t) = Cy(t)

ẏ(t) = f(y(t),u(t), t)

s(t) = g(y(t), t)

Replace y(t) ∈ Rn by Vŷ(t) =∑ri=1 viŷi(t), ŷ ∈ Rr where r � n

and multiply the state equation by WT . (Often W = V.)

˙̂y = WTAVŷ + WTBu + WT f(t)

ŝ = CVŷ

˙̂y = WT f(Vŷ,u)

ŝ = g(Vŷ)

Two main questions:

I Accuracy of the reduced order model? Approximation of theinput-to-output map u 7→ s.

I Efficiency of the reduced order model?


Projection Based Reduced Order Models (ROMs) -Overview

I Reduced Basis Method.See books [Hesthaven et al., 2015], [Patera and Rozza, 2007],[Quarteroni et al., 2016].

I Proper Orthogonal Decomposition (POD).See survey article [Hinze and Volkwein, 2005] and sections in books[Hesthaven et al., 2015], [Quarteroni et al., 2016].

I Balanced Truncation Model Reduction (BTMR) for linear timeinvariant problems.See book [Antoulas, 2005] and for connections with POD[Rowley, 2005].

I Interpolation Based Model Reduction.See survey articles [Antoulas et al., 2010], [Benner et al., 2015].


Reduced Order Model (ROM) of Parametric Elliptic PDEI Given Hilbert space V, bounded coercive bilinear form a(·, ·;µ) onV × V, and bounded linear functional f(·;µ) on V.

I Find y ∈ V that satisfies the variational formulation:

a(y, v;µ) = f(v;µ) ∀v ∈ V.

I Given bounded linear functional `(·) on V, we are often interested in

s(µ) = `(y(µ)) (quantity of interest)

I Example

−∇2y(µ) +[µ0

]· ∇y(µ) = 100e−5

√‖x‖2 , in Ω = [−1, 1]2, µ ∈ [−10, 10],

y(µ) = 0, on ∂Ω,

s(µ) =

∫Ω

y(x;µ)dx.

Here V = H10 (Ω) and f(v;µ) =∫

Ω100e−5

√‖x‖2v(x)dx,

a(y, v;µ) =∫

Ω∇y(x) · ∇v(x) +

[µ0

]· ∇y(x)v(x)dx.


Finite Element ApproximationI Vn = span{φ1, . . . , φn} ⊂ V.

Find y = y(µ) ∈ Vn such that

a(y, v;µ) = f(v;µ) ∀v ∈ Vn.I Linear system for y =

∑ni=1 yiφi:

A(µ)y = f(µ), (n× n)

where

A(µ)ij = a(φj , φi;µ),

f(µ)i = f(φi;µ).

I Well posedness: If there exists α > 0 such that

a(v, v;µ) ≥ α‖v‖2V for all v ∈ V and µ ∈ Γ,

then yTA(µ)y ≥ α‖y‖2V (norm ‖y‖2V =∑ni,j=1 yiyj〈φi, φj〉V) and

A(µ)y = f(µ)

⇒ α‖y‖2V ≤ yTA(µ)y = yT f(µ) ≤ ‖y‖V ‖f(µ)‖V−1⇒ ‖y‖V ≤ α−1‖f(µ)‖V−1 ⇒ ‖A(µ)−1‖V ≤ α−1.


Reduced Order Model (ROM)I Subspace Vr = span{ζ1, . . . , ζr} ⊂ Vn, r � n.

Find ŷ = ŷ(µ) ∈ Vr such that

a(ŷ, v;µ) = f(v;µ) ∀v ∈ Vr.I Linear system for ŷ =

∑ri=1 ŷiζi:

I Represent Vr basis: ζi =∑nk=1 vkiφk, V ≡ (vij) ∈ R

n×r.I Insert into bilinear/linear forms

a(ζj , ζi;µ) =

n∑k=1

n∑k̂=1

vkivk̂ja(φk̂, φk;µ) = (VTA(µ)V)ij ,

f(ζi;µ) =

n∑k=1

vkif(φk;µ) = (VT f(µ))i.

I ROM

a(ŷ, ζi;µ) =r∑j=1

ŷja(ζj , ζi;µ) = f(ζi;µ), i = 1, . . . , r,

is equivalent to

VTA(µ)Vŷ = VT f(µ). (r × r)

I Well posedness inherited: ŷTVTA(µ)Vŷ ≥ α‖Vŷ‖2V ≡ α‖ŷ‖2V.Matthias Heinkenschloss June 14, 2016 39

Basic ROM Algorithm1. Compute Snapshots: Given {µ1, . . . , µr} compute full solutions:

A(µi)y(µi) = f(µi)

2. Orthogonalize: Find V ∈ Rn×r where VTV = I and

Ran(V) = span{y(µ1), . . . ,y(µr)}

3. Construct Reduced Order System:

Â(µ) = VTA(µ)V ∈ Rr×r, f̂(µ) = VT f(µ) ∈ Rr

4. ROM Solution: Cheaply solve reduced order system forout-of-sample parameter choices µ:

Â(µ)ŷ = f̂(µ).

Approximation y(µ) ≈ Vŷ(µ).Above is basic algorithm

I At what parameters µi do we sample?

I ROM Â(µ), f̂(µ) is smaller, but evaluation of Â(µ), f̂(µ) not cheap


ROM Error Estimates

I Error in the solution

A(µ)(y −Vŷ) = f(µ)−A(µ)Vŷ)=⇒ ‖y −Vŷ‖ ≤ ‖A(µ)−1‖ ‖f(µ)−A(µ)Vŷ)‖.

I Error in output of interest s(µ) = lTy(µ):

|s(µ)− ŝ(µ)| = |lTy − l̂T ŷ|≤ ‖l‖ ‖A(µ)−1‖ ‖f(µ)−A(µ)Vŷ)‖.

But can do much better [Machiels et al., 2001].


ROM Error Estimate - Quantity of Interest

Primal:

A(µ)y(µ) = f(µ)

s(µ) = lTy(µ)

Primal RB:

Â(µ)ŷ(µ) = f̂(µ)

ŝ(µ) = l̂T ŷ(µ)

Primal Residual:

‖f(µ)−A(µ)Vŷ(µ)‖

Primal Bound:

‖y −Vŷ‖ = ‖A−1(f −AVŷ)‖≤ ‖A−1‖‖f −AVŷ‖

Dual:

p(µ)TA(µ) = −l(µ)T

Dual RB:

p̂(µ)T Â(µ) = −̂l(µ)

Dual Residual:

‖ − l(µ)−A(µ)TVp̂(µ)‖

Dual Bound:

‖p−Vp̂‖ = ‖A−T (−l−ATVp̂)‖≤ ‖A−T ‖‖ − l−ATVp̂‖


Petrov-Galerkin Reduced Order Model (ROM)

I Could also generate two subspaces

Vr = span{ζ1, . . . , ζr} ⊂ Vn,Wr = span{ξ1, . . . , ξr} ⊂ Vn, r � n.

For example

Vr = span{y(µ1), . . . , y(µr)} ⊂ Vn,Wr = span{p(µ1), . . . , p(µr)} ⊂ Vn.

Find ŷ = ŷ(µ) ∈ Vn such that

a(ŷ, w;µ) = f(vn;µ) ∀w ∈ Wr.

I Linear system for y =∑nj=1 ŷjφj :

WTA(µ)Vŷ = WT f(µ). (r × r)

I But well posedness no longer inherited.Not automatically guaranteed that WTA(µ)V is invertible orinverse is uniformly bounded in r.


Reduced Basis Method - Greedy SelectionRecall |s(µ)− ŝ(µ)| ≤ ∆(µ) ≡ ‖A−1‖‖f −AVŷ‖‖ − l−ATVp̂‖.Choose Γtrain ⊂ Γ, tolerance � > 0 and maximum ROM size rmax.

Given r > 0, V ∈ Rn×r.While r < rmax

1. Next sample point via greedy and error estimate

µr+1 = argmaxµ∈Γtrain∆(µ)

2. If ∆(µr+1) < � stop. Computed ROM of desired accuracy.3. Compute new basis vector y(µr+1) by solving

A(µr+1)y = f(µr+1)4. Update old basis: compute V ∈ Rn×(r+1) where VTV = I and

Ran(V) = span{y(µ1), . . . ,y(µr+1)}

5. Update Reduced Order System:

Â(µ) = VTA(µ)V ∈ R(r+1)×(r+1), f̂(µ) = VT f(µ) ∈ Rr+1

6. Set r ← r + 1.Constructed y(µ1), . . . ,y(µr+1) are linearly independent.Convergence of the greedy selection [Binev et al., 2011].


Example

−∇2y(µ) +[µ0

]· ∇y(µ) = 100e−5

√‖x‖2 , in Ω = (−1, 1)2, µ ∈ [−10, 10],

y(µ) = 0, on ∂Ω

s(µ) =

∫Ω

y(µ)

(a) µ = −10 (b) µ = 0 (c) µ = 10


Figure: Convection-Diffusion Equation: Output, s(µ) vs. Parameter, µ


Proper Orthogonal Decomposition

I Given snapshots y(µ1), . . . , y(µm) ∈ Vn, m > r.I Compute orthonormal basis v1, . . . , vr ∈ Vn as solution of

min

m∑k=1

‖yk −r∑i=1

〈yk, vi〉V vi‖2V

s.t. 〈vi, vj〉V = δij .

I SolutionI Compute eigenvectors v1, v2, . . . ∈ Vn and eigenvaluesλ1 ≥ λ2 ≥ . . . ≥ λm ≥ 0 of the linear operator

ψ 7→ Kψ =m∑k=1

yk〈yk, ψ〉V .

I Solution v1, v2, . . . , vr,

m∑k=1

‖yk −r∑i=1

〈yk, vi〉V vi‖2V =m∑

i=r+1

λi.


I Finite dimensional representation of snapshotsy(µ1), . . . ,y(µm) ∈ Rn, m > r.

I Inner product 〈v, w〉V = vTMw, M s.p.d. (not nec. mass matrix)I Solution

I Define Y = [y(µ1), . . . ,y(µm)] ∈ Rn×m.I Compute M-orthonormal eigenvecs. v1,v2, . . . ∈ Rn and eigenvals.λ1 ≥ . . . ≥ λm ≥ 0 of generalized n× n eigenvalue prob.

MYYTMvi = λiMvi.

I Alternatively, if n > m compute eigenvectors w1,w2, . . . ∈ Rm andeigenvalues λ1 ≥ λ2 ≥ . . . ≥ λmin{m,n} ≥ 0 of

YTMYwi = λiwi.

vi = λ−1/2i Ywi, i = 1, . . . ,m.

I Usually, fix tolerance � > 0. Compute eigenvectors v1,v2, . . . ∈ Rnand eigenvalues λ1 ≥ λ2 ≥ . . . ≥ λmin{m,n} ≥ 0.

I Find smallest r such that∑mi=r+1 λi < �.

I If only some of the largest eigenvals. and vecs. are computed:Find smallest r such that λr+1/λ1 < �.

Reduced order model V = [v1, . . . ,vr] ∈ Rn×r.Matthias Heinkenschloss June 14, 2016 55

I POD often used to ‘compress’ solution of (linear or nonlinear)dynamical system

My′(t) = Ay(t) + f(t), t ∈ (0, T ),y(0) = y0.

I Solutions y(t0), . . . ,y(tm) ∈ Rn at time steps0 = t0 < . . . < tm = T used as snapshots.

I Can combine Reduced Basis Method and POD for parameterizeddynamical systems

M(µ)y′(t;µ) = A(µ)y(t;µ) + f(t;µ), t ∈ (0, T ),y(0;µ) = y0,

s(µ) =

∫ T0

c(t;µ)Ty(0;µ)dt. (output of interest)

Corresponding dual

−M(µ)Tp′(t;µ) = A(µ)Tp(t;µ)− c(t;µ), t ∈ (0, T ),p(T ;µ) = 0,


Balanced Truncation Model Reduction (BTMR)I Consider

d

dty(t) = Ay(t) + Bu(t), t ∈ (0, T )

z(t) = Cy(t) + Du(t), t ∈ (0, T )y(0) = 0.

I Projection methods for model reduction produce n× r matrices V,Wwith r � n and with WTV = Ir.

I One obtains a reduced form by setting y = Vŷ and projecting so that

WT [Vd

dtŷ(t)−AVŷ(t)−Bu(t)] = 0, t ∈ (0, T ).

I This leads to a reduced order system of order n given by

d

dtŷ(t) = Âŷ(t) + B̂u(t), t ∈ (0, T )

ẑ(t) = Ĉŷ(t) + Du(t), t ∈ (0, T )ŷ(0) = 0.

with Â = WTAV, B̂ = WTB, and Ĉ = CV.


Controllability and Observability Gramians

I Recall

y′(t) = Ay(t) + Bu(t), t ∈ (0, T )z(t) = Cy(t) + Du(t), t ∈ (0, T ).

Assume the system is stable (Re(λ(A)) < 0), controllable and observable.

I Controllability Gramian.

I P =∫ ∞

0

eAtB BT eAT tdt.

I Eigenspaces corresponding to large eigenvalues are ‘easy’ to control(control has smaller energy).

I Controllability Gramian solves the Lyapunov equation

AP + PAT + BBT = 0.

I Observability Gramian.

I Q =∫ ∞

0

eAT tCTCeAtdt.

I Eigenspaces corresponding to large eigenvalues are ‘easy’ to observe.I Observability Gramian solves the Lyapunov equation

ATQ+QA + CTC = 0.


I Compute controllability and observability gramians P,Q P = UUT andQ = LLT in factored form, i.e., solve

AP + PAT + BBT = 0,

ATQ+QA + CTC = 0.

I Compute the SVD UTL = ZSYT ,where Sr = diag(σ1, σ2, . . . , σr) with S = Sn, and σ1 ≥ σ2 ≥ . . ..

I Set V = UZrS−1/2n , W = LYrS

−1/2n , where n is selected to be the

smallest positive integer such that σr+1 < τσ1. Here τ > 0 is aprespecified constant. The matrices Zr,Yr consist of the correspondingleading r columns of Z,Y.

I It is easily verified that PW = VSr and QV = WSr.I Hence

0 = WT (AP + PAT + BBT )W = ÂSr + SrÂT + B̂B̂T ,

0 = VT (ATQ+QA + CTC)V = ÂTSr + SrÂ + ĈT Ĉ.


Two important properties of balanced trunction model reduction:

I Â is stable

I For any given input u we have

‖z− ẑ‖L2 ≤ 2‖u‖L2(σn+1 + . . .+ σN )

where ẑ is the output (response) of the reduced model[Glover, 1984].


Empirical Interpolation Method

I VTA(µ)V ∈ Rr×r, but evaluation µ 7→ VTA(µ)V requiresevaluation µ 7→ A(µ) 7→ VTA(µ)V at cost dependent on n.

I If A(µ) = A0 +∑kj=1 µjAj , then

VTA(µ)V = VTA0V +

k∑j=1

µjVTAjV.

Precompute VTAkV ∈ Rr×r, j = 0, . . . , k, afterwardsevaluate µ 7→ VTA(µ)V at cost of O(r2).

I For example, finite element discretization of

−∇2y(µ) +[µ0

]· ∇y(µ) = 100e−5

√‖x‖2 , in Ω = (−1, 1)2,

y(µ) = 0, on ∂Ω

leads to A(µ) = Adiff + µAadv.


Empirical Interpolation Method

I Empirical Interpolation Method (EIM): [Barrault et al., 2004][Eftang et al., 2010].

I Discrete Empirical Interpolation Method (DEIM):[Chaturantabut and Sorensen, 2010].

I Application of DEIM for finite element approximations[Antil et al., 2014], [Tiso and Rixen, 2013].

I Element based (compared to nodal/point based) version:[Farhat et al., 2015].


Outline

Overview





Error Estimates






Recall Optimization Problem

Original problem

min J(y, u)

s.t. c(y, u) = 0,u ∈ Uad.

⇓y(u) unique sol. of c(y, u) = 0

⇓

min j(u)

s.t. u ∈ Uad

where j(u)def= J(y(u), u).

Reduced order problem

min Ĵ(ŷ, u)

s.t. ĉ(ŷ, u) = 0,u ∈ Uad.

⇓ŷ(u) unique sol. of ĉ(ŷ, u) = 0

⇓

min ĵ(u)

s.t. u ∈ Uad

where ĵ(u)def= Ĵ(ŷ(u), u).


Gradient Computation (Using Adjoints)

1. Given u, solve state PDE c(y, u) = 0 for y = y(u).

2. Solve the adjoint PDE cy(y(u), u)∗ p = −DyJ(y(u), u) for p = p(u).




Outline

Overview





Error Estimates






Overview

I Optimization problem: More parameters - controls u ∈ Uad;objective function j(u) is quantify of interest.

I Approximating just the objective function j(u) is not enough.Also need to approximate gradient information ∇j(u).


Error Estimate (Strongly Convex Function) I

I u∗ minimizer of original objective j, û∗ min. of reduced objective ĵ.Want to estimate error ‖û∗ − u∗‖U .

I Optimality conditions

〈∇j(u∗), u− u∗〉U ≥ 0 ∀u ∈ Uad,

〈∇ĵ(û∗), u− û∗〉U ≥ 0 ∀u ∈ Uad.

I Assume j is strongly convex function on convex set C ⊂ U : Thereexists κ > 0 such that

〈u− w,∇j(u)−∇j(w)〉U ≥ κ‖u− w‖2U for all u,w ∈ C ⊂ U .

I Let ξ ∈ U be such that

〈∇j(û∗) + ξ, u− û∗〉U ≥ 0 ∀u ∈ Uad.

ξ = ∇ĵ(û∗)−∇j(û∗) always works; sometimes can find better ξ.

(û∗ solves perturbed optimization problem minu∈Uad j(u) + 〈ξ, u〉U .)Matthias Heinkenschloss June 14, 2016 68

Error Estimate (Strongly Convex Function) II

I Combine optimality conds. & convexity: If u∗, û∗ ∈ C,

κ‖u∗ − û∗‖2 ≤ 〈u∗ − û∗,∇j(u∗)−∇j(û∗)〉≤ 〈u∗ − û∗,∇j(u∗)−∇j(û∗)〉+ 〈u∗ − û∗,∇j(û∗) + ξ〉= 〈u∗ − û∗,∇j(u∗)〉+ 〈u∗ − û∗, ξ〉 ≤ 〈u∗ − û∗, ξ〉.

I Hence ‖u∗ − û∗‖U ≤ κ−1‖ξ‖U(≤ κ−1‖∇ĵ(û∗)−∇j(û∗)‖U

)I Estimate error in gradients to get estimate for error in solution.

I Applies whenI j is strongly convex function on Uad admissible set, e.g.,

convex-linear quadratic problems.I j satisfies strong second order optimality conditions at u∗ and û∗ is

in neighborhood of u∗.


Error Estimate for Unconstrained ProblemsI û∗ = argminuĵ(u) minimizer of unconstrained reduced problem.

I Newton-Kantorovich Theorem: Let r > 0 and ∇2j ∈ LipL(Br(û∗)).∇2j(û∗) be nonsingular and constants ζ, η ≥ 0 such that

‖∇2j(û∗)−1‖ = ζ, ‖∇2j(û∗)−1∇j(û∗)‖ ≤ η.

If Lζη ≤ 12 , there is unique local minimum u∗ of j in ball around û∗with radius min

{r,(

1−√

1− 2Lζη)/(Lζ)

}≤ min

{r, 2η

}.

I Estimate η:

‖∇2j(û∗)−1(∇j(û∗)−∇ĵ(û∗)︸︷︷︸=0

)‖ ≤ ζ‖∇j(û∗)−∇ĵ(û∗)‖ = η.

I Hence ‖u∗ − û∗‖U ≤ 2ζ‖∇ĵ(û∗)−∇j(û∗)‖UI Estimate error in gradients to get estimate for error in solution.

I Need Lζη ≤ 12 , i.e., ∇j(û∗) small enough.I Can estimate error using convergence properties of Newton’s

method started with û∗ applied to original problem.


Outline

Overview





Error Estimates






Elliptic Linear-Quadratic Model Problem

I Original problem

min1

2yTQy + cTy +

α

2uTRu

s.t. Ay + Bu = b,

u ∈ Uad

where A ∈ Rn×n invertible, B ∈ Rn×m, b, c ∈ Rn,Q = QT ∈ Rn×n positive semidef., R = RT ∈ Rm×m positive def.n large.

I Reduced order problem

min1

2ŷTVTQVŷ + cTVŷ +

α

2uTRu

s.t. WTAVŷ + WTBu = WTb,

u ∈ Uad

where V,W ∈ Rn×r, r � n, Â def= WTAV invertible.


I Gradient original problem

solve Ay + Bu = b,

solve ATp = −Qy − d,∇j(u) = αRu + BTp.

I Gradient reduced order problem

solve WTAV ŷ + WTBu = WTb,

solve VTATWp̂ = −VTQVŷ −VTd,

∇ĵ(u) = αRu + BTWp̂.

I Error

∇ĵ(u)−∇j(u) = BT(Wp̂− p

)= BT (Wp̂− p̃ + p̃− p

),

p̃ solves AT p̃ = −QVŷ − d (full adjoint with reduced Vŷ input).

I Error bound

‖∇ĵ(u)−∇j(u)‖ ≤ ‖B‖(‖A−T ‖‖ATWp̂ + QVŷ + d‖

+ ‖A−T ‖ ‖A−1‖ ‖Q‖ ‖AVŷ + Bu− b‖)


I Basis V represents state information, W represents adjointinformation. In principle can construct V 6= W, but must haveWTAV invertible.Often compute V = W from samples/snapshots of states andadjoints.Careful, states and adjoints represent different objects and havedifferent scales (more later).

I Application to parameterized optimal control problems µ ∈ D

min1

2yTQ(µ)y + c(µ)Ty +

α

2uTR(µ)u

s.t. A(µ)y + B(µ)u = b(µ).

I Assume computable uniform bounds, e.g., ‖A(µ)−1‖ ≤ a, µ ∈ D.I Have error estimates, can now use ROM machinery developed for

equations.I Use greedy procedure to sample µ ∈ Γ. Use error estimate.I Add state y(µ) and adjoint p(µ) at sample µ to basis V = W.


Parabolic Linear-Quadratic Model ProblemI Consider optimal control probl. governed by advection diffusion PDE

∂

∂ty(x, t)−∇(k(x)∇y(x, t)) + a(x) · ∇y(x, t)) = f(x, t)

in Ω× (0, T ). Optimization variables are related to the right handside f or to boundary data.

I After (finite element) discretization in space the optimal controlproblems are of the form

min j(u) ≡ 12

∫ T0

‖Cy(t) + Du(t)− d(t)‖2dt,

where y(t) = y(u; t) is the solution of

My′(t) = Ay(t) + Bu(t), t ∈ (0, T ),y(0) = y0.

Here y(t) ∈ Rn, M ∈ Rn×n invert., A ∈ Rn×n, B ∈ Rn×m, n large.D ∈ Rm×m invertible. Strongly convex problem.


I Reduced optimal control problem

min ĵ(u) ≡ 12

∫ T0

‖=Ĉ︷︸︸︷CV ŷ(t) + Du(t)− d(t)‖2dt

where ŷ(t) = ŷ(u; t) solves

WTMV︸︷︷︸=M̂

ŷ′(t) = WTAV︸︷︷︸=Â

ŷ(t) + WTB︸︷︷︸=B̂

u(t), t ∈ (0, T ),

ŷ(0) = ŷ0.

Here ŷ(t) ∈ Rr, M̂, Â ∈ Rr×r, B̂ ∈ Rr×m, with r � n small.


I Gradient computation original problem

My′(t) = Ay(t) + Bu(t), t ∈ (0, T ), y(0) = y0,z(t) = Cy(t) + Du(t)− d(t), t ∈ (0, T )

−MTp′(t) = ATp(t) + CT z(t), t ∈ (0, T ), p(T ) = 0,∇j(u) = q(t) = BTp(t) + DT z(t), t ∈ (0, T )

I Gradient computation reduced problem

M̂ŷ′(t) = Âŷ(t) + B̂u(t), t ∈ (0, T ) ŷ(0) = ŷ0,

ẑ(t) = Ĉŷ(t) + Du(t)− d(t), t ∈ (0, T )

−M̂T p̂′(t) = ÂT p̂(t) + ĈT ẑ(t), t ∈ (0, T ) p̂(T ) = 0,

∇ĵ(u) = q̂(t) = B̂T p̂(t) + DT ẑ(t), t ∈ (0, T )

I ‘Duality’ in input-output maps u 7→ z, w 7→ q of state, adjoint sys.I Need to approximate input-to-output maps u 7→ z, w 7→ q.


I Balanced Truncation Model Reduction (BTMR) error bound:If system is stable (Re(λ(A)) < 0), controllable and observable (truefor model problem), can use BTMR to compute W,V ∈ RN×n:

‖z− ẑ‖L2 ≤ 2(σr+1 + . . .+ σn) ‖u‖L2 ∀u,‖q− q̂‖L2 ≤ 2(σr+1 + . . .+ σn) ‖w‖L2 ∀w,

where σ1 ≥ . . . ≥ σr ≥ σr+1 ≥ . . . σn ≥ 0 are Hankel singular vals.I Introduce auxiliary adjoint for error estimate

I Original problem

−Mp′(t) = ATp(t) + CT z(t), t ∈ (0, T ), p(T ) = 0,

∇j(u) = q(t) = BTp(t) + DT z(t), t ∈ (0, T )

I Reduced order problem

−p̂′(t) = ÂT p̂(t) + ĈT ẑ(t), t ∈ (0, T ) p̂(T ) = 0,

∇ĵ(u) = q̂(t) = B̂T p̂(t) + DT ẑ(t), t ∈ (0, T )

I BTMR bound requires same input in full and reduced adjoint system.I Easy to fix: Introduce auxiliary adjoint p̃ as solution of the original

adjoint, but with input ẑ instead of z.


I Assume that there exists γ > 0 such that

vTAv ≤ −γ vTMv, ∀v ∈ Rn.

(Satisfied for model problem).

I Gradient error:

‖∇j(u)−∇ĵ(u)‖L2 ≤ 2 (c‖u‖L2 + ‖ẑ(u)‖L2) (σr+1 + . . .+ σn)

for all u ∈ L2! (ẑ(u) output of reduced order state with input u.)I Solution error:

‖u∗ − û∗‖L2 ≤2

κ(c‖û∗‖L2 + ‖ẑ∗‖L2) (σr+1 + . . .+ σn).


Example Problem (modeled after [Dedé and Quarteroni, 2005]

Minimize1

2

∫ T0

∫D

(y(x, t)− d(x, t))2dx dt+ 10−4

2

∫ T0

∫U1∪U2

u2(x, t)dx dt,

subject to

∂

∂ty(x, t)−∇(κ∇y(x, t)) + a(x) · ∇y(x, t)

= u(x, t)χU1(x) + u(x, t)χU2(x) in Ω× (0, T ),

with boundary conditions y(x, t) = 0 on ΓD × (0, T ), ∂∂ny(x, t) = 0 onΓN × (0, T ) and initial conditions y(x, 0) = 0 in Ω.1032 L. DEDE’ AND A. QUARTERONI

0 0.2 0.4 0.6 0.8 1.0 1.2−0.4

−0.2

0

0.2

0.4

Ω

U1

U2

D

ΓD

ΓN

ΓN

ΓN

ΓN

ΓN

ΓN

(a)

0 0.2 0.4 0.6 0.8 1.0 1.2−0.4

−0.2

0

0.2

0.4

Ω Γ

Din

ΓD

ΓD

ΓD

ΓD

ΓN

ΓN

(b)

Figure 4. Test 1. Reference domain for the control problem. We report the boundary condi-tions for the advection–diffusion Equation (10) (a) and for the Stokes problem (64) (b).

where ρwK , ρpK and ρ

uK are defined in Equations (55) and (61) (for the sake of simplicity, we have dropped the

apex (j) on the error indicators). Results are compared with those obtained on fine grids, that we consider anaccurate guess of the exact solution.

4.1. Test 1: water pollution

Let us consider a first test case that is inspired to a problem of a water pollution. The optimal control problemconsists in regulating the emission rates of pollutants (rising e.g. from refusals of industrial or agricultural plants)to keep the concentration of such substances below a desired threshold in a branch of a river.

We refer to the domain reported in Figure 4a, that could represent a river that bifurcates into two branchespast a hole, which stands for, e.g., an island. Referring to Equation (10), we obtain the velocity field V as thesolution of the following Stokes problem:

−µ∆V + ∇p = 0, in Ω,V = (1 − ( y0.2 )

2, 0)T , on ΓinD ,V = 0, on ΓD,µ∇V · n− pn = 0, on ΓN ,

(64)

where p stands for the pressure, while ΓinD , ΓD and ΓN are indicated in Figure 4b. Adimensional quantitiesare used. Here the Stokes problem serves the only purpose to provide an appropriate velocity field for theadvection–diffusion problem; since the latter governs our control problem, the analysis provided in Section 1and Section 2 applies. Moreover, for the sake of simplicity, we adopt the method and the a posteriori errorestimate (54) proposed in Section 3. In fact, this approach is not fully coherent, being the velocity field Vcomputed numerically by means of the same grid adopted to solve the control problem, i.e. we consider Vhinstead of V.

For the Stokes problem we assume µ = 0.1 , for which the Reynolds number reads Re ≈ 10; we solve theproblem by means of linear finite elements with stabilization (see [16]), computed with respect to the same gridof the control problem. In Figure 5 we report the velocity field and its intensity as obtained by solving theStokes problem.

For our control problem we assume ν = 0.015, u = 50 in both the emission areas U1 and U2 and zd = 0.1 inthe observation area D. The initial value of the control function, u = 50, can be interpreted as the maximumrate of emission of pollutants (divided by the emission area), while the state variable w stands for the pollutant

Ω with boundary conditions for theadvection diffusion equation

0 0.2 0.4 0.6 0.8 1 1.2−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4Velocity

ξ1

ξ 2

the velocity field a


grid k m n r1 168 9 1545 92 283 16 2673 93 618 29 6036 9

Number k of observations,number m of controls,size n of full order system, andsize r of reduced order systemfor three discretizations.

0 10 20 30 40 50 6010

−20

10−15

10−10

10−5

100

Hankel Singular Values

Largest Hankel singular valuesand threshold 10−4σ1 (grid # 3)


0 0.5 1 1.5 2 2.5 3 3.5 4

10−6

10−4

10−2

100

102

Time

Integrals∫U1u2∗(x, t)dx (solid

blue line) and∫U1û2∗(x, t)dx

(dashed red line) of the optimalcontrols computed using the fulland and the reduced order model.

0 0.5 1 1.5 2 2.5 3 3.5 4

10−6

10−4

10−2

100

102

Time

Integrals∫U2u2∗(x, t)dx (solid

blue line) and∫U2û2∗(x, t)dx

(dashed red line) of the optimalcontrols computed using the fulland and the reduced order model.

Full and reduced order model sols. in excellent agreement:‖u∗ − û∗‖2L2 = 6 · 10−3.


0 2 4 6 8 10 12 1410

−10

10−9

10−8

10−7

10−6

10−5

Iteration

CG

Res

idua

l

Convergence histories of the Conjugate Gradient algorithm applied to full (+)and reduced (o) order optimal control problems.

Recall error bound for the gradients:

‖∇j(u)−∇ĵ(u)‖L2 ≤ 2 (c‖u‖L2 + ‖ẑ(u)‖L2) (σr+1 + . . .+ σn) ∀u ∈ L2!


Outline

Overview





Error Estimates






Shape Optimization with Local Parameter Dependence([Antil et al., 2010, Antil et al., 2011, Antil et al., 2012])

I Consider minimization problem

minθ∈Θad

j(θ) :=

∫ T0

∫Ω(θ)

`(y(x, t; θ), t, θ)dx dt

where y(x, t; θ) solves

∂

∂ty(x, t)−∇(κ(x)∇y(x, t))

+V (x) · ∇y(x, t)) = f(x, t) (x, t) ∈ Ω(θ)× (0, T ),κ(x)∇y(x, t) · n = g(x, t) (x, t) ∈ ΓN (θ)× (0, T ),

y(x, t) = d(x, t) (x, t) ∈ ΓD(θ)× (0, T ),y(x, 0) = y0(x) x ∈ ΩD(θ)

I Semidiscretization in space leads to

minθ∈Θad

j(θ) :=

∫ T0

`(y(t; θ), t, θ) dt

where y(t; θ) solves

M(θ)d

dty(t) + A(θ)y(t) = B(θ)u(t), t ∈ [0, T ],

M(θ)y(0) = M(θ)y0.


I We would like to replace the large scale problem

minθ∈Θad

j(θ) :=

∫ T0

`(y(t; θ), t, θ) dt


M(θ)d

dty(t) + A(θ)y(t) = B(θ)u(t), t ∈ [0, T ],

M(θ)y(0) = M(θ)y0

I by a reduced order problem

minθ∈Θad

Ĵ(θ) :=

∫ T0

`(ŷ(t; θ), t, θ) dt

where ŷ(t; θ) solves

M̂(θ)d

dtŷ(t) + Â(θ)y(t) = B̂(θ)u(t), t ∈ [0, T ],

M̂(θ)ŷ(0) = M̂(θ)ŷ0.

I Problem is that we need a reduced order model that approximates the fullorder model for all θ ∈ Θad! Cannot be done using BTMR. I am notaware of any MR method that can do this with guaranteed error bounds.


Localized parameters (nonlinearity)I Consider classes of problems where the shape parameter θ only

influences a (small) subdomain:

Ω̄(θ) := Ω̄1 ∪ Ω̄2(θ), Ω1 ∩ Ω2(θ) = ∅Γ = Ω̄1 ∩ Ω̄2(θ).

�

JJĴ

Γ

Ω1 Ω1Ω2(θ)

I The FE stiffness matrix times vector can be decomposed into

Ay =

AII1 AIΓ1 0AΓI1 AΓΓ(θ) AΓI2 (θ)0 AIΓ2 (θ) A

II2 (θ)

yI1yΓyI2

where AΓΓ(θ) = AΓΓ1 + A

ΓΓ2 (θ).

The matrices M, B admit similar representations.I Consider objective functions of the type∫ T

0

`(y(t), t, θ)dt =1

2

∫ T0

‖CI1yI1−dI1(t)‖22 + ˜̀(yΓ(t),yI2(t), t, θ)dt.Matthias Heinkenschloss June 14, 2016 87

Our Optimization problem

minθ∈Θad

j(θ) :=

∫ T0

`(y(t; θ), t, θ) dt


M(θ)d

dty(t) + A(θ)y(t) = B(θ)u(t), t ∈ [0, T ],

M(θ)y(0) = M(θ)y0

can now be written as

minθ∈Θad

j(θ) :=1

2

∫ T0

‖CI1yI1 − dI1(t)‖22 + ˜̀(yΓ(t),yI2(t), t, θ)dt.where y(t; θ) solves

MII1d

dtyI1(t) + M

IΓ1d

dtyΓ(t) + AII1 y

I1(t) + A

IΓ1 y

Γ(t) = BI1uI1(t)

MII2 (θ)d

dtyI2(t) + M

IΓ2 (θ)

d

dtyΓ(t) + AII2 (θ)y

I2(t) + A

IΓ2 (θ)y

Γ(t) = BI2(θ)uI2(t)

MΓI1d

dtyI1(t) + M

ΓΓ(θ)d

dtyΓ(t) + MΓI2 (θ)

d

dtyI2(t)

+AΓI1 yI1(t) + A

ΓΓ(θ)d

dtyΓ(t) + AΓI2 (θ)y

I2(t) = B

Γ(θ)uΓ(t)

Dependence on θ ∈ Θad is now localized. The fixed subsystem 1 is large. Thevariable subsystem 2 is small. Idea: Reduce subsystem 1 only.


First Order Optimality Conditions

I Lagrangian

L(y,p, θ) =

∫ T0

`(y(t), t, θ) dt+

∫ T0

p(t)T(M(θ)

d

dty(t)+A(θ)y(t)−B(θ)u(t)

)dt

I The first order necessary optimality conditions are

M(θ)d

dty(t) + A(θ)y(t) = B(θ)u(t)&t ∈ [0, T ],

M(θ)y(0) = y0,

−M(θ) ddt

p(t) + AT (θ)p(t) = −∇y`(y(t), t, θ) t ∈ [0, T ],

M(θ)p(T ) = 0.

∇θL(y,p, θ)(θ̃ − θ) ≥ 0, θ̃ ∈ Θad

I Gradient of j is given by ∇j(θ) = ∇θ`(y(t),p(t), θ).


Using DD structure, state and adjoint equations can be written as

MII1d

dtyI1(t) + M

IΓ1d

dtyΓ(t) + AII1 y

I1(t) + A

IΓ1 y

Γ(t) = BI1uI1(t)

MII2 (θ)d

dtyI2(t) + M

IΓ2 (θ)

d


I2(t) + A

IΓ2 (θ)y


MΓI1d

dtyI1(t) + M

ΓΓ(θ)d


d

dtyI2(t)

+AΓI1 yI1(t) + A

ΓΓ(θ)d


I2(t) = B

Γ(θ)uΓ(t),

−MII1d

dtpI1(t)−MIΓ1

d

dtpΓ(t) + AII1 p

I1(t) + A

IΓ1 p

Γ(t) = −(CI1)T (CI1yI1(t)− dI1)

−MII2 (θ)d

dtpI2(t)−MIΓ2 (θ)

d

dtpΓ(t) + AII2 (θ)p

I2(t) + A

IΓ2 (θ)p

Γ(t) = −∇yI2˜̀(.)

−MΓI1d

dtpI1(t)−MΓΓ(θ)

d

dtpΓ(t)−MΓI2 (θ)

d

dtpI2(t)

+AΓI1 pI1(t) + A

ΓΓ(θ)d

dtpΓ(t) + AΓI2 (θ)p

I2(t) = −∇yΓ ˜̀(.),

To apply model reduction to the system corresponding to fixed subdomain Ω1,we have to identify how yI1 and p

I1 interact with other components.


Model Reduction of Fixed Subdomain Problem

We need to reduce

MII1d

dtyI1(t) = −AII1 yI1(t)−MIΓ1

d

dtyΓ(t) + BI1u

I1(t)−AIΓ1 yΓ(t)

zI1 = CI1y

I1(t)− dI1

zΓ1 = −MΓI1d

dtyI1 −AΓI1 yI1,

−MII1d

dtpI1(t) = −AII1 pI1(t) + MIΓ1

d

dtpΓ(t)− (CI1)T zI1 −AIΓ1 pΓ(t)

qI1 = (BI1)TpI1

qΓ1 = MΓI1

d

dtpI1 −AΓI1 pI1

For simplicity we assume that

MIΓ1 = 0 MΓI1 = 0,


we get

MII1d

dtyI1(t) = −AII1 yI1(t) + (BI1 | −AIΓ1 )

(uI1yΓ

),(

zI1zΓ1

)=

(−CI1−AΓI1

)yI1 +

(I0

)dI1,

−MII1d

dtpI1(t) = −AII1 pI1(t) + (−(CI1)T | −AIΓ1 )

(zI1pΓ

),(

qI1qΓ1

)=

((BI1)

T

−AΓI1

)pI1.

System is exactly of form needed for balanced truncation model red.


Reduced Optimization ProblemI We apply BTMR to the fixed subdomain problem with inputs and output

determined by the original inputs to subdomain 1 as well as the interfaceconditions.

I In optimality conditions replace fixed subdomain problem by its reducedorder model.

I We can interpret the resulting reduced optimality system as the optimalitysystem of the following reduced optimization problem

min

∫ T0

1

2‖ĈI1ŷI1 − dI1(t)‖22 + ˜̀(yΓ(t),yI2(t), t, θ)dt

subject to

M̂II1d

dtŷI1(t) + M̂

IΓ1d

dtyΓ(t) + ÂII1 ŷ

I1(t) + Â

IΓ1 y

Γ(t) = B̂I1uI1(t)

MII2 (θ)d

dtyI2(t) + M

IΓ2 (θ)

d


I2(t) + A

IΓ2 (θ)y


M̂ΓI1d

dtyI1(t) + M

ΓΓ(θ)d


d

dtyI2(t)

+ÂΓI1 yI1(t) + A

ΓΓ(θ)d


I2(t) = B

Γ(θ)uΓ(t)

ŷI1(0) = ŷI1,0 y

I2(0) = y

I2,0, y

Γ(0) = yΓ0 ,

θ ∈ ΘadMatthias Heinkenschloss June 14, 2016 93

Error EstimateIf I there exists α > 0 such that

vTAv ≤ −αvTMv, ∀v ∈ RN ,

I the gradients ∇y

(2)I

˜̀(y(2)I ,yΓ, t, θ), ∇yΓ ˜̀(y(2)I ,yΓ, t, θ),∇θ ˜̀(y(2)I ,yΓ, t, θ), are Lipschitz continuous in y(2)I ,yΓ

I for all ‖θ̃‖ ≤ 1 and all θ ∈ Θ the following bound holds

max{‖DθM(2)(θ)θ̃‖, ‖DθA(2)(θ)θ̃‖, ‖DθB(2)(θ)θ̃‖

}≤ γ,

then there exists c > 0 dependent on u, ŷ, and p̂ such that

‖∇J(θ)−∇Ĵ(θ)‖L2 ≤c

α(σr+1 + ...+ σn).

If we assume the convexity condition

(∇J(θ̂∗)−∇J(θ∗))T (θ̂∗ − θ∗) ≥ κ‖θ̂∗ − θ∗‖2,

then we obtain the error bound

‖θ∗ − θ̂∗‖ ≤c

ακ(σr+1 + ...+ σn).


ExampleI Reference domain Ωref

−10 −8 −6 −4 −2 0 2 4 6 8 10−1

0

1

ΩA

ΩB

ΩH

ΓI

ΓI

ΩC

ΓR

ΓL

ΓT

ΓB

I Optimization problem

min

T∫0

∫ΓL∪ΓR

|y − yd|2dsdt+T∫

0

∫Ω2(θ)

|y − yd|2dxdt

subject to the differential equation

yt(x, t)−∆y(x, t) + y(x, t) =100 in Ω(θ)× (0, T ),n · ∇y(x, t) = 0 on ∂Ω(θ)× (0, T ),

y(x, 0) = 0 in Ω(θ)

and design parameter constraints θmin ≤ θ ≤ θmax.I We use kT = 3, kB = 3 Bézier control points to specify the top and the

bottom boundary of the variable subdomain Ω2(θ).The desired temperature yd is computed by specifying the optimalparameter θ∗ and solving the state equation on Ω(θ∗).


I We use automatic differentiation to compute the derivatives with respectto the design variables θ.

I The semi-discretized optimization problems are solved using a projectedBFGS method with Armijo line search. The optimization algorithm isterminated when the norm of projected gradient is less than � = 10−4.

I The optimal domain

−10 −8 −6 −4 −2 0 2 4 6 8 10

−1

0

1


N(1)dof Ndof

Reduced 147 581Full 4280 4714

Sizes of the full and thereduced order problems

0 50 100 150 200 250 30010

−10

10−8

10−6

10−4

10−2

100

The largest Hankel singular valuesand the threshold 10−4σ1

Error in solution between full and reduced order problem:‖θ∗ − θ̂∗‖2 = 2.325 · 10−4

Optimal shape parameters θ∗ and θ̂∗ (rounded to 5 digits)computed by minimizing the full and the reduced order model.θ∗ (1.00, 2.0000, 2.0000, -2.0000, -2.0000, -1.00)

θ̂∗ (1.00, 1.9999, 2.0001, -2.0001, -1.9998, -1.00)


The convergence histories of the projected BFGS algorithm applied to thefull and the reduced order problems.

0 5 10 15 20 25 3010

−6

10−4

10−2

100

102

k

J(x k)

convergence history of the objec-tive functionals for the full (+)and reduced (o) order model.

0 5 10 15 20 25 3010

−6

10−4

10−2

100

102

k

|| P

∇ J

(xk)

||

convergence history of the pro-jected gradients for the full (+)and reduced (o) order model.


Example - Stokes

Geometry motivated by biochip

Problems where the shape param. θ only influences a (small) subdomain:

Ω̄(θ) := Ω̄1 ∪ Ω̄2(θ), Ω1 ∩ Ω2(θ) = ∅, Γ = Ω̄1 ∩ Ω̄2(θ).

Here Ω2(θ) is the top left yellow, square domain.


minθmin≤θ≤θmax

J(θ) =

T∫0

∫Ωobs

12 |∇×v(x, t; θ)|

2dx+

∫Ω2(θ)

12 |v(x, t; θ)−v

d(x, t)|2dxdt

where v(θ) and p(θ) solve the Stokes equations

vt(x, t)− µ∆v(x, t) +∇p(x, t) = f(x, t), in Ω(θ)× (0, T ),∇ · v(x, t) = 0, in Ω(θ)× (0, T ),

v(x, t) = vin(x, t) on Γin × (0, T ),v(x, t) = 0 on Γlat × (0, T ),

−(µ∇v(x, t)− p(x, t)I)n = 0 on Γout × (0, T ),v(x, 0) = 0 in Ω(θ).

Here Ω(θ) = Ω1 ∪ Ω2(θ) and Ω2(θ) is the top left yellow, square domain.Observation region Ωobs is part of the two reservoirs.

Stokes equation requires additional care:I Domain decomposition ([Pavarino and Widlund, 2002]).I Balanced truncation ([Stykel, 2006], [Heinkenschloss et al., 2008])I See [Antil et al., 2011]


We have 12 shape parameters, θ ∈ R12.

Reference domain Ωref Optimal domain


grid m N(1)v,dof N

(1)v̂,dof Nv,dof Nv̂,dof

1 149 4752 23 4862 1332 313 7410 25 7568 1833 361 11474 26 11700 2524 537 16472 29 16806 363

The number m of observations in Ωobs, thenumber of velocities N

(1)v,dof , N

(1)v̂,dof in the

fixed subdomain Ω1 for the full and re-duced order model, the number of velocitiesNv,dof , Nv̂,dof in the entire domain Ω forthe full and reduced order model for fivediscretizations.

0 50 100 150 200 250 30010

−10

10−8

10−6

10−4

10−2

100

102

The largest Hankel singu-lar values and the threshold10−3σ1


I Error in optimal parameter computed sing the full and the reducedorder model (rounded to 5 digits)

θ∗ (9.8987, 9.7510, 9.7496, 9.8994, 9.0991, 9.2499, 9.2504, 9.0989)

θ̂∗ (9.9026, 9.7498, 9.7484, 9.9021, 9.0940, 9.2514, 9.2511, 9.0956)

I The convergence histories of the projected BFGS algorithm appliedto the full and the reduced order problems.

0 5 10 1510

−8

10−6

10−4

10−2

k

J(x k)

convergence history of the objec-tive functionals for the full (+)and reduced (o) order model.

0 5 10 15

10−4

10−3

10−2

10−1

k

|| P

∇ J

(xk)

||

convergence history of the pro-jected gradients for the full (+)and reduced (o) order model.


Recap of Part I

I Reviewed projection based model reduction for simulation.

I Reviewed adjoint eqn. approach for gradient and Hessiancomputation.

I Gradient computation requires solution of adjoint PDE.I Hessian time vector computation requires solution of linearized state

PDE and 2nd order adjoint PDE.

I Reduced order models for optimization must approximate theobjective function j(u) and its gradient ∇j(u).

I Considered two classes of optimization problemsI Parameterized linear quadratic problems.

Sample optimization problems to generate reduced order model thatallows fast on-line solution of linear quadratic problem at out ofsample parameter.

I Linear quadratic problems, or problems with localized nonlinearity forwhich reduced order models can be computed that are goodapproximations for all controls u.


Review Part I

I Overview

I Example Optimization Problems

I Optimization Problem

I Projection Based Model Reduction

I Back to Optimization

I Error Estimates

I Linear-Quadratic Problems

I Shape Optimization with Local Parameter Dependence


I Original problem

min j(u)

s.t. u ∈ Uad,

where j(u) = J(y(u),u), y(u) ∈ Rn solves c(y,u) = 0. n large.

I Reduced order problemConstruct V ∈ Rn×r, r � n, rank(V) = r.Reduced order problem:

min ĵ(u)

s.t. u ∈ Uad,

where ĵ(u) = J(Vŷ(u),u), ŷ(u) ∈ Rr solves reduced stateequation VT c(Vŷ,u) = 0 ∈ Rr.


Review Proper Orthogonal Decomposition I

I Finite dimensional representation of snapshotsy(t1), . . . ,y(tm) ∈ Rn, m > r.

I Inner product vTMw and norm ‖ · ‖M. M s.p.d. but not nec. massmatrix.

I Compute orthonormal basis v1, . . . ,vr as solution of

min

m∑k=1

‖y(tk)−r∑i=1

y(tk)TMvi vi‖2M

s.t. vTi Mvj = δij .

I SolutionI Define Y = [y(t1), . . . ,y(tm)] ∈ Rn×m.I Compute M-orthonormal eigenvecs. v1,v2, . . . ∈ Rn and eigenvals.λ1 ≥ . . . ≥ λm ≥ 0 of generalized n× n eigenvalue prob.

MYYTMvi = λiMvi.


Review Proper Orthogonal Decomposition II

I Alternatively, if n > m compute eigenvectors w1,w2, . . . ∈ Rm andeigenvalues λ1 ≥ λ2 ≥ . . . ≥ λmin{m,n} ≥ 0 of

YTMYwi = λiwi.

vi = λ−1/2i Ywi, i = 1, . . . ,m.

I Usually, fix tolerance � > 0. Compute eigenvectors v1,v2, . . . ∈ Rnand eigenvalues λ1 ≥ λ2 ≥ . . . ≥ λmin{m,n} ≥ 0.

I Find smallest r such that∑mi=r+1 λi < �.

I If only some of the largest eigenvals. and vecs. are computed:Find smallest r such that λr+1/λ1 < �.

Reduced order model V = [v1, . . . ,vr] ∈ Rn×r.Error

m∑k=1

‖y(tk)−r∑i=1

y(tk)TMvi vi‖2M =

min{m,n}∑i=r+1

λi


Review Error Estimate for Unconstrained ProblemsI û∗ = argminuĵ(u) minimizer of unconstrained reduced problem.

I Newton-Kantorovich Theorem: Let r > 0 and ∇2j ∈ LipL(Br(û∗)).∇2j(û∗) be nonsingular and constants ζ, η ≥ 0 such that

‖∇2j(û∗)−1‖ = ζ, ‖∇2j(û∗)−1∇j(û∗)‖ ≤ η.

If Lζη ≤ 12 , there is unique local minimum u∗ of j in ball around û∗with radius min

{r,(

1−√

1− 2Lζη)/(Lζ)

}≤ min

{r, 2η

}.

I Estimate η:

‖∇2j(û∗)−1(∇j(û∗)−∇ĵ(û∗)︸︷︷︸=0

)‖ ≤ ζ‖∇j(û∗)−∇ĵ(û∗)‖ = η.

I Hence ‖u∗ − û∗‖U ≤ 2ζ‖∇ĵ(û∗)−∇j(û∗)‖UI Estimate error in gradients to get estimate for error in solution.

I Need Lζη ≤ 12 , i.e., ∇j(û∗) small enough.I Can estimate error using convergence properties of Newton’s

method started with û∗ applied to original problem.


Outline

Overview





Error Estimates






Semilinear Parabolic Model ProblemsI Distributed control

min1

2

∫ T0

∫Ω

|y(x, t)− z(x, t)|2 dx dt+ α2

∫ T0

∫Ω

|u(x, t)|2 dx dt,

where y = y(u) solves

yt(x, t)− ν∆y(x, t) + y(x, t)3 + u(x, t) = f(x, t), (x, t) ∈ Ω× (0, T ),y(x, t) = 0, (x, t) ∈ Γ× (0, T ),y(x, 0) = y0(x), x ∈ Ω.

I Robin boundary control of Burgers equation

min1

2

∫ T0

∫ 10

|y(x, t)−z(x, t)|2 dx dt+ α2

∫ T0

(|u0(t)|2 +|u1(t)|2) dt,

where y = y(u) solves

yt(x, t)− νyxx(x, t) + y(x, t)yx(x, t) = f(x, t), (x, t) ∈ (0, 1)× (0, T ),νyx(0, t) + σ0y(0, t) = u0(t), t ∈ (0, T ),νyx(1, t) + σ1y(1, t) = u1(t), t ∈ (0, T ),

y(x, 0) = y0(x), x ∈ (0, 1).

Matt

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