modeling and quantification of uncertainties in numerical aerodynamics

Numerical Methods for UncertaintyQuantification in Aerodynamic

Alexander Litvinenko and Hermann G. Matthies

Abstract In this work we research the influence of uncertainties in parameters and geome-

try on the solution in numerical aerodynamic. Typical examples of parameters withuncertainties are the angle of attack, the Mach number and airfoil geometry. Wequantify presented uncertainties and define how they propagate out. The RANSsolver is TAU code with k-w turbulence model. Discretisation techniques whichwe used here are Karhunen-Loeve and polynomial chaos expansions. To integratehigh-dimensional integrals in probabilistic space we usedMonte Carlo simulationsand collocation methods. Probability density as well as cumulative distribution func-tions are computed by the usage of the response surface, constructed via polynomialchaos expansion.

1 Introduction

Very often mathematical models contain parameters, right-hand sides, initial andboundary conditions which are uncertain. Possible reasonsfor uncertainties are,e.g., lack of data, random external influences or a random or uncertain environment.Typical examples of uncertain values are the angle of attack, the Mach number,the Reynolds number, airfoil geometry, parameters in turbulence modelling and vis-cosity. All these uncertainties can affect the solution dramatically. In this paper weconcentrate on the angle of attack, the Mach number and the airfoil geometry. Oneof the most promising techniques to model such uncertainties is the usage of randomvariables and random fields. To solve the given system numerically, in our case, thestochastic Navier Stokes equation with a k-w turbulence model, one has to discretisethe deterministic operator as well as the stochastic part. The total dimension of the

Institute of Scientific Computing, Technische Universitat Braunschweig, Hans-Sommer str. 65,38106, Braunschweig, Germany, e-mail: [email protected],WWW home page:http://www.wire.tu-bs.de

1

2 Alexander Litvinenko and Hermann G. Matthies

resulting discrete system is the product of dimensions of the deterministic part andthe stochastic part. The dimension of the stochastic part may be large.

As soon as the coefficients of the model are uncertain the solution (velocity, pres-sure,CL, CD, CM etc) will also be uncertain. The information of the interestusuallyis not the whole set of the solutions (too much data), but someother stochastic in-formation — cumulative distribution function, density function, the mean value,variance, exceedance probability etc.

A recent, efficient and most easy to implement technique for solving non-linearsystems with stochastic coefficients is the stochastic collocation method. The greatadvantage in comparison with, e.g., the stochastic Galerkin method is that no modi-fications of the deterministic code are necessary. The advantage in comparison withMonte Carlo (MC) methods is a much smaller computational cost. But we alsonote that if the stochastic Galerkin method is implemented (especially for non-linearproblems) it can beat the stochastic collocation methods.

Additional aim is to speed up the stochastic collocation method by applying dif-ferent sparse data techniques for approximation of stochastic input data as well asfor computing sparse representation of the solution for further post-processing, that,in its turn, can also be very expensive numerically.

We have implemented two different strategies to quantify the influence of un-certainties in the angle of attackα and in the Mach numberMa. In the first one(Section 5.1) we assumed that the cumulative distribution functions, mean valuesand standard deviations of the random variablesα andMa are given. Then for eachpair αi andMai we compute the solution with the help of the TAU code. Since theexact stochastic solution is unknown, we compare the obtained results with the re-sults obtained via Monte Carlo simulations. In the second strategy (Section 5.2) weassume that the turbulence in the atmosphere randomly changes the velocity vec-tor and the angle of attack (see Fig. 2). We model turbulence in the atmosphereby two additionally axes-parallel velocity vectorsv1 andv2, which have Gaussiandistributions.

Numerical examples demonstrate the influence of the uncertainties in the Machnumber (Ma) and angle of attack (α) on the stochastic solution (CL andCD).

The structure of the paper is the following. Section 2 presents the short theoryabout direct computation of high dimensional integrals. Weremind the theory ofMonte Carlo simulations as well as the theory of sparse grid methods. In Section5 we describe two ways of modelling of uncertainties in parameters. Section 6 isdevoted to low-rank data formats and to the data compression. Numerical resultsare presented in Section 7.

2 Direct integration

In this section we remind Monte Carlo methods and sparse integration grids (seemore in [15, 23]). In problems with uncertainties the valuesof interest are the meanvalue, variance, density function of the solution etc. The mean value of a functional

Numerical Methods for Uncertainty Quantification 3

Ψ of the solutionu(x) can be computed as an integral over multidimensional domainΘ:

Ψu(x) = E(Ψ(x,θ ,u(x,θ ))) =

∫

ΘΨ(x,θ ,u(x,θ ))P(dθ ). (1)

To compute integral numerically, one approximates the “infinite dimensional” mea-sure spaceΘ by a “finite dimensional” one,ΘM, i.e. one uses only finitely many (M)random variables:

Ψu(x) ≈ ΨN =N

∑z=1

wzΨ(x,θ z,u(θ z)) =N

∑z=1

wzΨN(θ z), (2)

whereu(θ z) is the approximate solution produced by the deterministic solver forthe realisationθ z (e.g. from spaceRM). An example of the functionalΨ can be thelift coefficientCL depending on the angle of attackα and on the Mach numberMa

CL = CL(α,Ma) = CL(α(θ ),Ma(θ )). (3)

The evaluation points areθ z ∈ ΘM, andwz are the weights. Particularly, in MCmethods the weights arewz = 1/N. It is also this approximate solutionu(θ z) foreach realisationθz which makes the direct integration methods so costly.

To compute integral (2), proceed in the following way:Algorithm: (Computation of integral)

1. Select points{θz|z= 1, . . . ,N} ⊂ ΘM according to the integration rule.2. For eachθ z —a realisation of the stochastic system—solve the deterministic

problem with that fixed realisation, yieldingu(θ z).3. For eachθ z compute the integrandΨu(x,θ z,u(θ z)) in (2).4. Compute the sum in (2).

2.1 Monte Carlo simulations

Monte Carlo methods (MC methods) obtain the integration points asN independentrandom realisations ofθ z ∈ ΘM distributed according to the probability-measureΓon ΘM, and use constant weightswz = 1/N. MC methods are probabilistic as theintegration points are chosen randomly, and therefore the approximationΨN andthe error are random variables.

Monte Carlo is very robust, and almost anything can be tackled by it, providedit has finite variance. Its other main advantage is that its asymptotic behaviour forN → ∞ is not affected by the dimension of the integration domain asfor most othermethods.

Due to theO(‖ΨN‖L2 N−1/2) behaviour of the error, MC methods convergeslowly asN → ∞ —for instance, the error is reduced by one order of magnitudeif the number of evaluations is increased by two orders. The MC methods are wellsuited for integrands with small variance and low accuracy requirements. Monte


Carlo methods are well-parallelable and the easiest in implementation. These meth-ods do not require modification of the deterministic code.

2.2 Quadrature rules and sparse integration grids

The textbook approach to an integral like (2) would be to takea good one-dimensional quadrature rule, and to iterate it in every dimension; this is the fulltensor product approach.

Assume that we use one-dimensional Gauss-Hermite-formulas Qk with k ∈ N

integration pointsθ j ,k and weightswj ,k, j = 1, . . . ,k. As it is well-known, theyintegrate polynomials of degree less than 2k exactly, and yield an error of orderO(k−(2r−1)) for r-times continuously differentiable integrands, hence takes smooth-ness into full account.

If we take a tensor product of these rules by iterating themM times, we have

ΨN = QMk (Ψ) := (Qk⊗·· ·⊗Qk)(Ψ) =

M⊗

j=1

Qk(Ψ)

=k

∑j1=1

· · ·k

∑jM=1

wj1,k · · ·wjM ,kΨN(θ j1,k, . . . ,θ jM ,k).

This “full” tensor quadrature evaluates the integrand on a regular mesh ofN = kM

points, and the approximation-error has orderO(N−(2r−1)/M). Due to the exponen-tial growth of the number of evaluation points and hence the effort with increasingdimension, the application of full tensor quadrature is impractical for high stochasticdimensions. This has been termed the “curse of dimensions” [25].

Sparse grid, hyperbolic cross, or Smolyak quadrature [30, 4, 8] can be applied inmuch higher dimensions—for some recent work see e.g. [18, 25, 5, 26, 28] and thereferences therein. A software package is available in [27].

Like full tensor quadrature, a Smolyak quadrature formula is constructed fromtensor products of one-dimensional quadrature formulas, but it combines quadratureformulas of high order in only some dimensions with formulasof lower order in theother dimensions. For a multi-indexη ∈ N

M the Smolyak quadrature formula is

ΨN = SMk (Ψ) := ∑

k≤|η|≤k+M−1

(−1)k+M−1−|η|(

k−1|η |−k

) M⊗

j=1

Qη j (Ψ).

For a fixedk the number of evaluations grows significantly slower in the numberof dimensions than for full quadrature. The price is a largererror: full quadratureintegrates monomialsθη = θ η1

1 · · ·θ ηMM exactly if their partial degree maxjη j does

not exceed 2k−1. Smolyak formulasSMk integrate multivariate polynomials exactly

only if their total polynomial degree|η | is at most 2k−1. However the error is only


O(N−r(logN)(M−1)(r+1)) with N =O([

2k/(k!)]

Mk)

evaluation points for ar-timesdifferentiable function. This has been used up to several hundred dimensions.

It may be seen that smoothness of the function is taken into account very dif-ferently by the various integration methods. The “roughest” is Monte Carlo, it onlyfeels the variance, quasi Monte Carlo feels the function’s variation, and the Smolyakrules actually take smoothness into account.

A particular kind of Smolyak quadrature formulas is sparse Gauss-Hermite grids.The algorithms for construction of sparse Gauss-Hermite grids are well known (e.g.,[18]). An example of a sparse Gauss-Hermite grid (αi , Mai ), i = 1..137, is shown inFig. 1). This figure demonstrates a 2D sparse Gauss-Hermite grid for the perturbedangle of attackα ′

and the Mach numberMa′. Points of a sparse grid are collocation

Fig. 1 Sparse Gauss-Hermite grids for the perturbed angle of attack α ′and the Mach numberMa

′,

n = {13, 29, 137}.

points and in these points we compute deterministic code.

3 Karhunen-Loeve expansion

By definition, the Karhunen-Loeve expansion (KLE) ofκ(x,ω) is the followingseries [21]

κ(x,ω) = µκ(x)+∞

∑ℓ=1

√

λℓφℓ(x)ξℓ(ω), (4)

whereξℓ(ω) are uncorrelated random variables andµκ(x) = Eκ(x) is the mean valueof κ(x,ω), λℓ andφℓ are the eigenvalues and the eigenvectors of problem

Tφℓ = λℓφℓ, φℓ ∈ L2(G ), ℓ ∈ N, (5)

and operatorT is defined like follows

T : L2(G ) → L2(G ), (Tφ)(x) :=∫

Gcovκ(x,y)φ(y)dy.


For numerical purposes one truncates the KLE (4) to a finite numbermof terms. Inthe case of a Gaussian random field, theξℓ are independent standard normal randomvariables. In the case of a non-Gaussian random field, theξℓ are uncorrelated butnot necessary independent. Dependent random variables canbe approximated in aset of new independent Gaussian random variables [11, 31], e.g.

ξℓ(ω) = ∑α∈J

ξ (α)ℓ Hα(θ (ω)),

whereθ (ω) = (θ1(ω),θ2(ω), ...), ξ (α)ℓ are coefficients,Hα , α ∈ J , is a Hermitian

basis (see Appendix) andJ := {α|α = (α1, ...,α j , ...), α j ∈N0} a multi-index set.For the purpose of actual computation, truncate the polynomial chaos expansion(PCE) [11, 31] after finitely many terms.

After a finite element discretisation [17] the discrete eigenvalue problem (5)looks like

MCMφ ℓ = λ hℓ Mφ ℓ, Ci j = covκ(xi ,y j). (6)

Here the mass matrixM is stored in a usual data sparse format and the dense ma-trix C ∈ R

n×n (requiresO(n2) units of memory) is approximated in the sparseH -matrix format [17] (requires onlyO(nlogn) units of memory) or in the Kro-necker low-rank tensor format [16]. If not the complete spectrum is of interest, butonly a part of it then the needed computational resources canbe drastically reduced[1]. To computem eigenvalues (m≪ n) and corresponding eigenvectors we applyan iterative Krylov subspace (Lanczos) eigenvalue solver for symmetric matrices[20, 1, 19, 29]. This eigensolver requires only matrix-vector multiplications. Allmatrix-vector multiplications are performed in theH -matrix format which will costO(nlogn), wheren is number degrees of freedom.

For memory requirements and computing times ofH -matrix approximationssee [17].

4 Polynomial Chaos expansion and response surface

As suggested by Wiener [31], any random variable (e.g. the lift coefficientCL) maybe represented as a series of polynomials in uncorrelated and independent Gaussianvariablesθ = (θ1, ...,θm), this is thepolynomial chaos expansion(PCE) (see Ap-pendix). For this representation we take Hermitian basisHβ , β = (β1, ...,β j , ...) ∈J a multiindex,J a multiindex set (see Appendix):

CL(θ ) = ∑β∈J

Hβ (θ )CLβ . (7)

Decomposition (7) can be understood as a response surface for CL. As soon as theresponse surface is built, one can obtain the valueCL(θ ) for anyθ almost for free(only by evaluating the polynomial (7)). It can be very practical if one needs, e.g.


106 evaluations ofCL(·). Since Hermite polynomials are orthogonal, the coefficientsCLβ can be computed by projection:

CLβ =1

β !

∫

ΘHβ (θ )CL(θ )dP(dθ ), (8)

whereΘ the Gaussian probability space. This multidimensional integral can becomputed approximately, for example, on a sparse Gauss-Hermite grid

CLβ =1

β !

n

∑i=1

Hβ (θ i)CL(θ i)wi , (9)

where weightswi and pointsθ i are defined from sparse Gauss-Hermite integrationrule.

5 Statistical modelling of uncertainties

We have implemented two different strategies to research simultaneous propagationof uncertainties in the angle of attackα and in the Mach numberMa. In the firststrategy (Section 5.1) we assumed that the distributions, the mean values and stan-dard deviations for the random variablesα andMa are given. Then for each pairαi andMai of a sparse Gauss-Hermite grid we compute the solution with help ofthe deterministic code (TAU code) as well as different statistical functionals of in-terest. Since sparse Gauss-Hermite grid methods may be unstable, we compare theobtained results with the results of Monte Carlo simulations. In the second strategy(Section 5.2), we assume that the turbulence in the atmosphere randomly and simul-taneously changes the velocity vector and the angle of attack (see Fig. 2). We modelturbulence in the atmosphere by two additionally axes-parallel velocity vectorsv1

andv2, which have Gaussian distribution.

5.1 Distribution functions ofα and Ma are given

In real-life applications distribution functions of the Mach numberMa and the angleof attackα are unknown. As a start point we consider the uniform and the Gaussiandistributions. For our further numerical experiments we choose mean values andstandard deviations as in Table 5. The Reynolds number isRe= 6.5e+ 6 and thecomputational geometry is RAE-2822 airfoil.


5.2 Modelling of turbulence in the atmosphere

In this section we model influence of uncertainties in the turbulence in the atmo-sphere on the angle of attackα and on the Mach number (see Fig. 2). One shouldnot mix this kind of turbulence with the turbulence in the boundary layer reasonedby friction. We assume that turbulence vortices in the atmosphere are comparablewith the size of the airplane.We model the turbulence in the atmosphere via two vectors (in2D)

α

v

v

u

u’

α’v1

2

Fig. 2 Two random vectorsv1 andv2 model turbulence in the atmosphere. Airfoil, the old and newfreestream velocitiesu andu

′, the old and new angles of attackα andα ′

.

v1 =σθ1√

2and v2 =

σθ2√2

. (10)

whereθ1 andθ2 two Gaussian random variables with zero mean and unit variance,σ = Iu∞ the standard deviation of the turbulent velocity fluctuations, I the meanturbulence intensity andu∞ is the freestream velocity.

Denotingθ :=√

θ 21 + θ 2

2 , thenv :=√

v21 +v2

2 = Iu∞√2

θ andβ := arctgv2v1

. Aftereasy computations the new angle of attack will be as follows

α′= arctg

sinα +zsinβcosα −zcosβ

, wherez :=Iθ√

2(11)

and the new Mach number

Ma′= Ma

√

1+I2θ 2

2−√

2Iθ cos(β + α). (12)

By default, in the TAU code, the mean turbulence intensity isI = 0.001.Thus, alternatively to the way of modelling introduced in Sec. 5.1, uncertainties

in the angle of attackα ′= α ′

(θ1,θ2) and in the Mach numberMa′= Ma

′(θ1,θ2)

are described via two standard normal variablesθ1 andθ2. This is the second wayof modelling.


5.3 Uncertainties in geometry

We model uncertainties in the airfoil geometryG by the usage of a random fieldκ(x,ω):

∂Gε(ω) = {x+ εκ(x,ω)n(x) : x∈ ∂G }, (13)

where∂G is the surface of airfoil,n(x) a normal vector in pointx andε a smallparameter.To generateZ realisations of airfoils with uncertainties (e.g., for MC or collocationmethods) we follow to the Algorithm below:

1. Assume the covariance function cov(·, ·) for the random fieldκ(x,ω) is given2. Compute (in a sparse data format!)Ci j := cov(pi , p j) for all grid points3. Solve eigenproblem (6)4. Each random vectorξ = (ξ1(ω), ...,ξm(ω)) in KLE

κ(x,ω) ≈ ∑mi=1

√λiφiξi(ω) results new airfoil.

Sparse approximation of the dense matrixC is done in [17, 16].In Fig. 3 one can see 69 realisations of RAE-2822 airfoil withuncertainties in ge-ometry. The largest uncertainties in the airfoil geometry are in the positionx≈ 0.58.The used covariance function is of Gaussian type:

cov(p1, p2) = σ2 ·exp(−ρ2), (14)

whereσ is a parameter,p1 = (x1,x2), p2 = (y1,y2) ∈ ∂G two points,

ρ(p1, p2) =

√

2

∑i=1

|xi −yi |2/l2i , and l i are correlation length scales. (15)

The influence of uncertainties in the airfoil geometry on thesolution will be re-searched as follows:Algorithm: (Computation and usage of the response surface)

1. Computem eigenpairs of the discrete eigenvalue problem Eq. (6).2. Generate a sparse Gauss-Hermite grid inm-dimensional space. Denote the num-

ber of grid points byN.3. For each grid pointθ = (θ1, ...,θm) from item (2) compute KLE Eq. (4). Each

KLE is a new airfoil geometryγ(x,θ ).4. For each new airfoil solve the problem (call the TAU code).5. Using the solution from item (4) and Hermite polynomials,build response sur-

face (Sec. 4).6. Generate 106 points inm-dimensional space and evaluate response surface in

these points.7. Using the values form item (6) compute statistical functionals of interest.

The response surface from item (5) will be as follows


Fig. 3 69 RAE-2822 airfoils with uncertainties.

CL(θ1, ...,θm) = ∑β∈J

Hβ (θ1, ...,θm)CLβ .

6 Data compression

A large number of stochastic realisations requires a large amount of memory andcomputational resources. To decrease memory requirementsand the computing timewe offer to use a low-rank approximation for all realisations of input and output ran-dom fields. For each new realisation only corresponding low-rank update is com-puted (see, e.g. [3]). It can be practical when, e.g. many thousands Monte Carlosimulations are computed and stored.Let vi ∈ R

n be the solution vector (already centred), wherei = 1..Z a numberof stochastic realisations of the solution. Build from all these vectors the matrixW = (v1, ...,vZ) ∈ R

n×Z. Consider the factorization

W = ABT where A∈ Rn×k and B∈ R

Z×k. (16)

Definition 1. We say that matrixW is a rank-k matrix if the representation (16) isgiven. We denote the class of all rank-k matrices for which factorsA andBT in (16)exist byR(k,n,Z). If W∈R(k,n,Z) we say thatW has alow-rank representation.


The first aim is to compute a rank-k approximationW of W, such that

‖W−W‖ < ε, k≪ min{n,Z}.

The second aim is to compute an update for the approximationW with a linearcomplexity for every new coming vectorvZ+1. Below we present the algorithmwhich does this.

To get the reduced singular value decomposition we omit all singular values,which are smaller than some levelε or, alternative variant, we leave a fixed numberof largest singular values. After truncation we speak aboutreduced singular valuedecomposition (denoted by rSVD)W = U ΣVT , whereU ∈ R

n×k contains the firstkcolumns ofU , V ∈ R

Z×k contains the firstk columns ofV andΣ ∈ Rk×k contains

thek-biggest singular values ofΣ . There is theorem (see more in [24] or [6]) whichtells that matrixW is the best approximation ofW in the class of all rank-k matrices.The computation of such basic statistics as the mean value, the variance, the ex-ceedance probability can be done with a linear complexity. The following examplesillustrate computation of the mean value and the variance.Let W = (v1, ...,vZ) ∈ R

n×Z and its rank-k representationW = ABT , A ∈ Rn×k,

BT ∈ Rk×Z be given. Denote thej-th row of matrixA by a j ∈ R

k and thei-th col-umn of matrixBT by bi ∈ R

k.

1. One can compute the mean solutionv ∈ Rn as follows

v =1Z

Z

∑i=1

vi =1Z

Z

∑i=1

A ·bi = Ab, (17)

The computational complexity isO(k(Z+n)), besidesO(nZ)) for usual densedata format.

2. One can compute the mean value of the solution in a grid point x j as follows

v(x j) =1Z

Z

∑i=1

vi(x j) =1Z

Z

∑i=1

a j ·bTi = a jb. (18)

The computational complexity isO(kZ).3. One can compute the variance of the solution var(v) ∈ R

n by the computing thecovariance matrix and taking its diagonal. First, we compute the centred matrixWc := W−WeT , whereW = W ·e/Z ande= (1, ...,1)T . ComputingWc costsO(k2(n+ Z)) (addition and truncation of rank-k matrices). By definition, thecovariance matrix is cov= WcWT

c . The reduced singular value decompositionof Wc is Wc = UΣVT , Σ ∈ R

k×k, can be computed with a linear complexity viathe QR algorithm (Section 6.1). Now, the covariance matrix can be written like

cov=1

Z−1WcW

Tc =

1Z−1

UΣVTVΣTUT =1

Z−1UΣΣTUT . (19)

The variance of the solution vector (i.e. the diagonal of thecovariance matrix in(19)) can be computed with the complexityO(k2(Z+n)).


4. One can compute the variance value var(v(x j)) in a grid pointx j with a linearcomputational cost.

5. To compute minimum or maximum of the solution in a pointx j over all realisa-tions costO(kZ).

For further estimations of numerical complexity we recall the following theorem

Theorem 1.Table 1 shows storage requirements and computational complexitiesfor rank-k matrices.

Table 1 Storage requirements and computational complexities for rank-k matrices.

Operation Description Complexity

storage(W) W ∈ R(k,n,Z), W = ABT k(n+Z)Wx W∈ R(k,n,Z),W = ABT ,x∈ R

Z 2k(n+Z)−n−kW′ +W′′ W′ ∈ R(k′,n,Z),W′′ ∈ R(k′′,n,Z) (n+Z)(k′ +k′′)W′W′′ W′ ∈ R(k′,n,Z),W′′ ∈ R(k′′,n,Z) 2k′k′′(n+Z)−k′′(n+k′)rSVD(W) W = ABT ∈ R(k,n,Z) 6k2(n+Z)+22k3

Truncate the rankk→ k′ of W W∈ R(k,n,Z) 6k2(n+Z)+22k3

Proof: see [9], [7], [2], [10].

6.1 Low-rank update with linear complexity

LetW = ABT ∈ Rn×Z and matricesA andB be given. An rSVDW = UΣVT can be

computed efficiently in three steps (QR algorithm for computing the reduced SVD):

1. Compute (reduced) aQR-factorization ofA = QARA andB = QBRB, whereQA ∈ R

n×k, QB ∈ RZ×k, and upper triangular matricesRA,RB ∈ R

k×k.2. Compute an reduced SVD ofRART

B = U ′ΣV ′T .3. ComputeU := QAU ′, V := QAV ′T .

QR-decomposition can be done faster if a part of matrixA (or B) is orthogonal (see,e.g. [3]). The first and third steps needO((n+Z)k2) operations and the second stepneedsO(k3). The total complexity of rSVD isO((n+Z)k2+k3).Suppose we have already matrixW = ABT ∈R

n×Z containing solution vectors. Sup-pose also that matrixW

′ ∈ Rn×m contains newm solution vectors. For the small

matrix W′, computing the factorsC andDT , such thatW

′= CDT , is not expen-

sive. Now our purpose is to compute with a linear complexity the new matrixWnew∈ R

n×(Z+m) in the rank-k format. To do this, we build two concatenated ma-tricesAnew:= [AC]∈R

n×2k andBTnew= blockdiag[BT DT ]∈R

2k×(Z+m). Note thatthe difficulty now is that matricesAnewandBnewhave rank 2k. To truncate the rankfrom 2k to k we use the QR-algorithm above. Obtain


Wnew= UΣVT = U(VΣT)T = AnewBTnew,

whereAnew∈ Rn×k andBT

new∈ Rk×(Z+m). Thus, the “update” of the matrixW is

done with a linear complexityO((n+m+Z)k2+k3).

7 Numerics

We demonstrate the influence of uncertainties in the angle ofattack, the Mach num-ber and the airfoil geometry on the solution (the lift, drag,lift coefficient and skinfriction coefficient). As an example we consider two-dimensional RAE-2822 airfoil.The deterministic solver is the TAU code with k-w turbulencemodel. We assumethat α andMa are Gaussian with meansα = 2.79, Ma = 0.734 and the standarddeviationsσ(α) = 0.1 andσ(Ma) = 0.005.

In Fig.4 we compare the cumulative distribution and densityfunctions for the liftand drag, obtained via the response surface (PCE of order 1) and via 6360 MonteCarlo simulations. To get a large sample we use sparse Gauss-Hermite grids (with 13and 29 nodes) to build corresponding response surfaces and then perform 106 MCevaluations on each response surface. Thus, one can see thatvery cheap collocationmethod (13 or 29 deterministic evaluations) produces similar to MC method with6360 simulations. But, at the same time we can not say which result is more precise.The exact solution is unknown and 6360 MC simulations are toofew.

The graphics in Fig. 5 demonstrate error bars[mean−σ ,mean+ σ ], σ the stan-dard deviation, for the pressure coefficient cp and absoluteskin friction cf in eachsurface point of the RAE2822 airfoil. The data are obtained from 645 realisationof the solution. One can see that the largest error occur at the shock (x ≈ 0.6). Apossible explanation is that the shock position is expectedto slightly change withvarying parametersα andMa.


Fig. 4 Density functions (first row), cumulative distribution functions (second row) ofCL (left)andCD (right). PCE is of order 1 with two random variables. Three graphics computed with 6360MC simulations, 13 and 29 collocation points.

Fig. 5 Error bars[mean−σ , mean+ σ ], σ standard deviation, in each point of RAE2822 airfoilfor the cp and cf.


To decrease memory requirements we write allZ = 645 realisations of the so-lution as matrices∈ R

512×645 and compute their rank-k approximations. In Table2 one can see dependence of the accuracy of the rank-k approximations (in thespectral norm) on the rankk. Additionally, one can also see much smaller memoryrequirement (dense matrix format costs 2.6MB). In the two last rows we comparecomputing time needed for SVD-update Algorithm described in Section 6.1 withthe standard SVD of the global matrix∈ R

512×645. One can see that SVD-updateAlgorithm performs faster.

Table 2 Accuracy, computing time and memory requirements of the rank-k approximation of thesolution matricesD = [density], P = [pressure], CP= [cp]; CF = [cf] ∈ R

512×645.

rankk 2 5 10 20

‖D− Dk‖2/‖D‖2 6.6e-1 4.1e-2 3.5e-3 3.5e-4‖P− Pk‖2/‖P‖2 6.9e-1 8.4e-2 8.2e-3 7.2e-4‖CP−CPk‖2/‖CP‖2 6.0e-3 5.3e-4 3.2e-5 2.4e-6‖CF −CFk‖2/‖CF‖2 9.0e-3 7.7e-4 4.6e-5 3.5e-6memory, kB 18 46 92 185SVD-update time, sec 0.58 0.60 0.62 0.68usual SVD time, sec 0.55 0.63 2.6 3.8


Fig. 6 demonstrates decay of 100 largest eigenvalues of fourmatrices, corre-sponding to the pressure, density, pressure coefficient cp and absolute skin frictioncf. Each matrix belongs to the spaceR

512×645.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−20

−15

−10

−5

0

5

log, #eigenvalues

log,

val

ues

pressuredensitycpcf

Fig. 6 Decay (in log-scales) of 100 largest eigenvalues of the matrices constructed from 645 solu-tions (pressure, density, cf, cp) on the surface of RAE-2822airfoil.

In Table 3 one can see dependence of the relative error (in theFrobenious norm)on the rankk. Seven solution matrices contain pressure, density, turbulence kineticenergy (tke), turbulence omega (to), eddy viscosity (ev), x-velocity (xv), z-velocity(zv) in the whole computational domain with 260000 dofs. Additionally, one canalso see much smaller memory requirement (dense matrix format costs 1.25GB).

Table 3 Relative errors and memory requirements of rank-k approximations of the solution matri-ces∈ R

260000×600. Memory required for the storage of each matrix in the dense matrix format is1.25 GB.

rankk pressure density tke to ev xv zv memory,MB

10 1.9e-2 1.9e-2 4.0e-3 1.4e-3 1.4e-3 1.1e-2 1.3e-2 2120 1.4e-2 1.3e-2 5.9e-3 3.3e-4 4.1e-4 9.7e-3 1.1e-2 4250 5.3e-3 5.1e-3 1.5e-4 9.1e-5 7.7e-5 3.4e-3 4.8e-3 104


In Table 4 one can see corresponding computing times: time required for theSVD-update Algorithm described in Section 6.1 and the time required for the stan-dard SVD of the global matrix∈ R

260000×600. A possible explanation for the largecomputing time for the standard SVD is the lack of memory and expensive swap-ping of data.

Table 4 Computing times (for Table 3) of rank-k approximations of the solution matrices∈R

260000×600.

rankk SVD-update time, sec. usual SVD time, sec.

10 107 153720 150 208450 228 8236

7.1 α and Ma have Gaussian distribution

For further numerical experiments we choose mean values andstandard deviationsas in Table 5.

Table 5 Mean values and standard deviations

mean st. deviation,σ σ /mean

Angle of attack,α 2.790 0.1 0.036Mach number,Ma 0.734 0.005 0.007

Table 6 demonstrates application of sparse Gauss-Hermite two-dimensional gridswith n = {5, 13, 29} grid points. The Hermite polynomials (see Section 4) are oforder 1 with two random variables. In the last column of each table we computethe measure of uncertaintyσ/mean. For instance, forn = 5 it shows that 3.6% and0.7% (Table 5) of uncertainties inα and inMa correspondingly result in 2.1% and15.1% of uncertainties in the lift and drag coefficients (Table 6). These three gridsshow very similar results in the the mean value and in the standard deviation. Atthe same time the results obtained via 1500 MC simulations are very similar to theresults computed on sparse Gauss-Hermite grids above.

Thus, one can make conclusion that the sparse Gauss-Hermitegrid with a smallnumber of points, e.g.n = 13, produces similar to MC results.

Table 7 demonstrates statistics obtained for the case when random variablesαand Ma have uniform distribution. Comparing Table 7 with Table 6 one can seethat, in the case of uniform distribution of uncertain parameters, the uncertainties in


Table 6 Uncertainties obtained on sparse Gauss-Hermite grids with5 ,13, 29 points and with 1500MC simulations.

n mean st. dev.σ σ /mean

5 CL 0.8530 0.0180 0.021CD 0.0206 0.0031 0.151

13 CL 0.8530 0.0174 0.020CD 0.0206 0.0030 0.146

29 CL 0.8520 0.0180 0.021CD 0.0206 0.0031 0.151

MC 1500 CL 0.8525 0.0172 0.020CD 0.0206 0.0030 0.146

the lift and drag coefficients are smaller. Namely, 1.2% and 8.8% (for CL andCD)against 2% and 14.6% in the case of the Gaussian distribution. But, uncertaintiesin the input parametersα andMa, in the case of the uniform distribution, are alsosmaller: 2.1% and 0.4% against 3.5% and 0.7%.

Table 7 Statistic obtained from 3800 MC simulations,α andMa have uniform distribution.

mean st. dev.σ σ /mean

α 2.787 0.058 0.021Ma 0.734 0.003 0.004CL 0.853 0.0104 0.012CD 0.0205 0.0018 0.088

7.2 α(θ1,θ2), Ma(θ1,θ2), whereθ1, θ2 have Gaussian distributions

In this section we illustrate numerical results for the model described in Section 5.2.Table 8 shows statistics (the mean value, variance and standard deviation), com-

puted on sparse Gauss-Hermite grids withn = 137 grid points.Table 9 compares uncertainties computed on sparse Gauss-Hermite grids with

n = {137, 381, 645} nodes with the uncertainties computed by the MC method(17000 simulations). All three grids and MC forecast very similar uncertaintiesσ /mean in the drag coefficientCD and in the lift coefficientCL.

Table 10 compares the mean values computed on sparse Gauss-Hermite grids (nnodes) with 17000 MC simulations. One can see that the errorsare very small. Butthis table tells only that sparse Gauss-Hermite grid withn points can be successfullyused to compute the mean valuesCL andCD.


Table 8 Statistics obtained on sparse Gauss-Hermite grid with 137 points.


α 2.8 0.2 0.071Ma 0.73 0.0026 0.0036CL 0.85 0.0373 0.044CD 0.01871 0.00305 0.163

Table 9 Comparison of results obtained by a sparse Gauss-Hermite grid (n grid points) with 17000MC simulations.

n 137 381 645 MC, 17000

σCLCL

0.044 0.042 0.042 0.045σCDCD

0.163 0.159 0.16 0.1589|CL−CL0|

CL7.6e-4 1.3e-3 1.6e-3 4.2e-4

|CD−CD0|CD

1.66e-2 1.46e-2 1.4e-2 2.1e-2

Table 10 Comparison of mean values obtained by MC simulations and by sparse Gauss-Hermitegrid with n grid points.

n 137 381 645

|CLn−CLMC|CLMC

·100% 0.1% 0.1% 0.1%|CDn−CDMC|

CDMC·100% 0.4% 0.6% 0.7%

7.3 Uncertainties in the geometry

We follow the Algorithm described in Section 5.3. The numberof KLE terms (4) ism= 3. We assume that the covariance function is of Gaussian type

cov(p1, p2) = σ2 ·exp(−d2), d =√

|x1−x2|2/l21 + |z1−z2|2/l22,

where σ = 10−3, p1 = (x1,0,z1), p2 = (x2,0,z2), the covariance lengthsl1 =|maxi(x)−mini(x)|/10 andl2 = |maxi(z)−mini(z)|/10. Stochastic dimension is 3and number of sparse Gauss-Hermite points is 25. After the response surfaces forCLandCD are built (see Section 4), we generated 106 MC points and evaluate valuesof both response surfaces in these points. Table 11 demonstrate the computed statis-tics. Surprisingly small are uncertainties in theCL andCD — 0.58% and 0.65%correspondingly. A possible explanation can be a small uncertain perturbations inthe airfoil geometry.


Table 11 Statistics obtained for uncertainties in the geometry. Gaussian covariance function wasused. PCE of order 1 with 3 random variables. Sparse Gauss-Hermite grid contains 25 points.


CL 0.8552 0.0049 0.0058CD 0.0183 0.00012 0.0065

8 Conclusion

In this work we research how uncertainties in the input parameters (the angle ofattackα and the Mach numberMa) and in the airfoil geometry influence the solution(lift, drag, pressure coefficient and absolute skin friction). Uncertainties in the Machnumber and in the angle of attack weakly affect the lift coefficient (1%−3%) andstrongly affect the drag coefficient (around 14%). Uncertainties in the geometryinfluence both the lift and drag coefficients weakly (less that 1%), but changes inthe geometry were also very small. Results obtained via collocation method on asparse Gauss-Hermite grid are comparable with Monte Carlo results, but requiremuch less deterministic evaluations (and as a sequence - smaller computing time).

To compare the results computed on a sparse Gauss-Hermite grids we used MCsimulations. We note that to get reliable results with MonteCarlo methods oneshould perform 105-107 simulations, but it is impossible to do in a reasonable time(1 simulation with TAU code requires at least 10000 iterations and takes between 20and 80 minutes). We performed 17000 simulations and this allows us to make onlyapproximate comparison.

To make statistical computational more efficient (linear complexity and linearstorage besides quadratic or even cubic) an additional research in this work was de-voted to the low-rank approximation of the results. We foundout that all realisationsof the solution can be approximated and stored in the low-rank format. This formatallows us to compute all important statistics with linear complexity and drasticallyreduces memory requirements.

Acknowledgements It is acknowledged that this research has been conducted within the projectMUNA under the framework of the German Luftfahrtforschungsprogramm funded by the Ministryof Economics (BMWA). The authors would like also to thank Elmar Zandler for his matlab package“Stochastic Galerkin library” [32].

9 Appendix A — Multi-Indices

In the above formulation, the need for multi-indices of arbitrary length arises. For-mally they may be defined by [23]

β = (β1, . . . ,β j , . . .) ∈ J := N(N)0 , (20)


which are sequences of non-negative integers, only finitelymany of which are non-zero. As by definition 0! := 1, the following expressions are well defined:

|β | :=∞

∑j=1

β j , (21)

β ! :=∞

∏j=1

β j !, (22)

ℓ(β ) := max{ j ∈ N |β j > 0}. (23)

9.1 Appendix B — Hermite Polynomials

As there are different ways to define—and to normalise—the Hermite polynomials,a specific way has to be chosen. In applications with probability theory it seemsmost advantageous to use the following definition [14, 11, 12, 13, 22]:

hk(t) := (−1)ket2/2(

ddt

)k

e−t2/2; ∀t ∈ R, k∈ N0,

where the coefficient of the highest power oft —which is tk for hk —is equal tounity.

The first five polynomials are:

h0(t) = 1, h1(t) = t, h2(t) = t2−1,

h3(t) = t3−3t, h4(t) = t4−6t2+3.

The recursion relation for these polynomials is

hk+1(t) = t hk(t)−khk−1(t); k∈ N.

These are orthogonal polynomials w.r.t standard Gaussian probability measureΓ ,whereΓ (dt) = (2π)−1/2e−t2/2dt —the set{hk(t)/

√k! |k ∈ N0} forms a complete

orthonormal system (CONS) inL2(R,Γ ) —as the Hermite polynomials satisfy∫ ∞

−∞hm(t)hn(t)Γ (dt) = n! δnm.

Multi-variate Hermite polynomials will be defined for an infinite number of vari-ables, i.e. fort = (t1, t2, . . . ,t j , . . .) ∈ R

N, the space of all sequences. This uses themulti-indices defined in Appendix A. Forα = (α1, . . . ,α j , . . .) ∈ J remember thatexcept for a finite number all otherα j are zero; hence in the definition of the multi-variate Hermite polynomial


Hα(t) :=∞

∏j=1

hα j (t j ); ∀t ∈ RN, α ∈ J ,

except for finitely many factors all others areh0, which equals unity, and the infiniteproduct is really a finite one and well defined.

The spaceRN can be equipped with a Gaussian (product) measure, again de-noted byΓ . Then the set{Hα(t)/

√α! | α ∈ J } is a CONS inL2(R

N,Γ ) as themultivariate Hermite polynomials satisfy

∫

RN

Hα(t)Hβ (t)Γ (dt) = α! δαβ ,

where the Kronecker symbol is extended toδαβ = 1 in caseα = β and zero other-wise.

References

1. Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, editors. Templates for thesolution of algebraic eigenvalue problems - A practical guide, volume 11 ofSoftware, Envi-ronments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia,PA, 2000.

2. S. Borm, L. Grasedyck, and W. Hackbusch.Hierarchical Matrices, volume 21 ofLectureNote. Max-Planck Institute for Mathematics, Leipzig, 2003. www.mis.mpg.de.

3. Matthew Brand. Fast low-rank modifications of the thin singular value decomposition.LinearAlgebra Appl., 415(1):20–30, 2006.

4. H.-J. Bungartz and M. Griebel. Sparse grids.Acta Numer., 13:147–269, 2004.5. Th. Gerstner and M. Griebel. Numerical integration usingsparse grids.Numer. Algorithms,

18(3-4):209–232, 1998.6. G. H. Golub and Ch. F. Van Loan.Matrix computations. Johns Hopkins Studies in the

Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, third edition, 1996.7. L. Grasedyck. Theorie und anwendungen hierarchischer matrizen. Ph.D. Thesis, University

of Kiel, Germany, 2001.8. M. Griebel. Sparse grids and related approximation schemes for higher dimensional problems.

In Foundations of computational mathematics, Santander 2005, volume 331 ofLondon Math.Soc. Lecture Note Ser., pages 106–161. Cambridge Univ. Press, Cambridge, 2006.

9. W. Hackbusch. A sparse matrix arithmetic based onH -matrices. I. Introduction toH -matrices.Computing, 62(2):89–108, 1999.

10. W. Hackbusch.Hierarchische Matrizen - Algorithmen und Analysis.Springer, 2009.11. T. Hida, Hui-Hsiung Kuo, J. Potthoff, and L. Streit.White noise - An infinite-dimensional

calculus, volume 253 ofMathematics and its Applications. Kluwer Academic PublishersGroup, Dordrecht, 1993.

12. H. Holden, B. Øksendal, J. Ubøe, and T. Zhang.Stochastic partial differential equations.Probability and its Applications. Birkhauser Boston Inc., Boston, MA, 1996. A modeling,white noise functional approach.

13. S. Janson.Gaussian Hilbert spaces, volume 129 ofCambridge Tracts in Mathematics. Cam-bridge University Press, Cambridge, 1997.

14. G. Kallianpur. Stochastic filtering theory, volume 13 ofApplications of Mathematics.Springer-Verlag, New York, 1980.


15. A. Keese. Numerical solution of systems with stochasticuncertainties. A general purposeframework for stochastic finite elements.Ph.D. Thesis, TU Braunschweig, Germany, 2004.

16. B. N. Khoromskij and A. Litvinenko. Data sparse computation of the Karhunen-Loeve expan-sion. Numerical Analysis and Applied Mathematics: International Conference on NumericalAnalysis and Applied Mathematics, AIP Conf. Proc., 1048(1):311–314, 2008.

17. B. N. Khoromskij, A. Litvinenko, and H. G. Matthies. Application of hierarchical matricesfor computing the Karhunen-Loeve expansion.Computing, 84(1-2):49–67, 2009.

18. A. Klimke. Sparse grid interpolation toolbox: www.ians.uni-stuttgart.de/spinterp/. 2008.19. C. Lanczos. An iteration method for the solution of the eigenvalue problem of linear differen-

tial and integral operators.J. Research Nat. Bur. Standards, 45:255–282, 1950.20. R. B. Lehoucq, D. C. Sorensen, and C. Yang.ARPACK users’ guide. Solution of large-scale

eigenvalue problems with implicitly restarted Arnoldi methods., volume 6 ofSoftware, Envi-ronments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia,PA, 1998.

21. M. Loeve.Probability theory I. Graduate Texts in Mathematics, Vol. 45, 46.Springer-Verlag,New York, fourth edition, 1977.

22. Paul Malliavin. Stochastic analysis, volume 313 ofGrundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin,1997.

23. H. G. Matthies. Uncertainty quantification with stochastic finite elements. 2007. Part 1.Fundamentals. Encyclopedia of Computational Mechanics.

24. L. Mirsky. Symmetric gauge functions and unitarily invariant norms.Quart. J. Math. OxfordSer. (2), 11:50–59, 1960.

25. E. Novak and K. Ritter. The curse of dimension and a universal method for numerical integra-tion. In Multivariate approximation and splines (Mannheim, 1996), volume 125 ofInternat.Ser. Numer. Math., pages 177–187. Birkhauser, Basel, 1997.

26. E. Novak and K. Ritter. Simple cubature formulas with high polynomial exactness.Constr.Approx., 15(4):499–522, 1999.

27. K. Petras. Smolpack—a software for Smolyak quadrature with delayed Clenshaw-Curtisbasis-sequence. http://www-public.tu-bs.de:8080/ petras/software.html.

28. K. Petras. Fast calculation of coefficients in the Smolyak algorithm. Numer. Algorithms,26:93–109, 2001.

29. Y. Saad.Numerical methods for large eigenvalue problems. Algorithms and Architectures forAdvanced Scientific Computing. Manchester University Press, Manchester, 1992.

30. S. A. Smolyak. Quadrature and interpolation formulas for tensor products of certain classesof functions.Sov. Math. Dokl., 4:240–243, 1963.

31. N. Wiener. The homogeneous chaos.American Journal of Mathematics, 60:897–936, 1938.32. E. Zander. A malab/octave toolbox for stochastic galerkin methods:

http://ezander.github.com/sglib/. 2008.

modeling and quantification of uncertainties in numerical aerodynamics

Engineering