Rights / License: Research Collection In Copyright - …...The present thesis was partially supported by the Swiss National Science Foundation under grant No. 200021-120290 which is

Research Collection

Doctoral Thesis

Sparse tensor discretizations of elliptic PDEs with random inputdata

Author(s): Bieri, Marcel

Publication Date: 2009

Permanent Link: https://doi.org/10.3929/ethz-a-005910608

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Diss. ETH No. 18598

Sparse tensor discretizations of ellipticPDEs with random input data

A dissertation submitted to

ETH Zurich

for the degree of

Doctor of Sciences

presented by

MARCEL BIERI

Dipl. Math. ETH

born January 7, 1979

citizen of Schangnau BE, Switzerland

accepted on the recommendation of

Prof. Dr. Christoph Schwab, ETH Zurich, examiner

Prof. Dr. Hermann G. Matthies, TU Braunschweig, co-examiner

Prof. Dr. Jan S. Hesthaven, Brown University, co-examiner

2009

Acknowledgments

First and foremost I would like to express my gratitude to my advisor Prof. ChristophSchwab, for his guidance during the last four years. He was always accessible andsupported me with his broad knowledge while also giving me the freedom to develop myown ideas. I would also like to thank my co-examiners Prof. Hermann Matthies andProf. Jan Hesthaven who kindly agreed to be the readers of my thesis.

I owe special thanks to my colleagues Bastian Pentenrieder and Claude Gittelson, withwhom I shared my office during the last two years and had many fruitful and encouragingdiscussions (mathematical and otherwise), and to Roman Andreev, who programmedand carried out some of the numerical examples presented in this thesis. Besides mycolleagues mentioned above, I would also like to thank all the other members and formermembers of the Seminar for Applied Mathematics. All of you made my time at ETH awonderful and most enjoyable experience.

But most of all I would like to thank my parents for their constant support, encour-agement and trust throughout the past years. Without them none of this would bepossible.

The present thesis was partially supported by the Swiss National Science Foundationunder grant No. 200021-120290 which is gratefully acknowledged.

Marcel Bieri

iii

iv

Abstract

We consider a stochastic Galerkin and collocation discretization scheme for solving el-liptic PDEs with random coefficients and forcing term, which are assumed to depend ona finite, but possibly large number of random variables.

Both methods consist of a hierarchic wavelet discretization in space and a sequence ofhierarchic approximations to the law of the random solution in probability space. Inthe Galerkin setting, the stochastic approximations are conducted by a best-N -termpolynomial chaos approximation while in the collocation setting we use interpolationoperators based on a Smolyak grid of Gauss points. In both approaches, we cover thecase of bounded random variables as well as unbounded Gaussian random variables asinput parameters. In a sparse tensor product fashion, we then compose the levels ofspatial and stochastic approximations, resulting in a substantial reduction of overalldegrees of freedom.

Numerical analysis is then used to estimate the convergence rates of the sparse tensorstochastic Galerkin and collocation methods, depending on the regularity of the randominputs. Numerical examples illustrate the theoretical results and indicate superiority ofthis novel sparse tensor product approximation compared to the ‘full tensor’ approachesused so far and the Monte Carlo method.

v

vi

Zusammenfassung

Wir betrachten ein stochastisches Galerkin- und Kollokationsverfahren fur die Losungpartieller Differentialgleichungen mit zufalligen Koeffizienten und Kraften, von welchenman annimmt, dass sie von einer moglicherweise grossen aber endlichen Anzahl vonZufallsvariablen abhangen.

Beide Methoden bestehen aus einer hierarchischen Wavelet-Diskretisierung im Raumund einer Folge von hierarchischen Approximationen an das Verhalten der zufalligenLosung im Wahrscheinlichkeitsraum. Im Galerkinverfahren werden die stochastischenApproximationen mittels einer best-N -term Approximation basierend auf polynomialemChaos durchgefuhrt, wahrend wir im Kollokationsverfahren Interpolationsoperatoren,basierend auf einem Smolyak-Gitter von Gauß-Punkten, benutzen. In beiden Ansatzendecken wir sowohl den Fall beschrankter Zufallsvariablen sowie auch unbeschrankterGauß’scher Zufallsvariablen als Eingabeparameter ab. In Analogie zur Konstruktion vondunnen Tensorprodukten setzen wir dann die Niveaus der raumlichen und stochastischenApproximationen zusammen, was zu einer erheblichen Reduzierung der Gesamtfreiheits-grade fuhrt.

Mittels numerischer Analysis erhalten wir Abschatzungen fur die Konvergenzraten imGalerkin- und Kollokationsverfahren. Numerische Experimente illustrieren die theo-retischen Resultate und lassen die Uberlegenheit dieser neuartigen dunntensorprodukt-Approximationen im Vergleich zu den

’volltensorprodukt‘-Approximationen und der

Monte-Carlo Methode erkennen.

vii

viii

Contents

Introduction xiii

1 Preliminaries 1

1.1 Product probability spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Tensor products of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Bochner spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Uncertainty quantification 5

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Karhunen-Loeve expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Definition of the KL-expansion . . . . . . . . . . . . . . . . . . . . 7

2.2.2 KL eigenvalue decay . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 KL eigenfunction bounds . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.4 The lognormal case . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Polynomial Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Hermite polynomials of Gaussian random variables . . . . . . . . . 14

2.3.2 Wiener chaos expansions . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3 Generalized polynomial chaos . . . . . . . . . . . . . . . . . . . . . 15

3 Problem formulation 17

3.1 Problem setting and notation . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Assumptions on the problem . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.2 Finite-dimensional noise . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.3 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.4 Growth at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.5 Stochastic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.2 Continuous dependence on input data . . . . . . . . . . . . . . . . 24

3.3.3 Growth at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.4 Stochastic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . 27

ix

Contents

4 Sparse tensor discretizations 31

4.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Stochastic Galerkin formulation . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.1 Hierarchic subspace sequences . . . . . . . . . . . . . . . . . . . . . 33

4.2.2 Sparse tensor stochastic Galerkin formulation . . . . . . . . . . . . 34

4.3 Stochastic collocation formulation . . . . . . . . . . . . . . . . . . . . . . 37

4.3.1 Hierarchic sequence of collocation operators . . . . . . . . . . . . . 37

4.3.2 Sparse tensor stochastic collocation formulation . . . . . . . . . . . 38

5 Wavelet discretization in D 43

5.1 Wavelet construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Basic properties of wavelet Galerkin approximation . . . . . . . . . . . . . 44

5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6 Sparse tensor stochastic collocation 49

6.1 Smolyak’s construction of collocation points . . . . . . . . . . . . . . . . . 49

6.2 Error analysis of Smolyak’s collocation algorithm . . . . . . . . . . . . . . 51

6.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.4 Proof of Proposition 6.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 Sparse tensor stochastic Galerkin 61

7.1 Hierarchic polynomial chaos approximation in L2ρ(Γ;H1

0 (D)) . . . . . . . . 61

7.1.1 Preliminary conventions and notations . . . . . . . . . . . . . . . . 61

7.1.2 Best-N -term polynomial chaos approximation . . . . . . . . . . . . 62

7.1.3 Approximation results . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.2 Analysis of the sparse tensor stochasticGalerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.3 Proof of Proposition 7.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.3.1 Analytic continuations . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.3.2 Bounds on the Legendre coefficients . . . . . . . . . . . . . . . . . 73

7.3.3 Bounds on the Hermite coefficients . . . . . . . . . . . . . . . . . . 75

7.3.4 τ -summability of the PC coefficients . . . . . . . . . . . . . . . . . 77

7.3.5 Proof of Proposition 7.1.2 . . . . . . . . . . . . . . . . . . . . . . . 78

8 Implementation and numerical examples 81

8.1 Implementational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8.1.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8.1.2 KL-eigenpair computation . . . . . . . . . . . . . . . . . . . . . . . 82

8.1.3 Localization of quasi-best-N -term gPC coefficients . . . . . . . . . 82

8.1.4 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

x

Contents

8.2.1 Wavelet discretization . . . . . . . . . . . . . . . . . . . . . . . . . 848.2.2 Smolyak interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 858.2.3 Sparse tensor stochastic collocation method . . . . . . . . . . . . . 878.2.4 Sparse tensor stochastic Galerkin method . . . . . . . . . . . . . . 88

References 93

Curriculum Vitae 99

xi

Contents

xii

Introduction

Many engineering models of physical phenomena are subject to significant data uncer-tainties. We mention subsurface flow, soil mechanics, earthquake engineering, to namebut a few. These uncertainties are usually crudely categorized into aleatory and epis-temic uncertainties, see e.g. [13]. Aleatory uncertainties are understood as inherentvariabilities of the system data parameters due to unpredictable effects, such as atmo-spheric conditions or subsurface properties of an aquifer in the study of groundwaterflows. Epistemic uncertainties, on the other hand, are understood as model uncertain-ties, which are due to a fundamental lack of knowledge of the processes and quantitiesidentified with the system. Neglecting epistemic uncertainties, we will exclusively con-sider PDEs with inherent parameter uncertainties, which are then often modeled asrandom fields [3, 61], resulting in stochastic partial differential equations.

The goal of our computations will be the approximation of statistical quantities, suchas mean value and correlation, of the modeled process. Numerical solution strategiesfor PDEs with random field inputs follow three major steps. First, the random inputfields are approximately parametrized by a finite, but possibly large number of randomvariables. This can e.g. be achieved by expanding the random fields into a Karhunen-Loeve expansion which is eventually truncated. Then, a numerical scheme to solvethe resulting high-dimensional deterministic PDEs is used to approximate the solution,depending now on space and time variables as well as on the set of input parameters.Finally, the solution is reconstructed as a random field by some form of post-processingand the statistical quantities of interest are computed. In this work, we will primarilyfocus on the second step, hence on the development of efficient strategies to solve PDEsdepending on a large set of alterable input parameters. These algorithms can be groupedinto two broad classes.

First, so-called nonintrusive schemes: here, existing deterministic solvers of the PDE ofinterest are used without any modification as a building block in an outer loop, wheresome form of sampling of the random parameter space is used to generate a set of par-ticular input realizations to be processed by the deterministic PDE solver, leading tocorresponding outputs of the random solution from which the desired statistics are recov-ered. Here, we find the Monte Carlo (MC) sampling strategies, stochastic collocation, aswell as certain high-order polynomial chaos (PC) methods, which are based on spectralrepresentations of the random fields’ in- and outputs. In the Monte Carlo method [21],

xiii

Introduction

a large set of i.i.d. parameter samples is generated, based on their prescribed statistics,and the solution statistics derived from the set of computed solutions of the (deter-ministic) PDE obtained from inserting these data samples. Due to the generally slowconvergence of the Monte Carlo method, the collocation approach has recently attracteda lot of attention. Introduced independently in [4] and [67], the stochastic collocationmethod, unlike MC, doesn’t choose the samples randomly, but in a deterministic way,based on the random inputs’ probability density functions. To overcome the curse ofdimension imposed by the possibly large numbers of input parameters, the work [67] al-ready proposed the use of Smolyak grids to reduce the number of collocation points. Thiswas further analyzed and developed in [43] and [42]. A different collocation approachwas proposed in [9], using an ANOVA based selection of collocation points to handlethe high dimensionality. Here, the set of stochastic input parameters is divided into(overlapping) groups of much smaller cardinality, resulting in an efficient approximationof the solution’s random behavior.

Second, so called intrusive schemes: here, approximants to the law of the random so-lutions are intertwined with existing deterministic solvers at an earlier stage of the al-gorithm. Popular representatives of this class of algorithms are stochastic Galerkin andperturbation methods. In stochastic Galerkin, the solution is projected onto pairings ofspatial and stochastic discretization spaces. A fundamental mathematical framework forthe stochastic Galerkin has been laid in [5], using finite elements in space and polynomialchaos in the random parameter domain as discretizations, thus often also referred to as(generalized) polynomial chaos method (gPC), e.g. [69, 68]. It already substantiatedmathematically the potential superiority of the stochastic Galerkin approach over MCtype methods. Multi-element polynomial chaos methods have then been considered in[62, 63], which can be seen as an h-version of the gPC approach, where the probabilitydomain is partitioned into smaller cells. Perturbation approaches lead to deterministic,but high-dimensional equations for the approximation of the k-th moment of the mod-eled process, hence aiming directly at computing the statistical moments. Using sparsetensor finite element techniques, these equations can be solved efficiently in time andmemory, see [54, 55].

All of the above algorithms, except the perturbation method, consist of a sequence ofstochastic approximations, e.g. polynomial chaos or collocation interpolation operators,to the law of the random solution and a spatial approximation, e.g. by finite elements.They exhibit an overall complexity, i.e. total number of degrees of freedom, of O(ND ×NΩ), where ND denotes the number of degrees of freedom of the spatial discretizationand NΩ the number of stochastic degrees of freedom. This is very prohibitive, especiallyif a fine resolution of the spatial behavior is required, e.g. due to short correlationlengths in the input random fields. The main idea of this work is to choose suitablehierarchic approximations in space and random parameter domain and combine themin a sparse tensor product fashion, leading to algorithms of O(ND logNΩ +NΩ logND)

xiv

Introduction

overall complexity, and hence a considerable reduction in computation time and memoryrequirement. We will use numerical analysis to tailor the stochastic approximations tothe levels of the deterministic discretization for the stochastic Galerkin and stochasticcollocation approach and provide convergence rates for both schemes. Our analysis willcover inputs of bounded random variables as well as Gaussian ones, which are often usedin practical applications.

The outline of the thesis is as follows. In Chapter 1, we will provide some preliminaryconstructions and results, which will repeatedly be used throughout the rest of the work.The parametrization of random fields will then be the aim of Chapter 2. Here, we will inparticular discuss the Karhunen-Loeve expansion of a random field and give a short in-troduction into polynomial chaos. In Chapter 3, we will introduce the class of problemsunder consideration and impose some necessary assumptions on it, on which the numer-ical analysis in the following chapters will rely. An example of a problem meeting thesecriteria will be provided. In Chapter 4, we will formulate the sparse tensor stochasticGalerkin and sparse tensor stochastic collocation method and present a first result on theoverall complexity of these algorithms. The construction of hierarchic wavelet bases forthe spatial approximation is then provided in Chapter 5. The construction will explicitlybe carried out for piecewise linear wavelets in one and two dimensions. In Chapter 6 wewill then analyze the sparse stochastic collocation method. We will provide a hierarchicsequence of interpolation operators, based on a Smolyak grid of Gauss points, and dis-cuss their adaption to the wavelet levels introduced in the previous chapter. In a similarmanner, we will then discuss the sparse stochastic Galerkin method in chapter 7. Here,we will present a hierarchic stochastic approximation by polynomial chaos in the spirit ofa best-N -term approximation and prove algebraic convergence rates. Finally, in Chap-ter 8, we will discuss issues regarding the implementation of these algorithms and givenumerical examples, which confirm the theoretical results of the previous chapters.

xv

Introduction

xvi

1 Preliminaries

We will first briefly review some of the important constructions and notations usedthroughout the present work. In particular countable products of probability spaces todescribe the random input data and the concept of Bochner spaces. Bochner spaces andcountable tensor products of probability spaces will turn out to be the natural functionspaces of solutions to the stochastic PDEs under consideration.

1.1 Product probability spaces

In this section, we will review the theory of products of probability spaces. The materialpresented here follows closely the one provided in §9 of [6] and will therefore not furtherbe referenced.

We denote by (Ω,Σ, P ) a probability space with Ω denoting the outcomes, Σ the sigma-algebra of possible events and P a probability measure on Σ, hence satisfying P (Ω) = 1.Assume we have a sequence of probability spaces (Ωn,Σn, Pn) where n ∈ N. For a subsetJ ⊂ N define

ΩJ :=×n∈J

Ωn

as the Cartesian product of the Ωn’s with n ∈ J . In particular, if J = N we writeΩ := ΩJ . If J = j consists only of one single element, we write Ωj := Ωj. We definethe projection operator

pJ : Ω → ΩJ

as the restriction of ω ∈ Ω to ΩJ . The product

Σ :=⊗

n∈N

Σn

of sigma-algebras is then defined as the smallest sigma-algebra such that any of theprojections pj is Σ-Σj measurable. The product measure P is defined as the uniquemeasure on

⊗

n∈NΣn s.t. for any finite set J ⊂ N and arbitrary events Ej ∈ Σj (j ∈ J)

it holds that

P

(

p−1J

(

×j∈J

Ej

))

=∏

j∈JPj(Ej).

1

1 Preliminaries

P is then called the product measure of Pnn∈N and denoted by⊗

n∈NPn. The proba-

bility space⊗

n∈N

(Ωn,Σn, Pn) :=

(

×n∈N

Ωn,⊗

n∈N

Σn,⊗

n∈N

Pn

)

is then called the product probability space of the triplets (Ωn,Σn, Pn)n∈N.

Remark 1.1.1. We note here, that the above definition extends also to uncountableproducts of probability spaces, where J ⊂ I is then a subset of some uncountable indexset I.

1.2 Tensor products of Hilbert spaces

This section introduces the tensor product between two Hilbert spaces, following theconstruction given in [48, Ch. II.4].

Let (H1, < ·, · >H1) and (H2, < ·, · >H2) be two Hilbert spaces with associated innerproducts. For each ϕ1 ∈ H1, ϕ2 ∈ H2, let ϕ1 ⊗ ϕ2 denote the bilinear form, acting onH1 ×H2, by

(ϕ1 ⊗ ϕ2) < ψ1, ψ2 >:=< ψ1, ϕ1 >H1< ψ2, ϕ2 >H2 .

Let E then be the set of all finite linear combinations of such forms, and define an innerproduct on E by

< ϕ1 ⊗ ϕ2, ψ1 ⊗ ψ2 >E :=< ϕ1, ψ1 >H1< ϕ2, ψ2 >H2

and extending it through linearity to E . It can be shown, that < ·, · >E is well definedand positive definite. The tensor product of H1 and H2, denoted by

H1 ⊗H2,

is then defined as the completion of E w.r.t. the inner product < ·, · >E . It can beshown, that if ϕkk∈N and ψll∈N are orthonormal bases for H1 and H2 respectively,then ϕk ⊗ ψlk,l∈N is an orthonormal basis for H1 ⊗H2.

Remark 1.2.1. The construction given above immediately extends to the tensor productbetween n Hilbert spaces

n⊗

k=1

Hk = H1 ⊗H2 ⊗ · · · ⊗Hn.

2

1.3 Bochner spaces

1.3 Bochner spaces

Bochner spaces are a generalization of Lp-spaces to functions taking values in a Banachspace X rather than in R. To define the Bochner spaces, however, we first need toextend the notions of measurability and integrability to Banach-valued functions, seee.g. [10, 20, 71].

Let (T,Σ, µ) be a measure space with Σ being a sigma-algebra over T and µ a measureon Σ. Let X be a Banach space with associated norm ‖ · ‖X . A function s : T → X iscalled simple, if it has the form

s(t) =

n∑

i=1

χEi(t)ui, t ∈ T,

where each Ei is a µ-measurable subset of T and ui ∈ X for i = 1, . . . , n. A functionf : T → X is called (strongly) measurable, if there exist simple functions sk : T → X,s.t.

sk(t) → f(t) for µ− a.e. t ∈ TIf s(t) =

∑ni=1 χEi(t)ui is a simple function, we define

∫

Ts(t) dµ(t) :=

n∑

i=1

µ(Ei)ui.

We further say the (strongly) measurable function f : T → X is summable, if thereexists a sequence sk∞k=1 of simple functions, s.t.

∫

T‖sk(t) − f(t)‖X dµ(t) −→ 0 as k → ∞.

In that case, we define the Bochner integral

∫

Tf(t)dµ(t) = lim

k→∞

∫

Tsk(t) dµ(t)

of X-valued functions f over T . The following result will be used later on when we proveconvergence rates of our algorithms.

Theorem 1.3.1 (Bochner). A strongly measurable function f : T → X is summable ifand only if t→ ‖f(t)‖X is summable. In this case it holds

∥∥∥∥

∫

Tf(t)dµ(t)

∥∥∥∥X

≤∫

T‖f(t)‖Xdµ(t).

3

1 Preliminaries

For a proof we refer to [71, Ch. V.5].

Given 1 ≤ p ≤ ∞ we now define the Bochner spaces

Lpµ(T ;X) =

f : T −→ X :

∫

T‖f(t)‖pX dµ(t) <∞

(1.1)

with norm ‖f‖Lpµ(T ;X) :=

(∫

T ‖f‖pX dµ(t))1/p

in the case of 1 ≤ p <∞ and

L∞µ (T ;X) =

f : T −→ X : ess supt∈T

‖f(t)‖X <∞

(1.2)

with norm ‖f‖L∞µ (T ;X) := ess supt∈T ‖f(t)‖X if p = ∞, where the essential supremum is

taken w.r.t. µ. In the case where p = 2 and X is a separable Hilbert space, we havethat

L2µ(T ;X) ≃ L2

µ(T ) ⊗X, (1.3)

where ⊗ denotes the tensor product between Hilbert spaces, defined in Section 1.2. Inparticular, L2

µ(T ;X) is itself again a Hilbert space, see e.g. [36, Ch. 1].

4

2 Uncertainty quantification

The computational quantitative characterization and reduction of uncertainties in aphysical model, called uncertainty quantification (UQ), is a key ingredient in develop-ing numerical schemes to describe the random behavior of the process being modeledmathematically up to a given accuracy. As explained in the introduction, we distin-guish model uncertainties, i.e. incomplete knowledge about the physical process, andparameter uncertainties, i.e. incomplete knowledge about the system parameters. Math-ematical methods for quantifying model uncertainties include e.g. fuzzy set theory, e.g.[72, 74, 40], and possibility theory, e.g. [73, 19]. Here, we stay within Kolmogorov’smathematical formalism of probability. Parameter uncertainties, which are the focus ofthis thesis, are then very often modeled as random fields [3, 61]. One straightforward wayto quantify these are Monte Carlo (MC) and Quasi Monte Carlo (QMC) methods [21].However, spectral methods, such as polynomial chaos (PC) [65] and Karhunen-Loeve(KL) [38] expansions of random fields have recently gained more and more attraction.In the following, we will briefly review the theory of KL- and PC-expansions and provideresults, which will be used in the subsequent chapters.

2.1 Preliminaries

For the mathematical modeling of uncertainty in the model parameters, denote by(Ω,Σ, P ) a complete probability space with Ω denoting the outcomes, Σ the sigma-algebra of possible events and P a probability measure. Furthermore, let D ⊂ R

d ford = 1, 2, . . . be a bounded, physical domain. In the following denote by

a(ω,x) : Ω ×D −→ R

a random field, i.e. a jointly measurable function from D × Ω to R w.r.t. the Borelsigma-algebra in D and R and Σ in probability space.

Assumption 2.1.1. For the random field a, the mean field

Ea(x) =

∫

Ωa(ω,x) dP (ω) (2.1)

5


and covariance

Va(x,x′) =

∫

Ω(a(ω,x) − Ea(x))(a(ω,x′) − Ea(x

′)) dP (ω) (2.2)

are known.

An equivalent assumption would be that the mean field Ea and the 2-point-correlation

Ca(x,x′) =

∫

Ωa(ω,x)a(ω,x′) dP (ω) (2.3)

are known, sinceVa(x,x

′) = Ca(x,x′) − Ea(x)Ea(x

′).

Note that for the mean and covariance to exist, we must require that a has finite secondmoments, i.e. a ∈ L2

P (Ω;L2(D)). We will now define what we will later refer to as anadmissible covariance function.

Definition 2.1.2. A covariance function Va(x,x′) ∈ L2(D ×D), given by (2.2), is said

to be admissible, if it is symmetric and positive definite in the sense that for any n ∈ N

0 ≤n∑

k=1

n∑

j=1

ckVa(xk, xj)cj ∀xk, xj ∈ D, ck, cj ∈ C. (2.4)

This property will be used later on to ensure that the Carleman operator associatedto Va defined in (2.5) ahead has real positive eigenvalues. Many covariance functionsappearing in practice, such as Gaussian or exponential ones, are admissible. For anintroduction into the theory of positive definite functions it is referred to [49, 50] wherealso many examples of widely used covariances are given.

2.2 Karhunen-Loeve expansion

The Karhunen-Loeve expansion can be understood as a Fourier representation of a ran-dom field, in which the spatial and stochastic parts are naturally separated into aninfinite number of random variables αi(ω) and functions Ni : D ⊂ R

d −→ R (for exam-ple finite element shape functions), in the sense of

a(ω,x) =∑

m≥0

Nm(x)αm(ω).

Obviously, there exist infinitely many such representations, see e.g. [34]. The Karhunen-Loeve expansion, however, turns out to be an optimal approximation of the random fielda in the mean square sense if truncated after the first, say M , terms, see e.g. [27, 38]and also Section 2.2.2 ahead. It is therefore our preferred choice.

6


2.2.1 Definition of the KL-expansion

The covariance operator of a random field a ∈ L2P (Ω;L2(D)) is

Va : L2(D) −→ L2(D), (Vau)(x) :=

∫

DVa(x,x

′)u(x′) dx′. (2.5)

Given an admissible covariance function Va(x,x′) in the sense of Definition 2.1.2, the

associated covariance operator Va is a symmetric, non-negative and compact integraloperator. It therefore has a countable sequence of eigenpairs (λm, ϕm)m≥1

Vaϕm = λmϕm, m = 1, 2, ... (2.6)

where the sequence of real and positive KL-eigenvalues λm is enumerated with decreasingmagnitude and is either finite or tends to zero as m → ∞, i.e. λ1 ≥ λ2 ≥ . . . ≥ 0 (withmultiplicities counted). The KL-eigenfunctions ϕm(x) are assumed to be scaled, s.t.

∫

Dϕm(x)ϕn(x)dx = δmn, m, n = 1, 2, ... (2.7)

i.e. they are L2(D)-orthonormal. For special covariances, e.g. exponential or triangularones, and on simple domains, the eigenpairs are known explicitly [27]. For more generalcovariances or more complicated domains D, the computation of the KL eigenpairs (2.6)can e.g. efficiently be carried out by means of a (generalized) Fast Multipole Method,see [56].

Definition 2.2.1 (Karhunen-Loeve expansion). The Karhunen-Loeve (KL) expansionof a random field a(x, ω) with finite mean (2.1) and covariance (2.3), which is admissiblein the sense of Definition 2.1.2, is given by

a(ω,x) = Ea(x) +∑

m≥1

√

λmϕm(x)Ym(ω). (2.8)

The family of random variables (Ym)m≥1 is determined by

Ym(ω) =1√λm

∫

D(a(ω,x) − Ea(x))ϕm(x) dx. (2.9)

Once the ϕm(x) are available, probability densities ρm of the Ym in (2.9) may be esti-mated from sample input fields a(ω,x) — the KL-eigenfunctions are, in effect, a tool toprocess such sampling data via (2.9).

One immediately verifies that

E[Ym] = 0, E[YmYn] = δmn, ∀ m,n ≥ 1, (2.10)

7


i.e. the Ym’s are centered with unit variance and pairwise uncorrelated. In the case wherethe random variables are Gaussian, (2.10) implies that the Ym’s are independent.

In order to be able to numerically handle the KL-expansion, the series is truncated afterM terms, and we define

aM (ω,x) := Ea(x) +M∑

m=1

√

λmϕm(x)Ym(ω). (2.11)

2.2.2 KL eigenvalue decay

The (truncated) KL-series (2.11) converges in L2P (Ω;L2(D)) to a as shown in [38] and is

an optimal approximation of the random field a in the mean square sense, that is for anyother linear combination aM of M functions, the error ||a− aM ||L2

P (Ω;L2(D)) is not smaller

than for the KL-expansion, see e.g. [27]. Due to (2.7) and (2.10), the L2P (Ω;L2(D)) error

of truncating the KL expansion is given by

||a− aM ||L2P (Ω;L2(D)) = E

[∫

D(a(ω,x) − aM (ω,x))2 dx

]

=∑

m>M

λm. (2.12)

If the sequence (√λm‖ϕm‖L∞(D))m≥1 is summable and (Ym)m≥1 is uniformly bounded

in L∞(Ω), then the convergence can easily be shown to be even uniform in L2(Ω ×D).Precisely, we have the pointwise estimate

||a− aM ||L∞(Ω×D) ≤ C∑

m>M

‖ϕm‖L∞(D)

√

λm, (2.13)

where the constant C > 0 is independent of M . Estimates on the KL eigenvalue decayand L∞(D) bounds on the eigenfunctions are therefore crucial to obtain a good a-prioricontrol over the error of truncating the KL-expansion after M terms. These boundsare in turn highly dependent on the regularity of the covariance (2.2). First results onthe eigenvalue decay of integral operators with positive definite kernels in D ⊂ R

1 havebeen proved by J.B. Reade [46, 47, 37, 35]. The results presented in the following arean extension, and sometimes also sharpened version, of those results to the case d > 1.Parts of it can also be found in [56].

Definition 2.2.2. A covariance function Va : D × D → R is said to be piecewiseanalytic/smooth/Hp,q on D ×D (p, q ∈ [0,∞]), if there exists a partition D = DjJj=1

of D into a finite sequence of simplices Dj and a finite family G = GjJj=1 of open sets

in Rd such that

D =

J⋃

j=1

Dj , Dj ⊂ Gj ∀1 ≤ j ≤ J,

8


and such that Va|Dj×Dj′ has an extension to Gj × Gj′ which is analytic in Gj ×Gj′/is

smooth in Gj ×Gj′/is in Hp,q(Gj ×Gj′) := (Hp(Gj)⊗L2(D)) ∩ (L2(D)⊗Hq(Gj′)) forany pair (j, j′).

Proposition 2.2.3. Let Va ∈ L2(D × D) be an admissible covariance in the sense ofDefinition (2.1.2).

1. If Va is piecewise Ht,t(D×D) in the sense of Definition 2.2.2, then for the eigen-values of (2.6) it holds

λm ≤ Cm−t/d, m ≥ 1, (2.14)

with a constant C > 0 independent of m.

2. If Va is piecewise analytic in the sense of Definition 2.2.2, then the eigenvalues of(2.6) admit the bound

λm ≤ Ce−c1m1/d, m ≥ 1, (2.15)

with C > 0 independent of m.

3. If Va is a Gaussian covariance, i.e.

Va(x,x′) = σ2e

− |x−x′|2γ2diam(D)2 , (x,x′) ∈ D ×D, (2.16)

with σ, γ > 0 referring to the standard deviation and correlation length, respectively,then the eigenvalues satisfy

0 < λm ≤ Cσ2

γ2

(1/γ)m1/d

Γ(12m

1/d), ∀ m ≥ 1, (2.17)

where Γ(·) denotes the Gamma function [1, Ch. 6] and C > 0 is independent ofM .

The case of Gaussian covariances (2.16) is particularly interesting, since, as a consequenceof the central limit theorem, it is considered in many applications. The fact that thisfunction can be extended to an entire function in C

d leads to the faster than exponentialeigenvalue decay (2.17).

Corollary 2.2.4. If Va is piecewise smooth in the sense of Definition 2.2.2, then, dueto (2.14), the eigenvalues decay at least as

λm ≤ Cm−s, m ≥ 1 (2.18)

for any given s > 0, with C > 0 independent of m.

9


Proof of Proposition 2.2.3. The first two assertions (2.14) and (2.15) have already beenproved in [56, Prop. 2.18 and 2.21]. Although also the Gaussian eigenvalue decay (2.17)has already been stated in [56], a proof is missing there and therefore given here forcompleteness. Let H be a Hilbert space on D and denote by B(H) the set of boundedlinear operators in H. In [56, Lemma 2.16] it has been shown that for a symmetric,non-negative and compact operator C ∈ B(H) with eigenpair sequence (λm, ϕm)m≥1, itholds

λm+1 ≤ ‖C − Cm‖B(H),

where Cm ∈ B(H) denotes an operator of rank at most m. We define

C = σ2e|x−y|2

γ2diam(D)2 = σ2∞∑

k=0

1

k!

1

(−γdiam(D))2k|x − y|2k

and

Cm = σ2e|x−y|2

γ2diam(D)2 = σ2m∑

k=0

1

k!

1

(−γdiam(D))2k|x − y|2k. (2.19)

Denote by C and Cm the Carleman operators which are, by means of (2.5), associated toC and Cm, respectively. It follows that

‖C − Cm‖B(H) . ‖C − Cm‖L∞(D×D)

. σ2∞∑

k=m+1

1

k!

1

(γdiam(D))2kdiam(D)2k

.1

(m+ 1)!

1

γ2(m+1)e1/γ

2

.1

(m+ 1)!

1

γ2(m+1).

The rank of the Carleman operator defined by (2.19) can be estimated as

rank(Cm) ≤ dim spanxα11 · · · xαd

d : α1 + · · · + αd ≤ 2m =

(2m+ d

d

)

.

Hence, for a given m ∈ N we choose m such that

(2m+ d

d

)

≤ m ≤(

2(m+ 1) + d

d

)

,

from which it follows that m ∼ 12m

1/d as m→ ∞. This completes the proof of Proposi-tion 2.2.3.

10


2.2.3 KL eigenfunction bounds

The first result states that the regularity of the covariance Va implies the correspondingregularity of the KL eigenfunctions and can be found in [56, Prop. 2.23].

Proposition 2.2.5. Assume that the covariance Va is piecewise analytic/ smooth/Hp,q

w.r.t. D in the sense of Definition 2.2.2. Then the Karhunen-Loeve eigenfunctions areanalytic/smooth/Hp in every Dj ∈ D.

As motivated at the beginning of the previous paragraph, L∞(D) bounds on the KLeigenfunctions ϕm are crucial to obtain a pointwise control over the error stemming fromtruncating the KL expansion. We will first discuss the case of covariances Va with finiteSobolev regularity and then state the corresponding result for smooth/analytic/Gaussiancovariances as a corollary.

Proposition 2.2.6. Assume that the covariance Va is piecewise Ht,t(D ×D) w.r.t. Dwith t > d/2. Then ϕm ∈ Ht(Dj) for every Dj ∈ D and for every ε ∈ (0, t − d/2] thereexists a constant C > 0, depending on ε, d but not on m, such that

‖ϕm‖L∞(D) ≤ Cλ−(d/2+ε)/tm , m ≥ 1 (2.20)

Corollary 2.2.7. Assume that the covariance Va is piecewise smooth w.r.t. D. Then,for every s > 0 there exists a constant C > 0 depending on s, d, but not on m, such that

‖ϕm‖L∞(D) ≤ Cλ−sm , m ≥ 1 (2.21)

Proof of Proposition 2.2.6. The Ht(D) regularity of the eigenfunctions is due to Propo-sition 2.2.5. The Sobolev embedding theorem [2] for all t∗ ∈ (d/2, t] then yields

‖ϕm‖C0(D) ≤ C‖ϕm‖Ht∗ (D) ∀m ≥ 1, (2.22)

with a positive constant C depending on t∗, d. It remains to bound the norm on theright hand side of the inequality. It is well-known that fractional order Sobolev spacesHt∗(D) can be obtained by complex interpolation [60] between L2(D) and Ht(D) forany t > t∗ and we therefore have the Riesz-Thorin-type inequality [60, §1.9.3]

‖u‖Ht∗ (D) ≤ ‖u‖t∗/tHt(D)‖u‖

1−t∗/tL2(D)

for any function u ∈ Ht(D). Choosing now u = ϕm we have, due to the L2-normalizationthat

‖ϕm‖Ht∗(D) ≤ ‖ϕm‖t∗/tHt(D). (2.23)

11


From the eigenvalue equation

ϕm(x) =1

λm

∫

DCa(x,x

′)ϕm(x′) dx′,

it follows by differentiating and using the Cauchy-Schwartz inequality

‖∂αϕm‖L∞(D) ≤ C(α)λ−1m . (2.24)

Combining now (2.24), (2.23) and (2.22) yields

‖ϕm‖L∞(D) ≤ Cλ−t∗/t

m ,

which implies (2.20).

Combining now Proposition 2.2.3 with Proposition 2.2.6, allows us to get a pointwisecontrol of the error stemming from truncating the KL-expansion (2.13). Precisely,

Corollary 2.2.8. Assume that the covariance Va is piecewise Ht,t(D×D) w.r.t. D witht > d/2.

√

λm||ϕm||L∞(D) ≤ Cm−s (2.25)

where

s =1

2d(t− d− 2ε) (2.26)

for an arbitrary ε ∈ (0, t− d/2] and where the constant C in (2.25) is independent of m.

2.2.4 The lognormal case

Lognormal random fields are random fields of the form a(ω,x) = eg(ω,x), with g beingGaussian. Therefore, its logarithm can be expanded in a Karhunen-Loeve series asbefore:

ln a(ω,x) = Eloga (x) +

∑

m≥1

√

λmϕm(x)Ym(ω) (2.27)

with Ym ∼ N (0, 1). Similarly to (2.11), we then define the truncated exponential KLseries as

aM (ω,x) := eEloga (x)+

PMm=1

√λmϕm(x)Ym(ω). (2.28)

Since a− aM = aM ( aaM

− 1), we obtain for the L2P truncation error

‖a− aM‖L2P (Ω;L2(D)) = ‖aM‖L2

P (Ω;L2(D))‖eP

m>M

√λmϕmYm − 1‖L2

P (Ω;L2(D)).

Since the random variables Ym are Gaussian, they can take values on the whole real axis.Therefore, to obtain a pointwise estimate on the truncation error, we have to control

12

2.3 Polynomial Chaos

their behavior towards −∞ and ∞. To this end, we introduce the Gaussian measure,denoted by

Gσ :=∞⊗

m=1

Gm,σm , (2.29)

where σ = (σ1, σ2, . . .) and Gm,σm is univariate Gaussian measure with variance σ2m and

zero mean imposed on Ym, see Section 1.1 for the definition of product measures.

Remark 2.2.9. It can be proved that the countable product measure in (2.29) defines aGaussian measure on R

∞ = ×∞m=1 R and moreover that this is essentially the unique

Gaussian measure with prescribed mean and variance, see [45, Sec. 1.5].

Denote by ℓexp1 the space of all sequences σ ∈ R

N, with σm = esm where smm∈N is asummable sequence of positive real numbers. If σ ∈ ℓexp

1 , we have

L2Gσ

(Ω) ⊂ L2G1

(Ω)

with ‖u‖L2G1

(Ω) ≤ ‖u‖L2Gσ

(Ω). Then it can be shown, using a result from [33] that Gσis equivalent to the standard Gaussian measure G1 on R

∞ with zero mean and unitvariance, see [28, Prop. 2.11]. We denote by

dGσdG1

=

∞∏

m=1

1

σme−

12

P∞m=1(σ−2

m −1)y2m (2.30)

the respective Radon-Nikodym derivative. This allows us to define weighted L∞ spacesas

L∞Gσ

(Ω ×D) := f : Ω ×D −→ R : ess supΩ×D

∣∣∣∣∣f

(dGσdG1

)−1∣∣∣∣∣<∞ (2.31)

with ‖f‖L∞Gσ

:= ess supΩ×D

∣∣∣∣f(dGσdG1

)−1∣∣∣∣. where the essential supremum is taken w.r.t.

Gσ⊗λ, where λ denotes the Lebesgue measure in Rd. The pointwise KL truncation error

can then be written as

‖a− aM‖L∞Gσ

(Ω×D) = ‖aM‖L∞Gσ

(Ω×D)‖eP

m>M

√λmϕmYm − 1‖L∞

Gσ(Ω×D). (2.32)


We saw in the previous section that a prerequisite to write a square-summable randomfield into a Karhunen-Loeve expansion, is the knowledge of its mean and covariance.This can clearly be assumed for the stochastic input parameters but not for the response

13


statistics of the physical system. Hence, an alternative expansion is needed to describethe solution’s random behavior. One such alternative is polynomial chaos (PC), firstintroduced by N. Wiener [65] for Gaussian random processes. The idea of PC is to takepolynomial combinations of known random variables (e.g. the KL random variables ofthe inputs) as a basis for the expansion. Its deterministic coefficients have then to befound by minimizing some norm of the error resulting from a finite representation, e.g.by Galerkin projection.

2.3.1 Hermite polynomials of Gaussian random variables

Assume Ym(ω)∞m=1 is a set of Gaussian random variables satisfying (2.10) and P is thestandard Gaussian measure G1. Denote by N

Nc the space of all sequences ν = (ν1, ν2, . . .)

of non-negative natural numbers with compact support, i.e. where the set supp(ν) =m ∈ N : νm 6= 0 is finite. For ν ∈ N

Nc , define by

Hν(Y) = e12Y⊤Y(−1)ν

∂|ν|

∂Y ν11 ∂Y ν2

2 . . .e−

12Y⊤Y

the multivariate Hermite polynomial, where Y = (Y1, Y2, . . .). Note that this definitionis meaningful, since only a finite number of νm’s are different from zero. It is proved in[14, Thm. 9.1.5] that the system Hνν∈NN

cis orthogonal and complete in L2

P (Ω), i.e.

every function f ∈ L2P (Ω) can be written as a sum

f =∑

ν∈NNc

fνHν , fν ∈ R

converging in L2P (Ω).

2.3.2 Wiener chaos expansions

Denote by πp the space spanned by multivariate Hermite polynomials in Ym(ω)∞m=1

with total degree at most p, i.e.

πp =

f =∑

ν∈NNc

|ν|≤p

fνHν : fν ∈ R

.

Furthermore, let πp then represent the set of all polynomials in πp, L2P -orthogonal to

πp−1, called polynomial chaos of order p. The space πp = spanπp ⊂ L2P (Ω) is called

14


the p-th homogeneous chaos and is given by

πp =

f =∑

ν∈NNc

|ν|=p

fνHν : fν ∈ R

.

Consequently, πp is often also called Hermite chaos. It follows, [14, Thm. 9.1.7] that

L2P (Ω) =

⊕

p≥0

πp,

which is called the Wiener-Ito decomposition of L2P (Ω). Therefore, any square integrable

Gaussian random variable f ∈ L2P (Ω), can be represented as

f =∑

n≥0

∑

ν∈NNc

|ν|=n

fνHν , fν ∈ R

with convergence in L2P (Ω). Truncating the expansion, such that only polynomials with

total order less than p are taken into account, gives rise to the so-called polynomial chaosexpansion of order p of f , i.e.

f =

p∑

n=0

∑

ν∈NNc

|ν|=n

fνHν , fν ∈ R.

2.3.3 Generalized polynomial chaos

Although N. Wiener originally proposed the polynomial chaos in terms of Gaussianrandom variables, the same idea also carries over to non-Gaussian ones which is thenoften referred to generalized polynomial chaos (gPC), introduced in [69, 51]. Given aset of i.i.d. random variables Ym(ω)∞m=1 with product measure P = ⊗∞

m=1Pm, thespaces πp, πp are then constructed by means of the respective (multivariate) orthogonalpolynomials, e.g. Legendre polynomials

1

2p ∂p

∂Yi1···∂Yip

∏pj=1 Yij

∂p

∂Yi1 · · · ∂Yip

p∏

j=1

(Y 2ij − 1)

in case of uniform random variables Ym, leading to so-called Legendre chaos, and like-wise for other probability distributions. In the course of the discretization discussed inChapter 7, we shall in particular use Legendre and Hermite expansions. We emphasize,however, that also for other probability densities, other polynomial systems can be usedin exactly the same fashion, see e.g. [51] for more details.

15


16

3 Problem formulation

In the present chapter, we will introduce the class of elliptic problems under considerationand the notation used throughout the remainder of the thesis. We follow [4] and sharpenand generalize some of the results therein. The concepts of Bochner spaces and tensorproducts of Hilbert spaces, introduced in Chapter 1, appear as natural function spaces inthe given framework. We will then impose several assumptions on the problem, on whichthe algorithms developed in Chapters 4–7 are relying. The chapter is then concludedby providing an example of a problem satisfying all of the assumptions which will then,in a slightly simplified form, also serve as a model problem for the following chapters toillustrate the numerical methods developed in this work.

3.1 Problem setting and notation

The problem under consideration is a stochastic elliptic boundary value problem of theform

L(u) = f in D,B(u) = g on ∂D

(3.1)

where L is an elliptic differential operator, depending on one or more random parameters,e.g. a random diffusion coefficient a(ω,x), and B is a (deterministic) boundary operator.The forcing term f(ω,x) can also assumed to be random, whereas, on the other hand,we do not consider randomness of the Dirichlet data g or the domain D, as e.g. done in[29, 30, 70] and references therein.

In the following, we will often interpret the random coefficients a, f and the solutionu not as real-valued, measurable functions on Ω × D, but as random variables, takingvalues in a suitable Banach space W(D) of solutions, if we attempt to solve (3.1) forgiven realizations a(ω, ·) and f(ω, ·) of the stochastic inputs. Hence, we introduce theBochner spaces

L2P (Ω;W(D)) and L∞

P (Ω;W(D)), (3.2)

as defined in Section 1.3

17


Note that if W(D) is a separable Hilbert space, then so is L2P (Ω;W(D)) and, by virtue

of (1.3), there exists an isomorphism, s.t.

L2P (Ω;W(D)) ≃ L2

P (Ω) ⊗W(D) (3.3)

where ⊗ denotes the tensor product between separable Hilbert spaces, as introduced inSection 1.2.

3.2 Assumptions on the problem

The numerical methods presented in this work rely on a number of assumptions imposedon the problem (3.1), which will be introduced next. Some of these assumptions aremandatory while others may be relaxed. At some points below, we will give remarks onpossible generalizations and also refer to other works where these have been addressed.

3.2.1 Well-posedness

The first assumption ensures that the problem is well posed in the sense of uniquesolvability.

Assumption 3.2.1. The random coefficients a and the forcing term f are such that theexistence and uniqueness of a solution u(ω,x) ∈ L2

P (Ω;W(D)) to (3.1) is guaranteed.

3.2.2 Finite-dimensional noise

The next assumption restricts the model problem (3.1) to ones where the stochasticbehavior of the input parameters can be described by a finite-dimensional random vector(Y1, . . . , YM ).

Assumption 3.2.2. The stochastic parameters a and f depend only on a finite numberM of random variables Ym : Ω → R, i.e.

a(ω,x) = a(Y1, . . . , YM ,x) and f(ω,x) = f(Y1, . . . , YM ,x). (3.4)

This may seem very restricting at a first glance, since random fields are usually onlyproperly described by an infinite number of random variables. However, since our goalis an approximation to the statistical moments of the solution, we only need to describethe random field to a sufficiently high accuracy, see also Section 3.3.2 ahead, which isusually possible by only taking a finite number of random variables into account, as wehave seen in (2.12) and (2.13).

18

3.2 Assumptions on the problem

Remark 3.2.3. Assumption 3.2.2 can be omitted for the Galerkin method presentedin Chapter 7. In fact, the Galerkin method presented there turns out to be dimension-adaptive, meaning that it automatically selects the relevant, finitely many input variablesto achieve the desired target accuracy.

3.2.3 Independence

Assumption 3.2.4.

i) The family (Ym)m≥1 : Ω → R is independent,

ii) with each Ym(ω) is associated a probability space (Ωm,Σm, Pm), m ∈ N, with thefollowing properties:

a) the probability measure Pm admits a probability density function

ρm : Ran(Ym) := Γm −→ [0,∞),

such that dPm(ω) = ρm(ym)dym, m ∈ N, ym ∈ Γm and

b) the sigma algebras Σm are subsets of the Borel sets of the interval Γm, i.e.Σm ⊆ Σ(Γm).

Due to the independence we can interpret the Ym’s as different coordinates in probabilityspace and hence parametrize Ω by the vector

y = (y1, y2, . . .) ∈ Γ := Γ1 × Γ2 × · · ·

rather than (Y1(ω), Y2(ω), . . .). The vector y is then equipped with the product prob-ability density ρ =

∏

m≥1 ρm : Γ → R. In the following we will therefore write a(y,x)instead of a(ω,x) and likewise for u and f . We will then identify

L2ρ(Γ;W(D)) = L2

P (Ω;W(D)) and L∞ρ (Γ;W(D)) = L∞

P (Ω;W(D)). (3.5)

Remark 3.2.5. The random variables Ym will in general not be independent, exceptfor Gaussian random fields (2.16). One possible remedy has been proposed in [4], byintroducing auxiliary probability density functions ρm with ‖ρm/ρm‖L∞(Γ) < ∞, suchthat the random variables Ym are independent with respect to ρ =

∏

m≥1 ρm. Analternative way to treat the case of dependent random variables has been proposed in[57], where an orthonormal system of polynomials are constructed, which then serve asa basis for the PC representation of the response process.

19


3.2.4 Growth at infinity

In the previous paragraph, the parameter domains Γm = Ran(Ym) have been intro-duced. These may either be bounded, e.g. in the case of uniform random variables, orunbounded, which includes in particular the case of Gaussian or exponential randomvariables Ym. In the case of unbounded parameter domains, however, the growth atinfinity has to be controlled. We define the function space

C0χ(Γ;W(D)) :=

v : Γ −→ W(D) : v cont. in y, supy∈Γ

‖χ(y)v(y)‖W(D) <∞

, (3.6)

where χ(y) =∏Mm=1 σm(ym) ≤ 1 and

χm(ym) =

1 if Γm bounded

e−αm|ym| for some αm > 0 if Γm unbounded.(3.7)

The associated norm is then given by ‖v‖C0χ

:= supy∈Γ ‖χ(y)v(y)‖W(D).

Assumption 3.2.6.

i) u ∈ C0χ(Γ;W(D)) and

ii) the joint probability density ρ satisfies

ρ(y) ≤ Cρe−

PMm=1(δmym)2 , ∀y ∈ Γ (3.8)

for a constant Cρ > 0 and δm strictly positive if Γm is unbounded and zero other-wise.

Remark 3.2.7. The decay parameter χ (3.7) and the density ρ (3.8) are chosen in sucha way that

C0χ(Γ;W(D)) ⊂ L2

ρ(Γ;W(D)).

Therefore, for the inclusion still to hold, other choices of χ and ρ could be considered aswell. Relaxing the parameter χ would then impose stronger assumptions on ρ and viceversa.

3.2.5 Stochastic regularity

Finally we need to make some regularity assumption on the stochastic behavior of thesolution. To simplify the notation we write

y∗m := (y1, . . . , ym−1, ym+1, . . .) ∈ Γ∗

m, where, Γ∗m :=

∏

j 6=mΓj .

20

3.3 Model problem

Assumption 3.2.8. For each m ∈ 1, . . . ,M there exists τm > 0, such that the solutionu(ym,y

∗m,x), as a function of ym ∈ Γm, admits an analytic extension to the closed region

of the complex plane

Σ(Γm, τm) := z ∈ C : dist(z,Γm) ≤ τm. (3.9)

3.3 Model problem

To conclude this section, we will present one example where the above assumptions holdtrue. This example, in a simplified form, will serve as our model problem throughoutthe rest of the paper.

Given probability spaces (Ωi,Σi, Pi), i = 1, 2, 3 and random fields

a(ω1,x) : Ω1 ×D → R, c(ω2,x) : Ω2 ×D → R, f(ω3,x) : Ω3 ×D → R,

we consider the following stochastic diffusion-reaction problem

−div(a(ω,x)∇u(ω,x)) + c(ω,x)u(ω,x) = f(ω,x) D,u(ω,x)|x∈∂D = 0,

P−a.e. ω ∈ Ω, (3.10)

where (Ω,Σ, P ) :=⊗3

i=1(Ωi,Σi, Pi) denotes the product probability space as definedin Section 1.1. Here, a and c denote the diffusivity and reaction rate, respectively andf a stochastic source term. In accordance with Assumption 3.2.4, we assume that therandom coefficients are independent. We choose W(D) = H1

0 (D) and by multiplyingwith a test function and integrating by parts, we obtain the variational formulation:Find u ∈ L2

P (Ω;H10 (D)) s.t. ∀ v ∈ L2

P (Ω;H10 (D)) it holds

b(u, v) = l(v), (3.11)

with

b(u, v) = E

[∫

Da∇u∇v + cuv dx

]

, l(v) = E

[∫

Dfv dx

]

, (3.12)

where we suppress the dependence of the coefficients and functions on (ω,x) ∈ Ω × Dfor notational convenience.

3.3.1 Well-posedness

Assumption 3.2.1 is satisfied if for the coefficients a, c, f it holds

21


i) a ∈ L2P (Ω;L2(D)) is positive and bounded away from zero almost surely, i.e. there

exists a∗ > 0 such that

P

ω ∈ Ω : a∗ ≤ ess infx∈D

a(ω,x)

= 1. (3.13)

ii) c ∈ L2P (Ω;L2(D)) is non-negative almost surely, i.e.

P

ω ∈ Ω : 0 ≤ ess infx∈D

c(ω,x)

= 1. (3.14)

iii) f is square-integrable with respect to P , i.e.

E[‖f‖2L2(D)] <∞. (3.15)

Indeed, by introducing the Hilbert space

Ha,c := v : Γ −→ H10 (D) : E

[∫

Da|∇v|2 + cv2 dx

]

<∞ (3.16)

with the norm ‖v‖2Ha,c

:= E[∫

D a|∇v|2 + cv2 dx], from an application of the Lax-Milgram

Lemma it follows the existence and uniqueness of a solution u ∈ Ha,c to (3.11). Precisely,consider the bilinear form b(u, v) given in (3.12). It follows immediately

b(u, v) ≤ ‖u‖Ha,c‖v‖Ha,c and b(u, u) ≥ ‖u‖2Ha,c

,

i.e. b(·, ·) is continuous and coercive with continuity and coercivity constant equal toone. It remains to show that l(·) (3.12) is continuous w.r.t. Ha,c:

l(v) = E

[∫

Df(ω,x)v(ω,x) dx

]

≤ E[‖f(ω, ·)‖L2(D)‖v(ω, ·)‖L2(D)

]

≤ CP√a∗

E

[

‖f(ω, ·)‖L2(D)‖√

a(ω, ·)∇v(ω, ·)‖L2(D)

]

≤ CP√a∗

‖f‖L2P (Ω;L2(D))‖v‖Ha,c ,

where CP denotes the Poincare constant, i.e. ‖v‖L2(D) ≤ CP ‖∇v‖L2(D), and the norm‖f‖L2

P (Ω;L2(D)) is finite due to (3.15). Hence, by Lax-Milgram, the existence of a unique

solution u ∈ Ha,c to (3.11) is guaranteed P -almost surely. Furthermore, since

‖u‖L2P (Ω;H1

0 (D)) ≤CP√a∗

‖u‖Ha,c , (3.17)

22

3.3 Model problem

the space Ha,c is continuously embedded in L2P (Ω;H1

0 (D)), i.e.

Ha,c → L2P (Ω;H1

0 (D)) (3.18)

and it follows the existence of a solution u ∈ L2P (Ω;H1

0 (D)) to (3.11).

Remark 3.3.1. As pointed out in [4], it is possible to relax the condition (3.13) above toa lower bound which is itself again a random variable, i.e. there exists a∗(ω) s.t.

Pω ∈ Ω : 0 < a∗(ω) < ess infx∈D

a(ω,x) = 1. (3.19)

This case is of particular interest, since it covers the case of lognormal random coefficientsa (2.27). However, the fact that a∗(ω) can take values arbitrary close to zero, calls forstronger regularity assumptions. In the following, denote by P always the standard M -fold Gaussian measure G1, as constructed in Section 2.2.4. Given σ ∈ ℓexp

1 , consider thefollowing variational problem w.r.t. the measure Gσ: find u ∈ L2

P (Ω;H10 (D)), s.t.

bσ(u, v) = lσ(v), ∀v ∈ L2P (Ω;H1

0 (D)), (3.20)

where bσ and lσ are defined as

bσM (uM , v) = Eσ

[∫

Da∇uM · ∇v + cuv dx

]

and

lσ(v) = Eσ

[∫

Dfv dx

]

where we again suppressed the dependence on (ω,x) for reasons of readability and whereEσ denotes the expectation taken w.r.t. Gσ. Accordingly, define Hσ

a,c as in (3.16) w.r.t.Gσ. The well-posedness and hence the existence of a solution in Hσ

a,c can then be shown inessentially the same way as above, except that we need stronger regularity assumptionson f . We have

lσ(v) = Eσ

[∫

Df(ω,x)v(ω,x) dx

]

≤ Eσ[‖f(ω, ·)‖L2(D)‖v(ω, ·)‖L2(D)

]

≤ CPEσ

[

‖f(ω, ·)‖L2(D)√

a∗(ω)‖√

a(ω, ·)∇v(ω, ·)‖L2(D)

]

and by twice using Holder’s inequality

≤ CP‖f‖1/2p

L2pGσ

(Ω;L2p(D))‖1/√a∗‖1/2q

L2qGσ

(Ω)‖v‖Hσ

a,c, (3.21)

23


where 1/p + 1/q = 1 and p, q ≥ 1. Hence, if 1/√a∗ ∈ L2q

Gσ(Ω) for some q ≥ 1, then

we must require f ∈ L2pGσ

(Ω;L2p(D)) for p = (1 − 1/q)−1. If we assume that a can beexpanded in a lognormal KL expansion (2.27), then a lower bound to a is given by

a∗(ω) := emin Eloga (x)−

P

m≥1

√λmϕm(x)|Ym(ω)|. (3.22)

We then have

‖u‖L2P (Ω;H1

0 (D)) ≤∥∥∥∥

1

a∗

∥∥∥∥L∞Gσ

(Ω×D)

‖u‖Hσa,c, (3.23)

where P denotes the standard Gaussian measure and with L∞Gσ

(Ω × D) as defined in(2.31), which implies

Hσa,c → L2

P (Ω;H10 (D)),

i.e. Hσa,c is continuously embedded in L2

P (Ω;H10 (D)), where → denotes the inclusion

map.

3.3.2 Continuous dependence on input data

A main requirement for the stable numerical approximation of (3.10) is continuous de-pendence of the solution u on the random input data. Moreover, this also justifiesthe approximation of the random fields a, c, f by a truncated KL-expansion. We willabbreviate the notions of the norms as follows:

‖ · ‖LpP (W) = ‖ · ‖Lp

P (Ω;W(D)) and ‖ · ‖L∞ = ‖ · ‖L∞(Ω×D) (3.24)

unless otherwise indicated.

Proposition 3.3.2.

i) Denote by a, c, f and a, c, f two sets of bounded random fields satisfying (3.13)-(3.15) and by u and u the respective unique solution to (3.10). Then it holds

‖u− u‖L2P (H1

0 ) . (‖a− a‖L∞ + ‖c− c‖L∞) ‖f‖L2P (L2) + ‖f − f‖L2

P (L2), (3.25)

where the constant depends only on a∗ and diam(D).

ii) Denote by a, c, f and a, c, f two sets of lognormal random fields (2.27) satisfying(3.19), (3.14) and by u and u the respective unique solution to (3.10). By Gσ denotea Gaussian measure as constructed in Section 2.2.4 and let P be the standardGaussian measure G1. It holds

‖u− u‖L2P (H1

0 ) .

(∥∥∥∥

a− a√a

∥∥∥∥

1/2q

L2qGσ

(L2q)

+

∥∥∥∥

c− c√c

∥∥∥∥

1/2q

L2qGσ

(L2q)

)

‖f‖L2p′Gσ

(L2p′ )

+ ‖f − f‖1/2q

L2qGσ

(L2q),

(3.26)

24

3.3 Model problem

with p′, q > 1 satisfying 1p′ + 1

q < 1 and the constant depending only on a∗, q, p′ anddiam(D).

Proof. i) First, we treat the case of bounded random fields. Denote by b(·, ·) andb(·, ·), the bilinear forms (3.12) w.r.t. a, c and a, c, respectively, and by l(·), l(·)the linear forms (3.12) w.r.t. f and f , respectively. Hence we have the variationalproblem of finding u,u ∈ L2

P (Ω;H10 (D)), respectively, s.t.

b(u, v) = l(v)

b(u, v) = l(v)

for all test functions v ∈ L2P (Ω;H1

0 (D)). Due to (3.13) it holds

‖u− u‖2L2

P (H10 ) ≤

1

a∗b(u− u, u− u)

= b(u, u− u) − b(u, u− u) + b(u, u− u) − b(u, u− u)

= b(u, u− u) − b(u, u− u) + l(u− u) − l(u− u). (3.27)

For the linear functionals we obtain

l(u− u) − l(u− u) =

∫

Ω

∫

D(f − f)(u− u) dx dP (ω)

≤ CP‖f − f‖L2P (L2)‖u− u‖L2

P (H10 ), (3.28)

where CP denotes the Poincare constant. Similarly, for the bilinear forms we obtain

b(u, u−u)−b(u, u−u) ≤ CP (‖a−a‖L∞+‖c−c‖L∞)‖u‖L2P (H1

0 )‖u−u‖L2P (H1

0 ) (3.29)

By observing that

‖u‖L2P (H1

0 ) ≤1

a∗b(u, u) = l(u) ≤ CP‖f‖L2

P (L2)‖u‖L2P (H1

0 ), (3.30)

(3.25) follows from combining (3.27)-(3.30).

ii) For the unbounded case, denote by bσ, bσ the bilinear forms from Remark 3.3.1w.r.t. the lognormal coefficients a, c and a, c, respectively and by l and l the linearforms w.r.t. f and f , respectively. We consider the variational problem of findingu,u ∈ L2

P (Ω;H10 (D)), respectively, s.t.

bσ(u, v) = lσ(v)

bσ(u, v) = lσ(v)

25


for all test functions v ∈ L2P (Ω;H1

0 (D)). Note that due to (3.23) it is sufficient tofind a bound of u− u in the energy norm ‖ · ‖Hσ

a,cas defined in Remark 3.3.1. As

in (3.27) it follows

‖u− u‖2Hσ

a,c= bσ(u, u− u) − bσ(u, u− u) + lσ(u− u) − lσ(u− u). (3.31)

By similar arguments as in (3.21), we obtain

lσ(u− u) − lσ(u− u) ≤ ‖f − f‖1/2p

L2pGσ

(L2p)

∥∥∥∥

1√a∗

∥∥∥∥

1/2q

L2qGσ

(L2q)

‖u− u‖Hσa,c

(3.32)

For the error in the bilinear form we have

bσ(u, u− u) − bσ(u, u− u) =

∫

Ω

∫

D

(a− a√a

)

∇u(√a∇(u− u)) dx dGσ(ω)

+

∫

Ω

∫

D

(c− c√c

)

u(√a(u− u)) dx dGσ(ω)

and by twice using Holder’s inequality

≤ C

∥∥∥∥

a− a√a

∥∥∥∥

1/2q

L2qGσ

(L2q)

‖∇u‖1/2p

L2pGσ

(L2p)‖u− u‖Hσ

a,c

+

∥∥∥∥

c− c√c

∥∥∥∥

1/2q

L2qGσ

(L2q)

‖∇u‖1/2p

L2pGσ

(L2p)‖u− u‖Hσ

a,c(3.33)

It remains to estimate the L2pGσ

(Ω;L2p(D))-norm of ∇u. To this end we use a resultfrom [28, Thm. 3.14], stating that

‖∇u‖1/2p

L2pGσ

(L2p)≤ Ca∗,p,q‖f‖L2p′

Gσ(L2p′ )

, (3.34)

provided that u is a solution to a diffusion equation with lognormal coefficientsand p′ > p. Combining (3.31)-(3.34) completes the proof of (3.26).

3.3.3 Growth at infinity

As imposed by Assumption 3.2.6, in the case of unbounded random variables, we haveto control the behavior of u as the parameters ym tend towards −∞ and ∞. Define thepointwise (or parametric) bilinear form

B(y;u, v) =

∫

Da(y,x)∇u(y,x)∇v(x) dx, y ∈ Γ, u, v ∈ H1

0 (D) (3.35)

26

3.3 Model problem

and linear form

F (y; v) =

∫

Df(y,x)v(x) dx, y ∈ Γ, v ∈ H1

0 (D). (3.36)

Then the solution to the parametric deterministic problem of finding u(y) ∈ H10 (D),

such thatB(y;u, v) = F (y; v),

satisfies the pointwise estimate

‖u(y)‖H10 (D) ≤

CPa∗(y)

‖f(y)‖L2(D), y ∈ Γ.

Hence, if we assume f ∈ C0χ(Γ;L2(D)), then, by (3.22), we clearly have u ∈ C0

χ(Γ;L2(D))

with decay parameter αm = αm +√λm‖ϕ‖L∞(D) (3.7).

3.3.4 Stochastic regularity

The following proposition shows that under certain assumptions on the growth of thederivatives of a, c, f , the coordinate-wise analyticity as requested in Assumption 3.2.8 isensured.

Proposition 3.3.3. Assume that for every y = (ym, y∗m) there exists γm <∞ satisfying

∥∥∥∥∥

∂kyma(y)

a(y)

∥∥∥∥∥L∞(D)

≤ γkmk!,‖∂kym

c(y)‖L∞(D)

‖c(y)‖L∞(D)≤ γkmk!,

‖∂kymf(y)‖L2(D)

1 + ‖f‖L2(D)≤ γkmk!

(3.37)Assume further that there exists p, q ≥ 1 with 1/p + 1/q = 1, such that there holdsc ∈ C0

χ1/p(Γ;L∞(D)) and f ∈ C0χ1/q(Γ;L2(D)). Then the solution u(ym, y

∗m, x) to (3.10),

as a function of ym, u : Γm → C0χ∗

m(Γ∗m;H1

0 (D)), admits an analytic extension into the

region Σ(Γm, τm) ⊂ C with 0 < τm < 12γm

, uniformly for all y∗m ∈ Γ∗m.

Proof. We consider the problem (3.10) which can be stated in variational form as: Findu ∈ L2

ρ(Γ;H10 (D)) s.t. ∀ v ∈ H1

0 (D) it holds

B(y;u, v) = F (y; v) (3.38)

with

B(y;u, v) =

∫

Da∇u∇v + cuv dx F (y; v) =

∫

Dfv dx, (3.39)

where we suppress the dependence of the coefficients and functions on (y,x) ∈ Γ×D toshorten the formulae below.

27


We note here that Lemma 3.3.3 is a slight generalization of the result given in [4, Lemma3.2] and hence the following proof follows closely the one given there.

Differentiating (3.38) k times with respect to ym and using Leibniz’s rule we obtain

k∑

l=0

(k

l

)∫

D

(

∂lyma∇∂k−lym

u∇v + ∂lymc∂k−lym

uv)

dx =

∫

D∂kym

fv dx.

For the remaining part of the proof we denote by ‖ · ‖L2 and ‖ · ‖L∞ the norms of L2(D)and L∞(D), respectively. Setting v = ∂kym

u we obtain

∥∥∥√a∇∂kym

u∥∥∥

2

L2≤

k∑

l=1

(k

l

)∥∥∥∥∥

∂lyma

a

∥∥∥∥∥L∞

∥∥∥√a∇∂k−lym

u∥∥∥L2


u∥∥∥L2

+

k∑

l=1

(k

l

)∥∥∥∂lym

c∥∥∥L∞

∥∥∥∂k−lym

u∥∥∥L2

∥∥∥∂kym

u∥∥∥L2

+∥∥∥∂kym

f∥∥∥L2

∥∥∥∂kym

u∥∥∥L2.

Using the Poincare inequality and dividing both sides by ‖√a∇∂kymu‖L2 gives


u∥∥∥L2

≤k∑

l=1

(k

l

)∥∥∥∥∥

∂lyma

a

∥∥∥∥∥L∞

∥∥∥√a∇∂k−lym

u∥∥∥L2

+k∑

l=1

(k

l

)C2P

a∗

∥∥∥∂lym

c∥∥∥L∞

∥∥∥∂k−lym

u∥∥∥L2

+CP√a∗

∥∥∥∂kym

f∥∥∥L2.

Using now the assumptions (3.37) and setting Rk(y) = ‖√a∇∂kymu(y, ·)‖L2/k!, we obtain

the following pointwise bound on Rk

Rk(y) ≤k∑

l=1

γlmRk−l(y) +C2P

a∗‖c(y, ·)‖L∞γlmRk−l(y) +

CP√a∗

(1 + ‖f(y, ·)‖L2)γkm.

Setting C1(y) := 1 +C2

Pa∗

‖c(y, ·)‖L∞ and C2(y) := CP√a∗

(1 + ‖f(y, ·)‖L2), we deduce by

induction

Rk(y) ≤ C1(y)(2γm)k(R0(y) + C2(y)), ∀y ∈ Γ.

Since by (3.38)

R0(y) = ‖√a(y, ·)∇u(y, ·)‖L2 ≤ CP√a∗

‖f(y, ·)‖L2 ≤ C2(y), ∀y ∈ Γ,

28

3.3 Model problem

we finally obtain

∥∥∇∂kym

u(y, ·)∥∥L2

k!≤ 1√

a∗Rk(y) ≤ 2C1(y)C2(y)√

a∗(2γm)k, ∀y ∈ Γ. (3.40)

Now define the formal power series of u around ym ∈ Γm:

u(zm, y∗m,x) =

∞∑

k=0

∂kymu(ym, y

∗m, x)

k!(zm − ym)k.

Using now (3.40), we obtain

χm(ym)‖u(zm)‖C0χ∗

m(Γ∗

m;H10 ) ≤

∞∑

k=0

‖∂kymu(zm)‖C0

χ∗m

(Γ∗m;H1

0 )

k!χm(ym)|zm − ym|k

≤2‖C1‖C0

χ1/p(Γ,L∞)‖C2‖C0

χ1/q(Γ,L2)

√a∗

×∞∑

k=0

(|zm − ym|2γm)k.

Thus the formal power series converges for all zm ∈ C with dist(zm, ym) ≤ τm < 1/(2γm)and hence by a continuation argument in Σ(Γm, τm) where τm < 1/(2γm). By unique-ness, its limit is equal to u.

The assumptions of Proposition 3.3.3 are satisfied if the random fields can be expandedinto a linear (2.8) or exponential (2.27) Karhunen-Loeve series. In fact, in the linearcase we have that

∥∥∥∥∥

∂kyma

a

∥∥∥∥∥L∞(Γ×D)

≤√

λm‖ϕm‖L∞(D)/a∗ k = 1

0 k > 1,(3.41)

such that we can choose γm =√λm‖ϕm‖L∞(D)/a∗. In the lognormal case we have

∥∥∥∥∥

∂kyma

a

∥∥∥∥∥L∞(Γ×D)

≤(√

λm‖ϕm‖L∞(D)

)k, (3.42)

and we can safely take γm =√λm‖ϕm‖L∞(D).

29


30

4 Sparse tensor discretizations

In this chapter, sparse tensor formulations of the stochastic Galerkin and stochastic col-location methods are introduced and applied to the model problem under consideration.Here, as opposed to other works in the field of stochastic PDEs, the term ‘sparse’ refersto a sparse composition of hierarchic discretizations in both, random and spatial param-eter spaces. Here and throughout the rest of the thesis, the term ‘composition’ refers to asuperposition of tensor products of detail spaces in hierarchic discretizations. The chap-ter aims at providing a general description of the sparse tensor stochastic Galerkin andcollocation method. The specification of the discretization spaces along with a thoroughdiscussion of the resulting sparse stochastic Galerkin and collocation algorithms is thenprovided in Chapters 5–7. We do emphasize, however, that also other discretizations asthe ones proposed there could be considered instead, e.g. hp-discretization in space andANOVA-based discretizations in random parameter space [59, 9].

4.1 Model problem

To simplify the presentation in the present and following chapters, we will restrict ourdiscussion to the stochastic diffusion problem

−div(a(ω,x)∇u(ω,x)) = f(x) in D,u(ω,x)|x∈∂D = 0,

P − a.e. ω ∈ Ω (4.1)

for P−a.e. ω ∈ Ω, where the diffusion coefficient a and the solution u are random fieldsin the physical domain D, with a satisfying (3.13). The source term f(x) ∈ L2(D) isassumed to be deterministic. We further assume that the random field can be expandedin either a bounded linear or exponential Karhunen-Loeve series, where either of themis truncated after M terms, hence, according to (2.11):

aM (ω,x) = Ea(x) +M∑

m=1

ψm(x)Ym(ω), ψm(x) :=√

λmϕm(x), (4.2)

or

aM (ω,x) = eEa(x)+PM

m=1 ψm(x)Ym(ω), ψm(x) :=√

λmϕm(x), (4.3)

31


where we assume that ‖ψm‖L∞(D)m∈N is summable.

Remark 4.1.1. It follows directly from Corollary 2.2.8 that the summability of the ψm’sin the L∞(D)-norm is guaranteed, if we e.g. assume that for the covariance Va (2.2),associated with the random field a or its logarithm, respectively, it holds that Va ∈Ht,t(D ×D) for t > 3d.

With either of the finite expansions (4.2), (4.3) in hand, we replace in (4.1) the randominput a by its M -term truncated KL expansion aM and obtain the truncated problem

−div(aM (ω,x)∇uM (ω,x)) = f(x) in D,uM (ω,x)|x∈∂D = 0,

P − a.e. ω ∈ Ω (4.4)

where uM denotes the solution to this truncated problem. The variational formulationof (4.4) with a linear coefficient (4.2) is then given by: Find uM ∈ L2

ρ(Γ;H10 (D)), s.t.

bM (uM , v) = l(v), ∀ v ∈ L2ρ(Γ;H1

0 (D)), (4.5)

with

bM (uM , v) = E

[∫

DaM (y,x)∇uM (y,x) · ∇v(y,x) dx

]

=

∫

Γ

∫

DaM (y,x)∇uM (y,x) · ∇v(y,x)ρ(y) dxdy

(4.6)

and

l(v) = E

[∫

Df(x)v(y,x) dx

]

=

∫

Γ

∫

Df(x)v(y,x)ρ(y) dxdy. (4.7)

The unique solvability of (4.5), uniformly in the number M of terms retained in thetruncated Karhunen-Loeve expansion, follows from (3.13) and (2.13). More precisely,there exists M0 > 0 and γ > 0 (depending on a∗ and

∑

m≥1 ‖ψm‖L∞(D),but not on M),such that

∀M ≥M0 ∀v ∈ L2ρ(Γ;H1

0 (D)) : bM (v, v) ≥ γ‖v‖2L2

ρ(Γ;H10 (D)), (4.8)

∀v,w ∈ L2ρ(Γ;H1

0 (D)) : |bM (v,w)| ≤ γ−1‖v‖L2ρ(Γ;H1

0 (D))‖w‖L2ρ(Γ;H1

0 (D)) (4.9)

and by using Lax-Milgram, the existence of a unique solution to (4.5) follows. Note thatdue to the results established in Section 3.3, (4.4) satisfies all of the Assumptions 3.2.1- 3.2.8. According to Proposition 3.3.2, we obtain for the error between the originalsolution u to (4.1) and the truncated solution uM to (4.4)

‖u− uM‖L2ρ(Γ;H1

0 (D)) ≤ C‖a− aM‖L∞(Γ×D)‖f‖L2ρ(Γ;L2(D)).

32

4.2 Stochastic Galerkin formulation

In the case where aM is given by a truncated exponential KL series (4.3), we consider,as in Remark 3.3.1, the variational problem

bσM (uM , v) = lσ(v), (4.10)

where bσ and lσ are defined as in (4.6), (4.7) but with the expectation taken w.r.t. theGaussian measure Gσ, i.e.

bσM (uM , v) =

∫

Γ

∫

DaM (y,x)∇uM (y,x) · ∇v(y,x)ρσ(y) dxdy (4.11)

and

lσ(v) =

∫

Γ

∫

Df(x)v(y,x)ρ(y)ρσ (y) dxdy, (4.12)

where ρσ denotes the density of Gσ w.r.t. the Lebesgue measure on Γ. If σ = 1, we writeρ = ρ1. Given σ, σ ∈ ℓexp

1 with σ > σ component-wise, we obtain

∀M ≥M0 ∀v ∈ L2ρ(Γ;H1

0 (D)) : |bσM (v, v)| ≥ ess infy∈Γ

a∗dGσdG1

‖v‖2L2

ρ(Γ;H10 (D)), (4.13)

∀v,w ∈ L2ρσ

(Γ;H10 (D)) : |bσM (v,w)| ≤ ess sup

y∈Γa∗dG1

dGσ‖v‖L2

ρσ(Γ;H1

0 (D))‖w‖L2ρσ

(Γ;H10 (D))

(4.14)Existence and uniqueness of a solution u ∈ L2

ρ(Γ;H10 (D) can then be shown as in Remark

3.3.1.


4.2.1 Hierarchic subspace sequences

In the stochastic Galerkin FEM (sGFEM), we discretize the variational formulation (4.5)by Galerkin projection onto a sequence of finite dimensional subspaces of

L2ρ(Γ;H1

0 (D)) ≃ L2ρ(Γ) ⊗H1

0 (D). (4.15)

Specifically, we choose two hierarchic families of finite dimensional subspaces

V Γ0 ⊂ V Γ

1 ⊂ . . . ⊂ V Γl1 ⊂ V Γ

l1+1 ⊂ . . . ⊂ L2ρ(Γ) (4.16)

and

V D0 ⊂ V D

1 ⊂ . . . ⊂ V Dl2 ⊂ V D

l2+1 ⊂ . . . ⊂ H10 (D), (4.17)

33


with l1, l2 being the levels of refinement. We introduce detail spaces WΓl1

and WDl2

, suchthat

WΓ0 := V Γ

0 and V Γl1 = V Γ

l1−1 ⊕WΓl1 for l1 = 1, 2, . . . (4.18)

andWD

0 := V D0 and V D

l2 = V Dl2−1 ⊕WD

l2 for l2 = 1, 2, . . . (4.19)

where the sums are direct so that the (finite-dimensional) approximation spaces V Γl1

and

V Dl2

admit a multilevel decomposition

V Γl1 =

L⊕

k1=0

WΓk1 and V D

l2 =L⊕

k2=0

WDk2 . (4.20)

In the lognormal case (4.10) we proceed analogously, except that we choose subspacesV Γl1

⊂ Lρσ(Γ) such that we have continuity and coercivity both w.r.t. functions in V Γ

l1,

taking values in V Dl2

, guaranteeing then for quasi-optimality of the Galerkin formulationintroduced below.

The specific choice of multilevel discretizations in space and random domain will besubject of the following chapters, where we will propose a wavelet discretization in thephysical domain D and a polynomial chaos discretization in L2

ρ(Γ).

4.2.2 Sparse tensor stochastic Galerkin formulation

Given the subspace hierarchies V Γl1l1≥0 ⊂ L2

ρ(Γ) and V Dl2l2≥0 ⊂ H1

0 (D) in (4.15),which admit splittings (4.18) and (4.19), we denote by

V ΓL ⊗ V D

L =⊕

0≤l1,l2≤LWΓl1 ⊗WD

l2 ⊂ L2ρ(Γ) ⊗H1

0 (D) (4.21)

the (full) tensor product space of the finite dimensional component subspaces V DL and

V ΓL , respectively. The (full) stochastic Galerkin formulation of (4.5) is then given by:

FinduM ∈ V Γ

L ⊗ V DL : bM (uM , v) = l(v) ∀ v ∈ V Γ

L ⊗ V DL . (4.22)

Denoting by NΓL and ND

L the number of stochastic and deterministic degrees of freedom,respectively, the stochastic Galerkin formulation w.r.t. V Γ

L ⊗ V DL clearly uses

dim(V ΓL ⊗ V D

L

)= NΓ

L ×NDL (4.23)

degrees of freedom. Due to the generally large number of KL terms M and thereforehigh dimensionality of the space L2

ρ(Γ;H10 (D)), a lot of recent works were focused on

34


?

-

WΓl1

WΓ3

WΓ2

WΓ1

WΓ0

WD0 WD

1 WD2 WD

3 WDl2

WΓ3 ⊗WD

0

WΓ2 ⊗WD

0

WΓ1 ⊗WD

0

WΓ0 ⊗WD

0

WΓ2 ⊗WD

1

WΓ1 ⊗WD

1

WΓ0 ⊗WD

1

WΓ1 ⊗WD

2

WΓ0 ⊗WD

2 WΓ0 ⊗WD

3

Figure 4.1: Illustration of the sparse tensor product space V ΓL ⊗V D

L in terms of the com-ponent detail spaces WΓ

l1and WD

l2.

the reduction of NΓL , by using sparse approximation techniques [22, 59, 12]. Here, we

will consider these approaches and, in addition, approximate the solution to (4.5) byGalerkin projection onto sparse tensor product spaces defined by

V ΓL ⊗V D

L :=⊕

0≤l1+l2≤LWΓl1 ⊗WD

l2 , (4.24)

see also Figure 4.1 for an illustration of V ΓL ⊗V D

L in terms of the component detailspaces.

Hence, we obtain the sparse stochastic Galerkin formulation: find

uM ∈ V ΓL ⊗V D

L : bM (uM , v) = l(v) ∀ v ∈ V ΓL ⊗V D

L . (4.25)

35


As a consequence of (4.8), (4.9), if M is sufficiently large, (4.25) defines, for every L ≥ 0a unique sGFEM approximation uM ∈ V Γ

L ⊗V DL which, by Galerkin orthogonality, is a

quasi-optimal approximation in L2ρ(Γ;W(D)) of uM defined in (4.5):

‖uM − uM‖L2ρ(Γ;W(D)) ≤ C‖uM − v‖L2

ρ(Γ;W(D)) ∀v ∈ V ΓL ⊗V D

L , y ∈ Γ (4.26)

Similarly, we obtain the sparse stochastic Galerkin formulation for the lognormal problem(4.10): find

uM ∈ V ΓL ⊗V D

L : bσM (uM , v) = lσ(v) ∀ v ∈ V ΓL ⊗V D

L . (4.27)

Since V ΓL ⊂ L2

ρσ(Γ) ⊂ L2

ρ(Γ), the bilinear form bσM (·, ·) is by virtue of (4.13), (4.14)

coercive and continuous, both w.r.t. u, v ∈ V ΓL ⊗V D

L . As a consequence, we have quasi-optimality as in (4.26), see also [28, Thm. 4.4].

‖uM − uM‖L2ρ(Γ;W(D)) ≤ C‖uM − v‖L2

ρσ(Γ;W(D)) ∀v ∈ V Γ

L ⊗V DL . (4.28)

Using the sparse tensor product space (4.24) instead of the full tensor product space(4.21), results in a reduction of the complexity (4.23) to essentially log-linear complexity.Precisely, we have

Lemma 4.2.1. Assume that the dimensions of the detail spaces WΓl1

and WDl2

grow

exponentially with respect to the levels l1, l2, i.e. there exist two geometric sequences bl1Γand bl2D, l1, l2 = 1, 2, 3, ..., with bases bΓ and bD such that

dim(WΓl1) ∼ bl1Γ and dim(WD

l2 ) ∼ bl2D.

Then the sparse tensor product space V ΓL ⊗V D

L is of cardinality

dim(V ΓL ⊗V D

L ) ∼ O(Lθ max(bΓ, bD)L+1), (4.29)

where θ = 1 if bΓ = bD and zero otherwise.

Proof. Observe that

dim(V ΓL ⊗V D

L ) ≤ cL∑

l1=0

L−l1∑

l2=0

bl1Γ bl2D ≤ c

L∑

l1=0

bl1Γ bL−l1+1D . (4.30)

In the case where bΓ = bD, we therefore have

dim(V ΓL ⊗V D

L ) ≤ cL∑

l1=0

bL+1Γ = c(L+ 1)bL+1

Γ , (4.31)

36

4.3 Stochastic collocation formulation

which leads to (4.29) with θ = 1. Now we assume w.l.o.g. bD > bΓ and it follows from(4.30)

dim(V ΓL ⊗V D

L ) ≤ c

L∑

l1=0

bl1Γ bL−l1+1D

= cbL+1D

L∑

l1=0

bl1Γ

(bΓbD

)l2

= cbL+1D

(bD/bΓ)L+1 − 1

(bD/bΓ) − 1.

Since the denominator is positive it can be absorbed in the constant and we find

dim(V ΓL ⊗V D

L ) ≤ cbL+1D .

This completes the proof.


4.3.1 Hierarchic sequence of collocation operators

Recall the definition of the spatial discretization spaces V Dl2

and their respective de-

tails WDl2

from (4.17) and (4.19). In addition, define the H10 -projection operator PDl2 :

H10 (D) → V D

l2by

(∇(uM − PDl2 uM ),∇v)L2 = 0 ∀ v ∈ V Dl2 , (4.32)

where (·, ·)L2 denotes the L2 inner product on D.

For each (stochastic) level l1 ∈ N0, choose a discrete finite set of collocation pointsYl1 ⊂ Γ and a set of multiindices Λl1 ⊂ N

M , such that the sequence is nested withincreasing l1, i.e.

Λ1 ⊂ Λ2 ⊂ · · · ⊂ Λl1 ⊂ Λl1+1 ⊂ · · ·NM (4.33)

and such that the interpolation problem for the points Yl1 is well-posed in V Γl1

:=

spanyα : α ∈ Λl1. Naturally, the V Γl1

are also nested and we define the interpola-tion operators

IΓl1 : C0

χ(Γ;H10 ) −→ V Γ

l1 (4.34)

and the difference operators

∆Γ0 := IΓ

0 , ∆Γl1 := IΓ

l1 − IΓl1−1, (4.35)

where C0χ has been defined in (3.6).

37


Remark 4.3.1. Note that for the index sets Λl1 , and hence the spaces V Γl1

, to be nested,the sets of collocation points Yl1 do not necessarily have to share this property. Thiscan easily be seen by considering a one-dimensional Lagrange interpolation operator onΓ1, based on the sets Yl1 of the l1 + 1 abscissae of the Gauss-Legendre quadrature ruleof order l1. Evidently, the collocation points are not nested, but the associated spacesV Γ1l1

= Pl1(Γ1) clearly are, where Pl1 denotes the space of polynomials up to order l1.

4.3.2 Sparse tensor stochastic collocation formulation

We first introduce the parametric variational formulation of (4.1): Find

uM : Γ → H10 (D) : BM (y;uM , v) = F (v) ∀ v ∈ H1

0 (D), (4.36)

where the parametric bilinear form BM (y, ·, ·) is defined as

BM (y;u, v) :=

∫

DaM (y,x)∇u(y,x)∇v(x) dx, (4.37)

and the linear form is given by

F (v) =

∫

Df(x)v(x) dx.

Note that due to (3.13), the variational problem (4.36) has a unique solution for everyfixed y ∈ Γ. This also holds true in the presence of a lognormal coefficient, sincea∗(y) ≤ aM (y,x) ≤ a∗(y), ∀x ∈ D, where

a∗(y) = emin Ea(x)−PM

m=1 ψm(x)|ym|, a∗(y) = emax Ea(x)+PM

m=1 ψm(x)|ym|. (4.38)

Then, we have, for y ∈ Γ,

BM (y;u, v) ≤ a∗(y)‖u(y, ·)‖H10 (D)‖v(y, ·)‖H1

0 (D),

BM (y;u, u) ≥ a∗(y)‖u(y, ·)‖2H1

0 (D).(4.39)

Proposition 4.3.2. Let uM : Γ → H10 (D) be the solution to (4.36), with aM in (4.37)

given either by a linear (4.2) or lognormal (4.3) truncated KL expansion. Then, uM ∈L2ρ(Γ;H1

0 (D)).

Proof. In the following, we mean by a∗ either the lower bound of aM given in (3.13) or(4.38). By (4.39) for every y ∈ Γ we have

‖uM (y, ·)‖2H1

0 (D) ≤1

a∗(y)BM (y;uM , uM ) =

1

a∗(y)F (uM )

38


≤ CPa∗(y)

‖f‖L2(D)‖uM (y, ·)‖H10 (D),

and hence

‖uM (y, ·)‖H10 (D) ≤

CPa∗(y)

‖f‖L2(D).

Therefore,

‖uM‖2L2

ρ(Γ;H10 (D)) =

∫

Γ‖uM (y, ·)‖2

H10 (D) ρ(y)dy

≤ C2P ‖f‖2

L2(D)‖1

a∗(y)‖2L2

ρ(Γ;H10 (D)).

With the finite dimensional subspaces V Dl2

at hand, we solve problem (4.36) for eachpoint y ∈ Yl1 by discretization in the spatial variable x, giving rise to the semidiscreteparametric formulation: Find, for a y ∈ Yl1 ,

uM (y, ·) ∈ V Dl2 : BM (y; uM (y, ·), v) = F (v) ∀ v ∈ V D

l2 . (4.40)

By a standard argument, the pointwise quasi-optimality of the Galerkin solution uM to(4.40) follows from (4.39):

‖uM (y, ·) − uM (y, ·)‖2H1

0 (D) ≤a∗(y)

a∗(y)infv∈VD

l2

‖uM (y, ·) − v‖2H1

0 (D). (4.41)

The random solution uM ∈ L2ρ(Γ;V D

l2) is then recovered by interpolating over the col-

located solutions by means of (4.34). We can describe the (full) tensor collocationapproximation to the solution u of (4.36) as

uL := IΓL(PDL u) =

∑

0≤l1,l2≤L∆Γl1((P

Dl2 − PDl2−1)u), (4.42)

where PDl2 denotes the H10 (D)-projection (4.32) and ∆Γ

l2the incremental interpolation

operators (4.35). Consequently, as in (4.23), this approach uses a total number of

NL = NΓL ×ND

L

degrees of freedom where NDL denotes the number of spatial basis functions and NΓ

L thenumber of collocation points in the random parameter space, each at level L. Since eachcollocation point entails one solution of a deterministic problem (4.36), numerous recentworks have been focused on reducing the number of collocation points by e.g. a Smolyakconstruction [43, 42, 67, 23] or ANOVA based construction [56, 9]. In addition to these

39


techniques, which will be discussed in more detail in Chapter 6, we will also consider asparse tensor approximation instead of (4.42):

uM :=∑

0≤l1+l2≤L∆Γl1((P

Dl2 − PDl2−1)u). (4.43)

The total number of degrees of freedom will then, as in Lemma 4.2.1, reduce to essentiallylog-linear complexity.

Lemma 4.3.3. Assume that the number of collocation points between each level l1 andthe dimension of the detail space WD

l2do not grow faster than exponential with respect

to the levels l1, l2, i.e. there exist two geometric sequences bl1Γ and bl2D, l1, l2 = 1, 2, 3, ...,with bΓ > 1 and bD > 1, such that

(|Yl1 | − |Yl1−1|) . bl1Γ and dim(WDl2 ) . bl2D.

Then the sparse tensor approximation of the solution u to (4.36) by means of (4.43) uses

NL . Lθ maxbΓ, bDL+1 (4.44)

degrees of freedom, where θ = 1 if bΓ = bD and zero otherwise.

Proof. The proof is analogous to the one given for Lemma 4.2.1. We observe that

NL ≤ c

L∑

l1=0

L−l1∑

l2=0

bl1Γ bl2D ≤ c

L∑

l1=0

bl1Γ bL−l1+1D . (4.45)

In the case where bΓ = bD, we therefore have

NL ≤ c

L∑

l1=0

bL+1Γ = c(L+ 1)bL+1

Γ ,

which leads to (4.44) with θ = 1. Now we assume w.l.o.g. bD > bΓ and it follows from(4.45)

NL ≤ cL∑

l1=0

bl1Γ bL−l1+1D

= cbL+1D

L∑

l1=0

bl1Γ

(bΓbD

)l2

= cbL+1D

(bD/bΓ)L+1 − 1

(bD/bΓ) − 1.

40


Since the denominator is positive it can be absorbed in the constant and we find

NL ≤ cbL+1D .

This completes the proof.

41


42

5 Wavelet discretization in D

As we have seen in the previous chapter, a hierarchic sequence of finite-dimensionalapproximation spaces V D

l2⊂ H1

0 (D) is a key component of the sparse tensor stochasticGalerkin and collocation algorithms. In this chapter, we will propose hierarchic finiteelement wavelets as a basis for the spatial discretization. The construction presentedhere, is based on [41], which, in principle, allows us to construct wavelets on a regularsimplicial triangulation of D ⊂ R

d. Here, we will treat the case of d = 1 and d = 2explicitly. For the case d = 3 see [41] and for a more general and detailed introductioninto wavelets we refer to [11, 15].

5.1 Wavelet construction

The starting point of the wavelet discretization presented here is based on a nestedsequence Tl2l2 of regular simplicial triangulations of D, obtained by successive refine-ment of an initial mesh T0. By I(Tl2) we denote the index set of vertices of the mesh Tl2,denoted by V(Tl2), and by I(Tl2+1) the index set of vertices of the mesh Tl2+1, which donot belong to Tl2. Hence, I(Tl2+1) = I(Tl2) ∪ I(Tl2+1), where the union is disjoint, andit therefore holds

|I(Tl2)| = |I(T0)|2dl2 , |I(Tl2+1)| = |I(T0)|2dl2(2d − 1). (5.1)

We then define

V Dl2 := Sp(D,Tl2) =

u ∈ H1

0 (D) : u|T ∈ Pp(T ) for T ∈ Tl2, (5.2)

i.e. the space of continuous piecewise polynomials of degree p w.r.t. the triangulationTl2. Clearly, the spaces V D

l2are hierarchic in the sense of (4.17). Denote by φkl2k,

k ∈ ∇l2 a basis of V Dl2

= Sp(D,Tl2), with ∇l2 being a suitable index set.

Given this basis of V Dl2

, we then construct a basis ψil2i, i ∈ ∇l2 of the detail spaces

WDl2

(4.19). First, the so-called scaling functions on the level l2 = 0 are defined as the

basis functions on the coarsest mesh T0, i.e. ψk0 = φk0 and ∇0 = ∇0. The construction ofwavelets on a higher level l2 > 0 is based on the basis functions on the meshes Tl2 and

43


Tl2−1. First, we construct a family of functions θil2(x) ∈ Sp(D,Tl2), i ∈ ∇l2 satisfying

(θil2 , φkl2−1)L2(D) ≃ δik. Such functions for p = 1 are given in d = 1 by

θil2(v) =

3 v = vi ∈ V(Tl2−1)

−12 v ∈ V(Tl2) is neighbor of vi

0 any other v ∈ V(Tl2)(5.3)

and in d = 2 by

θil2(v) =

14 v = vi ∈ V(Tl2−1)

−1 v ∈ V(Tl2) is neighbor of vi

0 any other v ∈ V(Tl2).(5.4)

The ensemble of functions θil2 : i ∈ I(Tl2−1) ∪ φil2 : i ∈ I(Tl2) forms a L2-Riesz basis

of V Dl2

satisfying (θil2 , φkl2−1)L2(D) = 0 if i 6= k. The wavelets on level l2 are then obtained

by a Gram-Schmidt orthogonalization process:

ψil2(x) = φil2(x) −∑

k∈I(Tl2−1)

(φil2 , φkl2−1)L2(D)

(θkl2 , φkl2−1)L2(D)

θkl2(x), i ∈ I(Tl2). (5.5)

By construction, (ψil2 , φkl2−1) = 0, and the functions ψil2 , i ∈ ∇l2 form a uniform L2-

Riesz basis for WDl2

, see [41, Prop. 3.2.10]. Scaling the wavelets with a factor 2−l2 , i.e.

defining ψil2(x) = 2−l2ψil2(x), then yields a Riesz basis forH1(D). In case of homogeneousDirichlet boundary conditions, the above construction can be modified as follows: Forvi ∈ V(Tl2), the corresponding φil2−1, φ

il2

and θil2 are excluded from the ensembles and

the resulting wavelets ψl2j then form a uniform Riesz bases for H10 (D). We refer to

Section 5.3, where we will construct the wavelets explicitly in the case p = 1, d = 1.

5.2 Basic properties of wavelet Galerkin approximation

In this section, basic results on the wavelet discretization proposed in the previous sectionare presented. In particular, we will give an approximation result and a bound on thecondition number of stiffness matrices obtained by wavelet discretization.

Recall the H10 (D) Galerkin projection PDl2 : H1(D) −→ V D

l2from (4.32), where, in the

present setting, V Dl2

= Sp(D,Tl2). We have the following approximation result, which isproved e.g. in [53, Thm. 3.17] for d = 1 and [52, Prop. 2.32] for d = 2, 3.

44

5.2 Basic properties of wavelet Galerkin approximation

Proposition 5.2.1. Assume u ∈ H1+t(D) with t ∈ [0, p] and l2 ≥ 0.

‖u− PDl2 u‖H1(D) ≤ C2−l2tp−t‖u‖H1+t(D), t ∈ [0, p], (5.6)

where the constant C > 0 depends only on the smoothness t and the quasi-uniformityconstant of the mesh Tl2 .

Next, the approximation result (5.6) is related to the number of degrees of freedom.Since ND

l2∼ pd2l2d we easily derive

Corollary 5.2.2. Assume u ∈ H1+t(D) with t ∈ [0, p] and l2 ≥ 0. Then, as l2 → ∞,

‖u− PDl2 u‖H1(D) . (NDl2 )−t/d‖u‖H1+t(D), t ∈ [0, p]. (5.7)

Denote by Ψl2 = ψikk,i k = 0, . . . , l2 and i ∈ ∇l2 the ensemble of wavelets up to levell2, forming a H1

0 (D) Riesz basis of V Dl2

= Sp(D,Tl2). Let us further consider the (purely

spatial) variational problem of finding a solution u to V Dl2

b(u, v) = l(v) on D ⊂ Rd

with zero Dirichlet boundary conditions, where b(·, ·), in the present context, is a con-tinuous and coercive symmetric bilinear form on V D

l2× V D

l2. Discretizing the problem

w.r.t. Ψl2 results in a linear systemAu = l (5.8)

to solve, where the symmetric and positive definite matrix A is given by A = b(Ψl2 ,Ψl2).Any function v ∈ V D

l2can be written as

v(x) = v⊤Ψl2 =∑

j

vjψj(x).

From the uniform Riesz basis property we have the norm equivalence

‖v‖H10 (D) = ‖v⊤Ψl2‖H1

0 (D) ∼ ‖v‖ℓ2 , ∀v ∈ V Dl2 (5.9)

with a constant that is independent of the level l2. Observing that b(u, v) = u⊤Av itfollows from the continuity and coercivity of b(·, ·) together with (5.9) that

c1 ≤ v⊤Av

‖v‖2ℓ2

≤ c2 ∀v ∈ ℓ2 (5.10)

with positive constants c1, c2. In other words, the Rayleigh quotient (5.10) of the matrixA is bounded for any vector v ∈ ℓ2, which in turn implies that λmax(A) ≤ c2 andλmin(A) ≥ c1. Hence the condition number of the matrix A with respect to the 2-normis bounded by

cond2(A) ≤ c1c2,

independent of the discretization level l2. This motivates the use of wavelets, if iterativesolvers, such as conjugate gradients, are used to solve the linear system (5.8).

45


-

l2 = 1

-

l2 = 2

-

l2 = 3

Figure 5.1: The hat functions φkl2 , k ∈ I(Tl2) for d = 1 on different levels as indicated,starting from an initial “one-element” mesh T0 with Dirichlet boundary con-ditions assumed

5.3 Example

In this section, we will conduct the construction presented in Section 5.1 in the cased = 1 and p = 1, hence resulting in piecewise linear wavelets. As a basis φkl2k, k ∈ ∇l2

for the discretization spaces V Dl2

= S1(D,Tl2), we take the standard hat functions, i.e.the piecewise linear polynomials with value 1 at the vertex k and zero at the other nodes,see Figure 5.1. We assume that Tl2 consists of equivalent elements, i.e. intervals, withmeshwidth hl2 and that the nodes are numbered in ascending order from left to right,i.e. v0 lies on the left boundary of the interval and vNl2

on the right boundary, whereNl2 denotes the number of elements of Tl2 .

Given the functions θil2 from (5.3), see also Figure 5.2, we easily compute that

(θil2 , φkl2−1)L2(D) = 2hl2δik and (φjl2 , φ

kl2−1)L2(D) =

hl22

if vj ∈ N (Tl2) is a neighbor of the vertex vk ∈ N (Tl2−1).

From formula (5.5) it then follows that for points vi ∈ N (Tl2), which are not adjacent

46

5.3 Example

-................................................................

........................................................

φk

l2

-....................................................................................................................................................................

...............................

.................................................................................................................................................

............

θi

l2+1

Figure 5.2: The hat functions φkl2 on level l2 (top) and θil2+1 on the finer level l2 + 1

(bottom)

to the boundary (2 ≤ i ≤ Nl2 − 2) that

ψil2(v) =

54 v = vi

−34 v = vi±1 ∈ V(Tl2)18 v = vi±2 ∈ V(Tl2)0 any other v ∈ V(Tl2)

For the points vi ∈ N (Tl2) adjacent to the boundary, we obtain the boundary wavelets

ψil2(v) =

98 v = v1 or v = vNl2

−1

−34 v = v2 or v = vNl2

−2

18 v = v3 or v = vNl2

−3

0 any other v ∈ V(Tl2)

Figure 5.3 shows the wavelets and boundary wavelets forming a L2(D) Riesz basis forWDl2

.

The same construction in the case p = 1 and d = 2 yields wavelets as shown in Figure5.4.

47


-

φl2

-.......................................................................................................................

................................................................

....................................................................................

.............................

ψl2+1

-

ψl2+1

Figure 5.3: Top: Hat function basis φkl2k for V Dl2

. Middle: Wavelet basis ψil2+1of WD

l2. Bottom: boundary wavelets in the case of Dirichlet boundary

conditions.

Figure 5.4: Left: The function θil2 as defined in (5.4). Right: The corresponding piecewise

linear finite element wavelet on a triangular mesh in 2-d

48

6 Sparse tensor stochastic collocation

We recall from Section 4.3 that the sparse tensor stochastic collocation method (4.43)is based on a hierarchic sequence of discretization spaces V D

l2l2 of D ⊂ R

d (4.17)

and a hierarchic sequence of collocation operators IΓl1

(4.34). After the choice of the

deterministic discretization spaces V Dl2

was subject of Chapter 5, we are now concernedwith the choice of the stochastic collocation operators

Il1 : C0χ(Γ;H1

0 (D)) −→ V Γl1 ⊗H1

0 (D),

with the space C0χ(Γ;H1

0 (D)) as defined in (3.6).

One possible choice is a full tensor Lagrange interpolation operator: Given sequencesym,0, . . . , ym,µm ⊂ Γm, µm(l1) ∈ N, 1 ≤ m ≤M , we denote by

Y fulll1 := yj = (y1,j1, . . . , yM,jM ) : 0 ≤ jm ≤ µm(l1), 1 ≤ m ≤M (6.1)

the so-called full tensor grid of collocation points. Then the corresponding full tensorLagrange interpolation operator is defined as

I fulll1 v =

∑

yj∈Y fulll1

v(yj,x)ℓj(y), (6.2)

where ℓj(y) denotes the tensorized Lagrange interpolation polynomial at the point yj.The application of this interpolation formula, however, requires the evaluation of v(·,x)at NΓ =

∏Mm=1(µm+1) points, which, as we have seen in Chapter 4, corresponds to NΓ

deterministic PDEs to be solved. In cases where M is large, this can be prohibitive, dueto the so-called ‘curse of dimension’, meaning that NΓ grows exponentially w.r.t. M .

6.1 Smolyak’s construction of collocation points

To keep the number of collocation points moderate, following [43, 42, 67], we will use aSmolyak-type construction. Contrary to those works, however, we will focus on Gaus-sian instead of Clenshaw-Curtis quadrature rules as the underlying univariate numerical

49


integration schemes, since their approximation properties allow us to avoid the intro-duction of a Lebesgue constant in each subspace Γm, resulting in an overall constantwhich would depend exponentially on M , see e.g. [4, 43, 42]. Moreover, by renouncingnestedness, we gain the flexibility to increase the number of collocation points in eachdirection linearly w.r.t. the level of refinement instead of doubling it in each step.

For any 1 ≤ m ≤M , any real number γ and a level km let ym,0, . . . , ym,µm ⊂ Γm be theabscissae of the Gauss quadrature rule of order µm(km) with respect to the probabilitydensity ρm, where

µm(km) = ⌈γkm⌉. (6.3)

The parameter 0 < γ ∈ R serves as a steering parameter for the growth of the numberof collocation points. It allows us to precisely tailor the stochastic refinements to theresolution of the wavelets in space. Its precise value will be determined later in (6.29),based on the expected convergence rates of the spatial discretization.

The one-dimensional interpolation operators I(m)km

(v) are then given by

I(m)km

(v) =

µm(km)∑

jm=0

v(ym,jm)ℓjm(ym), (6.4)

where ℓjm denotes the Lagrange interpolation polynomial of degree µm(km) in the pointym,jm, defined by

ℓm,jm(ym) ∈ Pµm(Γm), ℓm,jm(ym,im) = δjmim , 0 ≤ jm, im ≤ µm. (6.5)

Remark 6.1.1. Standard probability density functions, such as Gaussian or uniform ones,lead to well-known abscissae which are tabulated to full accuracy. For non-standarddensities we refer to [24], Theorem 3.1 where an algorithm for the derivation of theirassociated Gauss nodes and weights is described.

Next, we define the univariate differences

∆(m)km

:= I(m)km

− I(m)km−1, ∆

(m)0 = I(m)

0 . (6.6)

Using the multiindex notation k = (k1, . . . , kM ), the (isotropic) M -dimensional Smolyakinterpolation operator is then defined by

IΓl1 :=

∑

0≤|k|≤l1(∆

(1)k1

⊗ · · · ⊗ ∆(M)kM

) (6.7)

or, equivalently, by

IΓl1 =

∑

0≤|k|≤l1(−1)l1−|k|

(M − 1

l1 − |k|

)

(I(1)k1

⊗ · · · ⊗ I(M)kM

), (6.8)

50

6.2 Error analysis of Smolyak’s collocation algorithm

0 1 2 3 4 5 6 7 8 910

0

105

1010

1015

1020

1025

1030

1035

M = 10 fullM = 10 sparseM = 20 fullM = 20 sparseM = 40 fullM = 40 sparse

l1

Figure 6.1: Number of collocation points in a full tensor approach (6.1) compared to thenumber of collocation points in the Smolyak approach (6.7) for dimensionsM = 10, 20, 40

see [64].

Figure 6.1 illustrates the number of collocation points of the Smolyak approach (6.7)compared to the number of collocation points in a full tensor approach (6.1).


For the error analysis of the Smolyak collocation algorithm described above, we assumefor now that for each point yj the associated elliptic problem (4.36) can be solved exactlyin H1

0 (D). The numerical analysis of the respective full discretization is subject of thenext section.

We first provide a result on the approximation properties of the univariate interpolationoperators (6.4):

Proposition 6.2.1. If v solves (4.5) and I(m)km

denotes the operator defined in (6.4),then the interpolation error admits the following coordinate-wise bound:

‖v − I(m)km

(v)‖L2ρm (Γm;H1

0 (D)) ≤ C(rm)e−rmkm‖v‖C0χm (Σ(Γm,τm);L2

ρ∗(Γ∗m;H1

0 (D))) (6.9)

51


where ‖v‖C0χm

(Σ(Γm,τm);V ) = supz∈Σ(Γm,τm) χm(Re z)‖v‖V denotes the complex extensionof the norm and

• if Γm is bounded

rm = ln

(

2τm|Γm| +

√

1 +(

2τm|Γm|

)2)

C(rm) = C11

erm−1

• if Γm is unbounded

rm = ln(

√2δmτm)

C(rm) = C2(αm, δm) 1erm−1

where C1 is independent of m,km, rm and C2 = eα2m/(4δ

2m)(

1 + Erf(αm2δm

))

, where τm

defined as in Lemma 3.3.3 and δm,αm given by Assumption 3.2.6. Erf denotes the Gausserror function, see e.g. [1] Chapter 7.

We note here that the result of the bounded case has already been proposed in [4], butin the unbounded case, Proposition 6.2.1 provides a much sharper bound. In particularwe see in the above proposition that in both cases, the constants are decaying to zerosince the size of the analyticity regions τm is growing to infinity as m → ∞, see alsoLemma 3.3.3. The proof of Proposition 6.2.1 will be given at the end of this chapter.

Following the lines of [64, 44] we will now estimate the interpolation error of the Smolyakoperator (6.7). First we note that

IΓl1 =

∑

0≤|k∗|≤l1∆

(1)k1

⊗ · · · ⊗ ∆(M−1)kM−1

⊗l1−|k∗|∑

kM=0

∆(M)kM

=∑

0≤|k∗|≤l1∆

(1)k1

⊗ · · · ⊗ ∆(M−1)kM−1

⊗ I(M)l1−|k∗|, (6.10)

where k∗ = (k1, . . . , kM−1) In the following denote by Id(m) the identity operator on Γmand IdM =

∏Mm=1 Id(m). The interpolation error can be written as

IdM − IΓl1 = (IdM−1 − IΓ∗

Ml1

) ⊗ Id(M) +∑

0≤|k∗|≤l1∆

(1)k1

⊗ · · · ⊗ ∆(M−1)kM−1

⊗ (Id(M) − I(M)l1−|k∗|),

(6.11)where Γ∗

M = Γ1 × . . .× ΓM−1. In the following, unless otherwise stated, denote by ‖ · ‖the L2

ρ(Γ;H10 (D))-norm or its associated operator norm. From

‖∆(m)km

‖ ≤ ‖Id(m) − I(m)km

‖ + ‖Id(m) − I(m)km−1‖ ≤ 2Cme

−rmγkm , (6.12)

where Cm = C(rm) as in Proposition 6.2.1, it follows together with (6.11)

‖IdM − IΓl1‖ ≤ ‖IdM−1 − IΓ∗

Ml1

‖ +∑

0≤|k∗|≤l1‖∆(1)

k1‖ · · · ‖∆(M−1)

kM−1‖ ‖Id(M) − I(M)

l1−|k∗|‖

52


≤ ‖IdM−1 − IΓ∗M

l1‖

+∑

0≤|k∗|≤l1

(M−1∏

m=1

2Cme−rmkm

)

CMe−rMγkM

= ‖IdM−1 − IΓ∗M

l1‖ +

∑

0≤|k∗|≤l1

M∏

m=1

2Cme−rmγkm . (6.13)

Setting rmin = minr1, . . . , rM we arrive at the following recursive estimate

‖IdM − IΓl1‖ ≤ ‖IdM−1 − IΓ∗

Ml1

‖ + (M∏

m=1

2Cm)e−rminγl1

(M + l1M

)

. (6.14)

We observe from Proposition 6.2.1 that Cm → 0 if m→ ∞, hence the product in (6.14)can be bounded from above by a generic constant C which is independent of M . Using(6.13) repeatedly we arrive at

‖IdM − IΓl1‖ ≤ ‖Id(1) − I(1)

l1‖ +

1

2

M∑

m=2

Ce−rminγl1

(m+ l1m

)

≤ Ce−rminγl1

(M + l1M

)

(6.15)

Next, we want to estimate the number of collocation points used.

Lemma 6.2.2.

NΓl1 =

∑

|k|≤l1

M∏

m=1

(1 + γkm) =

(γl1 + 2M

2M

)

(6.16)

Proof. We first note that

NΓl1 =

∑

|k|≤γl1+M

M∏

m=1

km. (6.17)

Now define αj =∑

|k|=j∏Mm=1 km and observe that

∞∑

j=0

αjxj = (x+ 2x2 + 3x3 + . . .)M . (6.18)

The generating function of the series ak = k for k ≥ 1 is x(1−x)2 , see [66], hence

∞∑

j=0

αjxj =

(x

(1 − x)2

)M

. (6.19)

53


This in turn means that αj is the (j −M)-th coefficient of the power series of 1(1−x)2M ,

which, by [66] Section 2.5, is equal to(j+M−12M−1

). Summing the αj ’s therefore leads to

∑

|k|≤γl1+M

M∏

m=1

km =

γl1+M∑

j=0

αj =

(γl1 + 2M

2M

)

, (6.20)

which completes the proof.

Inserting the estimate for NΓl1

into (6.15) we obtain

Lemma 6.2.3. Let uM be the solution to (4.5). Then the collocation error admits thefollowing bounds with respect to the level l1 and the number of collocation points:

‖uM − IΓl1uM‖L2

ρ(Γ;H10 (D)) ≤ Ce−rminγl1

(M + l1M

)

‖uM‖C0χ(Σ(Γ,τ);H1

0 (D)) (6.21)

and


ρ(Γ;H10 (D)) ≤ C

√

NΓ exp(−rmin(N1/(2M)Γ − 1)

2M

1 + ln 2M)

× ‖uM‖C0χ(Σ(Γ,τ);H1

0 (D)),(6.22)

where C > 0 is independent of M and NΓ.

Proof. From (6.16) and using the inequality between geometric and arithmetic mean,we have

NΓl1 ≤

(γl1 + 2M

2M

)

=(γl1 + 2M)(γl1 + 2M − 1) · · · (γl1 + 1)

(2M)!

= (1 +γl12M

)(1 +γl1

2M − 1) · · · (1 +

γl11

)

≤(

2M + γl1∑2M

m=1 1/m

2M

)2M

≤(

1 +γl1(1 + ln 2M)

2M

)2M

. (6.23)

Hence, l1 ≥ ((NΓl1

)1/2M − 1) 2Mγ(1+ln 2M) from which (6.22) follows.

54

6.3 Error Analysis

Remark 6.2.4. As it can be seen in Lemma 6.2.3, the convergence rates with respect tothe number of collocation points are exponential but depending on M . Since M canpossibly be very large in applications, it is of interest to examine this case in some moredetail. From (6.23) we have for large M

NΓl1 ≤ eγl1(1+ln 2M) (6.24)

which, inserted into (6.21) and noting that

(l1 +M

M

)

≤(

(l1 +M)e

M

)M

≤ eM(

1 +l1M

)M

≤ el1+M

gives


ρ(Γ;H10 (D)) ≤ C(NΓ

l1)−(rmin−1)/(1+ln 2M)‖uM‖C0

χ(Σ(Γ,τ);H10 (D)).

Hence, for M ≫ 1, the exponential convergence rate in Lemma 6.2.3 behaves alge-braically in the preasymptotic range, i.e. for l1 ≪M .

Remark 6.2.5. So far, we only considered an isotropic Smolyak algorithm, i.e. with anequal number of quadrature points in each direction. However, one could also consideran anisotropic Smolyak algorithm, i.e. with

µm(km) = ⌈γξmkm⌉ (6.25)

instead of (6.3), where ξmMm=1 ∈ RM is a sequence of weights. Hence, the isotropic

formula is the special case ξm = 1, m = 1, . . . ,M . The (anisotropic) Smolyak operatorIΓl1

(6.7) is then defined analogously by replacing (6.3) with (6.25), featuring a hierarchyw.r.t. the level l1. Choosing the weights ξm = 1/rm, rm given by Proposition 6.2.1, weobtain instead of (6.15)

‖IdM − IΓl1‖ ≤ Ce−γl1

(M + l1M

)

The techniques used in Lemma 6.2.2, however, to compute the number of collocationpoints, do not work any more in the anisotropic case, since it is no longer a problem ofcounting integers.

6.3 Error Analysis

The main focus of this section aims at proving convergence rates for the sparse tensorstochastic collocation method. We note that these results hold for both, bounded andunbounded domains Γm. We assume in the following that the spatial discretizationis given by piecewise polynomials of degree p on a quasi-uniform mesh of width h, asdefined in (5.2).

55


Proposition 6.3.1. Let uM be the solution to (4.5) and denote by uM its sparse tensorapproximation given by (4.43). We have the following error estimate

‖uM − uM‖L2ρ(Γ;H1

0 ) ≤ CN−βL ‖uM‖C0

χ(Σ(Γ,τ);H1+t∩H10 (D)) (6.26)

in terms of the total number of degrees of freedom NL with rate

β(M) = min rmin − 1

1 + ln 2M,t ln 2

d (6.27)

and t ∈ [0, p].

In other words, by the sparse composition of hierarchic interpolation operators and amultilevel basis in space we retrieve the smaller of the two convergence rates providedby the spatial discretization (5.7) and stochastic interpolation, see Remark 6.2.4.

Remark 6.3.2 (Full tensor collocation). The main motivation for the sparse tensor com-position (4.43) of stochastic interpolation operators and spatial Galerkin projectors liesin the reduction of degrees of freedom to O(Lθ maxNΓ

L , NDL ) as shown in Lemma 4.2.1,

opposed to O(NΓL × ND

L ) in the full tensor approach (4.42). Using the approximationresults (5.6) and (6.21), we can easily derive the full tensor error estimate

‖uM − IΓL(PDL uM )‖L2(Γ;H1

0 (D)) ≤ C(NL)−β‖uM‖C0χ(Σ(Γ,τ);H1+t∩H1

0 (D)) (6.28)

with rate β = (d/t + (1 + ln 2M)/(rmin − 1))−1.

Proof. First, we will estimate the approximation error in terms of the global refinementlevel L. To simplify the notation, we will drop the subscript in the norms, hence ‖ · ‖ =‖ · ‖L2(Γ;H1

0 ) unless otherwise indicated. Using (4.43) we obtain

‖uM − uM‖ ≤ ‖L∑

l1=0

∞∑

l2=L−l1+1

∆l1(PDl2 − PDl2−1)uM‖

+ ‖∞∑

l1=L+1

∞∑

l2=0

∆l1(PDl2 − PDl2−1)uM‖

= ‖L∑

l1=0

∆l1(IdD − PDL−l1)u‖ + ‖(IdΓ − IΓ

L)IdDuM‖

≤L∑

l1=0

‖(IdΓ − IΓl1)(Id

D − PDL−l1)uM‖

56

6.4 Proof of Proposition 6.2.1

+

L∑

l1=0

‖(IdΓ − IΓl1−1)(Id

D − PDL−l1)uM‖ + ‖(IdΓ − IΓL)IdDuM‖

≤

L∑

l1=0

C

(M + l1M

)

e−rminγl1e− ln 2t(L−l1)

+C

(M + L

M

)

e−rminγL

)

‖uM‖C0χ(Σ(Γ,τ);H1+t∩H1

0 (D)),

where we used the estimates (5.6) and (6.21) in the last line. Absorbing now the binomialfactor into the exponent by

(M + l1M

)

≤(

(M + l1)e

M

)M

≤ eM(

1 +l1M

)M

≤ el1+M

and choosing

γ =t ln 2

rmin − 1(6.29)

we arrive at

‖uM − uM‖L2(Γ;H10 (D)) ≤ Ce−t(ln 2)L‖uM‖C0

χ(Σ(Γ,τ);H1+t∩H10 (D)) (6.30)

Next, we estimate the total number of degrees of freedom. We have from (6.24) thatNΓl1

≤ exp( t ln 2rmin−1(1 + ln 2M)l1), which satisfies the assumptions of Lemma 4.2.1 by

setting bΓ = 2t(1+ln 2M)/(rmin−1). Since moreover bD = 2d, we obtain

NL ≤ max2d, 2t(1+ln 2M)/(rmin−1)L (6.31)

The proof is completed by combining (6.31) together with (6.30).


In the following, to keep the results as general as possible, denote by W(D) a Banachspace. It has already been proved in [4] (cf. Lemma 4.3) that the L2

ρm(Γm) interpolation

error based on Gaussian abscissae can be bounded from above by a best-approximationerror, i.e.

Lemma 6.4.1. For every function v ∈ C0χm

(Γm;W(D)), the interpolation error satisfies

‖v − I(m)µm

(v)‖L2ρm (Γm;W(D)) ≤ C inf

w∈Pµm(Γm)⊗W(D)‖v − w‖C0

χm (Γm;W(D)), (6.32)

where C > 0 is independent of µm and I(m)µm is given by (6.4) based on the zeros of

orthogonal polynomials w.r.t. ρm.

57


It therefore remains to bound the best approximation error in the norm C0χ. To this end,

we distinguish the cases where Γm is bounded or unbounded. For the bounded case werefer to [4], Lemma 4.4

Lemma 6.4.2. For every function v ∈ C0([−1, 1],W(D)) which admits an analyticextension to the region Σ([−1, 1], τm) ⊂ C as in (3.9) for some τm > 0, it holds that

infw∈Pµm ([−1,1])⊗W(D)

‖v − w‖C0([−1,1];W(D)) ≤2

erm − 1e−µmrm max

zm∈Σ([−1,1],τm)‖v(z)‖W(D)

(6.33)

for rm = ln(

τm +√

1 + τ2m

)

.

By rescaling the domain [−1, 1] to Γm, Proposition 6.2.1 therefore follows in the boundedcase by combining Lemma 6.4.1 and 6.4.2 with µm = ⌈γkm⌉.

In the unbounded case we will, as in [4], first refer to a result from [31] but then proceedin a slightly different way to obtain the result claimed in Proposition 6.2.1. We denoteby Hn the Hermite polynomial of order n:

Hn(t) = (−1)net2/2 ∂

n

∂tne−t

2/2, t ∈ R (6.34)

satisfying∫

R

Hm(t)Hn(t)e−t2/2 dt =

√2πn!δmn.

Furthermore we define by hn(t) = e−t2/4 1√√

2πn!Hn(t) the L2(R)-orthonormal Hermite

functions. The following Lemma from [31] (cf. Theorem 1) then provides a necessaryand sufficient condition for Fourier-Hermite series to converge in a strip around the realaxis.

Lemma 6.4.3 (Hille 1940). Let f : Σ(R, τm) → C be analytic. A necessary and sufficientcondition in order that the Fourier-Hermite series

∞∑

k=0

fkhk(zm), fk =

∫

R

f(ym)hk(ym)dym

exists and converges to the sum f(zm) in Σ(R, τm), is that for every βm ∈ [0, τm) thereexists a finite constant C(βm) > 0, such that

|f(ym + iwm)| ≤ C(βm)e−|ym|√β2

m−w2m, ym ∈ R, wm ∈ [−βm,+βm]. (6.35)

58


Recall the measure χm(ym) = e−αm|ym| from (3.7) and furthermore define the Gaussianmeasure Gm(ym) = e−(δmym)2 for δm as in Assumption 3.2.6. Note that Lemma 6.4.1holds for both of them, see [4] for details. By v : Γm → R denote a function which isanalytic in the strip Σ(R, τm).

In order to apply Lemma 6.4.3, we have to rescale the measure G to the standardGaussian measure by the change of variables tm =

√2δmym. We then define v(tm) =

v(ym(t)). Since v is the composition of two analytic functions, it is itself again an analyticfunction and its domain of analyticity rescales to Σ(R,

√2τmδm).

Next, we show that the formal Hermite series of v

v(tm) =

∞∑

k=0

vkHk(tm),

with coefficients vk ∈ W(D) given by

vk =1√2πk!

∫

R

v(tm)Hk(tm)e−t2m/2 dtm

is well-defined and converges in Σ(R,√

2τmδm). To be in the setting of Lemma 6.4.3, we

set f(zm) = v(zm)e−z2m4 and observe that

fk =

∫

R

f(tm)hk(tm) dtm =1

√√2πk!

∫

R

v(tm)Hk(tm)e−t2m/2 dtm =

√√2πk!vk. (6.36)

f is clearly analytic in the strip Σ(R,√

2τmδm) as it is a product of two analytic functionswhich are each analytic in Σ(R,

√2τmδm). Furthermore, we have that

‖f(tm + iwm)‖W(D) = |e−(tm+iwm)2

4 |‖v(zm)‖W(D)

≤ e−t2m−w2

m4 e

α√2δm

|tm|‖v‖C0χm

(Σ(R,τm);W(D)).

The function f thus satisfies the hypotheses of Lemma 6.4.3 by setting

C(βm) = maxtm∈R

wm∈[−βm,βm]

exp

− t2m − w2

m

4+

α√2δm

|tm| + |tm|√

β2m − w2

m

.

Hence, the expansion of f into Hermite functions hk is converging pointwise in the stripΣm(R,

√2τmδm) and therefore, by (6.36) and

∞∑

k=0

vkHk(zm) =∞∑

k=0

fket2m/4hk(zm) = et

2m/4f(zm) = v(zm),

59


also the Hermite series of v.

Next, we will establish a bound for the Hermite coefficients vk. We define rm :=√

2τmδmand denote by Ctm(rm) the circle around tm with radius rm. Integrating by parts andusing Cauchy’s integral formula we obtain

‖vk‖W(D) =1√2πk!

∥∥∥∥

∫

R

v(tm)Hk(tm)e−t2m/2 dtm

∥∥∥∥W(D)

=1√2πk!

∥∥∥∥

∫

R

v(tm)∂k

∂tkme−t

2m/2 dtm

∥∥∥∥W(D)

=1√2πk!

∥∥∥∥

∫

R

∂k

∂tkmv(tm)e−t

2m/2 dtm

∥∥∥∥W(D)

=1√

2π2π

∥∥∥∥∥

∫

R

∮

Ctm(rm)

v(zm)

(zm − tm)k+1dzme

−t2m/2 dtm

∥∥∥∥∥W(D)

≤ 1√2πrkm

‖v‖C0χm

(Σ(R,τm);W(D))

∫

R

e−t2m/2+αm

|tm|√2δm e

rmαm√2δm dtm.

Hence, the Fourier Hermite coefficients satisfy the bound

‖vk‖W(D) ≤ C(αm, δm)erm

αm√2δm e−k ln rm‖v‖C0

χm(Σ(R,τm);W(D)), (6.37)

where C(αm, δm) = eα2m/(4δ

2m)(

1 + Erf(αm2δm

))

. Therefore, we can estimate the best

approximation error in Lemma 6.4.1 by

Eµm(v) := infw∈Pµm⊗W(D)

‖v − w‖C0Gm

(Γm;W(D)) ≤ maxtm∈R

‖∞∑

k=µm+1

vkhk(tm)‖W(D).

Since |hk(tm)| < 1 for all tm ∈ R, k ∈ N0 and due to (6.37), the truncated Hermite seriescan be bounded by

Eµm(v) ≤∞∑

k=µm+1

‖vk‖W(D) ≤ C(αm, δm)erm

αm√2δ ‖v‖C0

χm(Σ(R,τm);W(D))

∞∑

k=µm+1

e−k ln rm.

For the series on the r.h.s. we use the summation formula for geometric series to obtain

Eµm(v) ≤ C(αm, δm)eτmαme−µm ln rm

rm − 1. (6.38)

From Section 3.3.3 and (3.42) we have that αm ∼ τ−1m , and hence the term eτmαm remains

constant. Setting rm := ln rm, the claim now follows from (6.38) and Lemma 6.4.1.

60

7 Sparse tensor stochastic Galerkin

We recall from Section 4.2 that we discretize the variational formulation

bM (uM , v) = l(v), (7.1)

bM (·, ·) and l(·) given by (4.6) and (4.7), respectively, with respect to the sparse tensorproduct space V Γ

L ⊗V DL where

V Γl1 l1 ⊂ L2

ρ(Γ) and V Dl2 l2 ⊂ H1

0 (D)

are two hierarchic families of subspaces, see also (4.16) and (4.17). The choice of thespatial discretization spaces V D

l2= Sp(D,Tl2) has been subject of Chapter 5 and we are

now concerned with the choice of the stochastic spaces V Γl1

which is the crucial part inthe stochastic Galerkin FEM due to the possibly arbitrary high dimension M of theparameter domain Γ.

7.1 Hierarchic polynomial chaos approximation in

L2ρ(Γ; H1

0(D))

It is therefore imperative to reduce the number of stochastic degrees of freedom to aminimum while still retaining accuracy. To this end, we will, in the spirit of a best-N -term approximation, approximate the solution by a polynomial chaos expansion inwhich only the ‘most important’ terms are retained.

7.1.1 Preliminary conventions and notations

In the discussion that follows, we will distinguish the cases where the stochastic param-eter domains Γm are bounded and unbounded. In the first case, we can w.l.o.g. assumethat Γm = [−1, 1], ∀m. Moreover, we will then restrict our discussion to the case of auniform probability density ρm(ym) = 1

2 . In the case of unbounded domains we assumeΓm = R and, motivated by Assumption 3.2.6 and after a coordinate transformation,that ρm(ym) = 1√

2πe−(ym)2/2.

61


By NM0 , we denote the set of all multiindices of length M , i.e. all sequences ν =

(ν1, ν2, ..., νM ) of non-negative integers. For ν ∈ NM0 , we denote

|ν|0 = #supp(ν) and |ν| =∑

m≥1

|νm|,

where supp(ν) := j ∈ N : νj 6= 0. For ν ∈ NM0 , we denote the tensorized Legendre

polynomial of degree ν by

Lν(y) := Lν1(y1)Lν2(y2) · · ·LνM(yM ) y ∈ Γ, (7.2)

where Lνm(ym) denotes the Legendre polynomial of degree νm on [−1, 1]

Lνm(ym) =1

2nn!

∂νm

∂yνmm

((y2m − 1)

), (7.3)

normalized such that Lνm(1) = 1, i.e. they satisfy the orthogonality relation

∫ 1

−1Lνm(ym)Lµm(ym)ρm(ym) dym =

δνmµm

2νm + 1. (7.4)

Likewise, we define the tensorized Hermite polynomial of degree ν by

Hν(y) := Hν1(y1)Hν2(y2) · · ·HνM(yM ) y ∈ Γ, ν ∈ N

M0 , (7.5)

where Hνm(ym) denotes the Hermite polynomial of degree νm on R

Hνm(ym) = (−1)νmey2m/2

∂νm

∂yνmme−y

2m/2, (7.6)

satisfying the orthogonality relation∫

R

Hνm(ym)Hµm(ym)1√2πe−y

2m/2 dym = νm!δνmµm . (7.7)

7.1.2 Best-N-term polynomial chaos approximation

In the following, to keep the notation simple and to avoid repetitions, we denote by Pνeither of the polynomials Lν or Hν , depending on whether we consider the case wherethe Γm are bounded or unbounded.

From Section 2.3 we have that L2ρ(Γ) = spanPν : ν ∈ NM

0 , where the closure is w.r.tL2ρ(Γ) and the solution uM to the parametric, deterministic problem (4.36) can be rep-

resented in terms of ρ-orthogonal polynomials as

uM (y,x) =∑

ν∈NM0

uν(x)Pν(y), (7.8)

62

7.1 Hierarchic polynomial chaos approximation in L2ρ(Γ;H1

0 (D))

where the ‘coefficients’ uν ∈ H10 (D) are given by

uν =

(M∏

m=1

2νm + 1

)∫

ΓuM (y, ·)Lν(y)ρ(y) dy (7.9)

in the Legendre case and by

uν =1

ν!

∫

ΓuM (y, ·)Hν(y)ρ(y) dy (7.10)

in the Hermite case, where the equalities hold in H10 (D).

Given any index set Λ ⊂ NM0 of cardinality |Λ| <∞, we denote by

uM,Λ(y,x) =∑

ν∈Λ

uν(x)Pν(y) (7.11)

the Λ-truncated orthogonal PC expansion (7.8). In the Galerkin approximation below,we aim at finding an increasing sequence Λ(1) ⊂ Λ(2) ⊂ Λ(3) ⊂ · · · ⊂ N

M0 of index sets

Λ(l1), such that the approximations uM,Λ(l1) converge at a certain rate 0 < r ∈ R interms of |Λ(l1)|.For exponential eigenvalue decay (2.15) in the KL-expansion (2.8), this is indeed possiblewith arbitrary high algebraic convergence rates r as it has been shown in [9, 59]. There,an algorithm has been presented to a-priori locate index sets Λ, based only on the inputdata a, f , such that for any prescribed rate r > 0, there exists a constant C(r) > 0,independent of l1 and M and

‖uM − uM,Λ(l1)‖L2ρ(Γ;H1

0 (D)) ≤ C(r)|Λ(l1)|−r. (7.12)

The case of algebraic decay (2.14) of the Karhunen-Loeve eigenvalues however, is moredifficult since, as it will turn out, they imply a slow decay of the ‘coefficients’ in (7.11).Here, we use best N -term approximations of uM in (7.8) as a benchmark and prove aresult similar to (7.12) in the case of algebraic Karhunen-Loeve eigenvalue decay for acertain range of rates r, by imposing sufficient conditions on the input data.

To this end, for any finite set Λ ⊂ NNc and for any Banach space W(D), we denote by

Π(Λ;W(D)) := v =∑

ν∈Λ

vνPν : vν ∈ W(D), ν ∈ Λ ⊂ L2ρ(Γ;H) (7.13)

the linear space of W(D)-valued polynomials, which can be expressed as a finite lin-ear combination of |Λ| many ρ(dy) orthogonal polynomials Pν (if Λ = ∅, we defineΠ(Λ;W(D)) = 0). They are a countable ρ(dy)-orthonormal basis of L2

ρ(Γ) and any

63


v ∈ L2ρ(Γ;W(D)) is determined by its coefficient vector (vν : ν ∈ N

Nc ) ⊂ W(D) via

v =∑

ν∈NNcvνPν . The best N -term semidiscrete approximation error of u ∈ L2

ρ(Γ;W(D))is defined as

σN (u) := infΛ⊂NM

0 ,|Λ|≤Ninf

v∈Π(Λ;W(D))‖u− v‖L2

ρ(H). (7.14)

Note that σN (u) is uniquely defined even though v might not be unique.

Remark 7.1.1. As is well-known, bestN -term approximations are generally not computa-tionally accessible. We therefore present in Section 8.1 an algorithm to locate for certainsequences λm∞m=1 of Karhunen-Loeve -eigenvalues corresponding sequences Λl∞l=0

of index sets Λl ⊂ NM0 of ‘active’ indices in the chaos expansions (7.11) which realize

‘quasi-best N -term approximations’.

We define Ar(W(D)) as the set of those u ∈ L2ρ(Γ;W(D)), for which

|u|Ar(W(D)) := supN≥1

N rσN (u) (7.15)

is finite. This class consists of functions in L2ρ(Γ;W(D)), which can be best N -term

approximated at a rate r. For a parameter γ > 0 we introduce the index sets

Λγ(l1) := arg maxΛ⊂N

M0

|Λ|=⌈2γl1 ⌉

(∑

ν∈Λ

‖uν‖W(D)

)

⊂ NM0 , l1 = 0, 1, 2, ... (7.16)

of the ⌈2γl1⌉ coefficients uν in (7.8), with the largest W(D)-norm. Thus, Λγ(l1) providesa best ⌈2γl1⌉-term approximation of u ∈ L2

ρ(Γ;H).

To apply these concepts to (4.4), we choose W(D) = H10 (D). Again, as in the pre-

vious chapter, γ serves as a steering parameter for the stochastic resolution and willbe determined in Section 7.2, based on the expected convergence rate of the spatialdiscretization. The corresponding approximation spaces are then given by

V Γl1 := Π(Λγ(l1); R). (7.17)

By definition, the index sets Λγ(l1) are nested for a fixed γ > 0 and hence also the V Γl1

in the sense of (4.16). The detail spaces WΓl1

then consist exactly of the multivariateorthogonal polynomials Pν corresponding to the indices in Λγ(l1) \ Λγ(l1 − 1), i.e.

WΓl1 = Π(Λγ(l1) \ Λγ(l1 − 1); R).

64

7.2 Analysis of the sparse tensor stochastic Galerkin method

7.1.3 Approximation results

Define the L2ρ projection PΓ

l1: L2

ρ(Γ) −→ V Γl1

by PΓl1uM := uM,Λγ(l1) as in (7.11). The

following approximation properties rely on the decay of the Karhunen-Loeve coefficientsψm (4.2), i.e. on the parameter s > 0, where

‖ψm‖L∞(D) =√

λm‖φm‖L∞(D) ≤ Cm−s.

Proposition 7.1.2. Let s be the decay rate of ‖ψm‖L∞(D). If uM solves (4.5), then for

each 0 < r < s− 32 exists a constant C(r) such that for every γ > 0 and for the sequence

of projections PΓl1

corresponding to the index sets Λγ(l1) in (7.16) it holds

‖uM − PΓl1uM‖L2

ρ(Γ;H10 (D)) ≤ C(r)(NΓ

l1)−r|uM |Ar(H1

0 (D)) (7.18)

where NΓl1

:= |Λγ(l1)| → ∞ as l1 = 0, 1, 2, .... Since NΓl1

= ⌈2γl1⌉, we can express (7.18)also as

‖uM − PΓl1uM‖L2

ρ(Γ;H10 (D)) ≤ C(r)2−l1γr|uM |Ar(H1

0 (D)) (7.19)

Here, the constant C(r) is independent of the KL truncation order M , which equals thedimension of the stochastic parameter space. The proof will be provided at the end ofthis chapter.

Remark 7.1.3. To have r > 0 in (7.18) requires s > 32 in (2.25). Such a decay can be

expected, for example, if Va ∈ Ht,t(D ×D) with t > 4d as it can be easily derived fromCorollary 2.2.8. In other words, the regularity of the 2-point covariance Va (2.2) impliesthe range of possible rates r of best N -term approximations of uM .

Remark 7.1.4. Proposition 7.1.2 indicates only the existence and the convergence rate rof a semidiscrete best-NΓ-term approximation. These approximations are not construc-tive: the proof uses the (a-priori unknown) values of ‖uν‖H1

0 (D). However, in Section 8.1we will present a strategy based on the Karhunen-Loeve eigenvalues of the input datato a-priori locate sequences of sets Λγ(l1) that appear to be close to the optimal sets innumerical experiments. Moreover, the algorithm to find those sets scales linearly (up tologarithms) in time and memory w.r.t. the number of elements |Λγ(l1)|.

7.2 Analysis of the sparse tensor stochastic

Galerkin method

After having separately discussed the convergence rates of the polynomial chaos approx-imation in Γ and the multilevel discretizations in D of our model problem (4.4), we will

65


now derive approximation results for the sparse tensor stochastic Galerkin formulation(4.25).

To obtain estimates on the convergence rate for the sGFEM approximation uM of u itremains, due to the quasi-optimality (4.26), to estimate the best-approximation error ofuM from the sparse tensor product space V Γ

L ⊗V DL .

To this end, we define the sparse tensor product projection operator PL : L2ρ(Γ) ⊗

H10 (D) −→ V Γ

L ⊗V DL by

(PLv)(y,x) :=∑

0≤l1+l2≤L

(PΓl1 − PΓ

l1−1

)⊗(PDl2 − PDl2−1

)v(y,x) (7.20)

To simplify the notation, we denote by ‖ · ‖ the norm taken w.r.t. L2ρ(Γ;H1

0 (D)) unlessotherwise indicated. We denote by uM the solution to the problem (4.5) and by uM thediscrete solution to (4.25). To estimate the discretization error eL := ‖uM − PLuM‖, westart from the definition of PL in (7.20) and write

eL ≤ ‖L∑

l1=0

∞∑

l2=L−l1+1

(PΓl1 − PΓ

l1−1

)⊗(PDl2 − PDl2−1

)uM‖

+ ‖∞∑

l1=L+1

∞∑

l2=0

(PΓl1 − PΓ

l1−1

)⊗(PDl2 − PDl2−1

)uM‖

= ‖L∑

l1=0

(PΓl1 − Id + Id − PΓ

l1−1

)⊗(Id − PDL−l1+1

)uM‖

+ ‖(Id − PΓ

L

)⊗ (Id)uM‖

≤L∑

l1=0

(‖(Id − PΓ

l1

)⊗(Id − PDL−l1+1

)uM‖

+ ‖(Id − PΓ

l1−1

)⊗(Id − PDL−l1+1

)uM‖

)

+ C(r)2−Lγr‖uM‖Ar(H1+t∩H10 (D))

≤ (C(r, t)L2−Lmin(γr,t) + C(r)2−Lγr)‖uM‖Ar(H1+t∩H10 (D)).

Note that we used the approximation results (7.19) and (5.6) in the last two lines of theestimates above. We equilibrate the two error bounds by choosing

γ :=t

r(7.21)

66

7.2 Analysis of the sparse tensor stochastic Galerkin method

and we obtain for the error eL the estimate

eL = ‖uM − uM‖L2ρ(Γ;H1

0 (D)) ≤ C(r, t)L2−Lt‖uM‖Ar(H1+t∩H10 (D)) (7.22)

with a constant possibly depending on the stochastic approximation rate r.

Since the cardinality of the detail spacesWΓl1

, WDl2

both satisfy the assumptions of Lemma

4.2.1, setting bΓ = 2γ and bD = 2d, we arrive at our main result on convergence ratesfor the sGFEM discretization of sPDEs.

Proposition 7.2.1. Let the solution uM to the model problem (4.5) satisfy

uM ∈ Ar((H1+t ∩H1

0 )(D)) for some 0 < r < s− 3

2, 0 < t ≤ p (7.23)

with s > 32 as in (2.25). Let uM denote the sparse tensor approximation to the problem

(4.25) with the sparse tensor product spaces sequence V ΓL ⊗V D

L , defined in (4.24).

Then, there exists a constant C > 0, independent of M and L, such that

‖uM − uM‖L2ρ(Γ;H1

0 (D)) ≤ CL1+β(NL)−β‖u‖Ar(H1+t∩H10 (D)) (7.24)

where NL := dim(V ΓL ⊗V D

L ) and β = min(r, t/d).

We can see from the above definition of β that by choosing p = ⌈rd⌉ as the polynomialdegree in the spatial approximation space (5.2), we can obtain the maximum possiblerate r, provided that uM is sufficiently regular.

Remark 7.2.2 (full tensor product). The main motivation for using the sparse tensorproduct (4.24) between hierarchic sequences of spatial and stochastic discretizationspaces lies in the reduction of degrees of freedom to O(Lθ max(dim(V Γ

L ),dim(V DL ))),

shown in Lemma 4.2.1, as opposed to O(dim(V ΓL )× dim(V D

L )) in a full tensor approach.Using the approximation results (7.19) and (5.6), one can derive the full tensor errorestimate

‖uM − PΓL ⊗ PDL uM‖ ≤ ‖(Id − PΓ

L ) ⊗ (Id − PDL )uM‖ + ‖(Id − PΓL ) ⊗ IduM‖

+ ‖(Id ⊗ (Id − PDL )uM‖≤ (C(r, t)2−2Lmin(γr,t) + C(t)2−Lt

+ C(r)2−Lγr)‖uM‖Ar((H10∩Ht+1)(D)).

Choosing, as in (7.21), γ = t/r, yields

‖uM − PΓL ⊗ PDL uM‖ ≤ C(r, t)2−Lt‖uM‖Ar((H1

0∩Ht+1)(D)), t ∈ [0, p].

67


Since the total number of degrees of freedom in the full tensor case equals NL = 2(γ+d)L =2(t/r+d)L, we obtain the full tensor estimate

‖uM − PDL ⊗ PΓLuM‖ ≤ C(NL)−β‖u‖Ar((H1

0∩Ht+1)(D)), t ∈ [0, p]. (7.25)

with rate β = (d/t + 1/r)−1 and where PDL and PΓL denote the H1

0 - and L2-projectionsonto the spatial and stochastic discretization spaces, respectively.

A proof of the regularity (7.23) for p = 1 is given in [12].


To prove Proposition 7.1.2, we will have to provide a bound on the coefficients uν(x) in(7.8) in both, the Legendre and the Hermite case. To this end, we will essentially use theanalyticity property of the solution, cf. Assumption 3.2.8, and therefore first extend theparametric formulation of our problem (4.36) to complex valued arguments. We thenestablish some results on approximation of sequences of such coefficients and completethe section by the actual proof of Proposition 7.1.2.

7.3.1 Analytic continuations

Recall the definition of Σ(Γm, τm) from (3.9) and define the Cartesian product domain

Σ(Γ, τ) =M×m=1

Σ(Γm, τm) ⊂ CM , (7.26)

where τ = (τ1, . . . , τM ) ∈ RM .

We adopt the notation from Section 3.2.5 and write y∗m = (y1, ..., ym−1, ym+1, ...) and

y = (y∗m, ym) ∈ Γ∗

m×Γm if the dependence of y-dependent quantities on the coordinateym is to be emphasized. Likewise, we denote by z = (z∗m, zm) ∈ Σ(Γ, τ) = Σ(Γ∗

m, τ∗m) ×

Σ(Γm, τm) the corresponding partitioning of the complex vector z. Recall the definitionof the real-valued parametric bilinear form bM (y; ·, ·) from (4.37), which we will nextextend to complex-valued arguments: abusing notation, we denote by V D = H1

0 (D) alsothe space of complex-valued functions which belong to H1

0 (D) and define, for z ∈ Σ(Γ, τ)the complex extension of the (truncated) diffusion coefficient aM (2.11),(2.28) by either

aM (z,x) = Ea(x) +

M∑

m=1

√

λmϕm(x)zm (7.27)

68


in the case of bounded domains Γm, or by

aM (z,x) = eEloga (x)+

PMm=1

√λmϕm(x)zm (7.28)

in the case of unbounded domains. For u, v ∈ V D the sesquilinear extension of BM (y; ·, ·)is then given by

BM (z;u, v) =

∫

DaM (z, x)∇u · ∇v dx. (7.29)

The complex valued parametric problem is then given by

−div(aM (z,x)∇uM (z,x)) = f(x) in D,uM (z,x)|x∈∂D = 0,

(7.30)

and, in weak form: Find uM ∈ L2(Σ(Γ, τ);H10 (D)) s.t.

BM (z;uM , v) =

∫

Dfv dx ∀v ∈ H1

0 (D) ∀z ∈ Σ(Γ, τ). (7.31)

We will now prove that for certain r = (r1, . . . , rM ), problem (4.5) with bM as in (7.31)is uniquely solvable in Σ(Γ, r) and its solution is ‘coordinate-wise’ analytic. We startwith the case of bounded domains Γm:

Lemma 7.3.1. Let aM be given as in (7.27). Define the vector of radii r(δ)by

rm(δ) :=τm

C(δ)m1+δ, where 0 < τm <

a∗min√λm‖ϕm‖L∞(D)

(7.32)

and

C(δ) :=∑

m≥1

1

m1+δ, a∗min = min

x∈DEa(x) − a∗, (7.33)

with a∗ given in (3.13). Then, for any z ∈ Σ(Γ, r(δ)) with δ > 0, (4.5) admits a uniquesolution uM ∈ L2(Σ(Γ, r(δ));H1

0 (D)). Moreover, uM (z∗m, zm, ·) : Σ(Γ, r(δ)) → H10 (D)

is, for fixed coordinates z∗m ∈ Σ(Γ∗m, r

∗m(δ)), a H1

0 (D)-valued analytic function of zm ∈Σ(Γm, rm(δ))

Proof. To prove the existence and uniqueness of a solution to (7.31), we show that thereal part Re a(z,x) of the random field, as in (7.27), with z ∈ Σ(Γ, r(δ)), is boundedaway from zero:

Re aM (z,x) = Ea(x) +∑

m≥1

√

λmϕm(x)Rezm

≥ Ea(x) −∑

m≥1

bm|zm|

69


≥ Ea(x) −∑

m≥1

bma∗min

C(δ)bmm1+δ

≥ minx∈D

Ea(x) − amin =: a∗min > 0,

where bm := ‖ψm‖L∞(D). In a similar fashion we can conclude that

|aM (z,x)| ≤ a∗max := ‖Ea‖L∞(D) + a∗min

for z ∈ Σ(Γ, r(δ)) and x ∈ D. It follows that ∀ z ∈ Σ(Γ, r(δ)) and ∀u, v ∈ H10 (D)

ReB(z;u, u) ≥ 1

a∗min

‖u‖2H1

0 (D) and |B(z;u, v)| ≤ a∗max‖u‖H10‖v‖H1

0. (7.34)

Hence, the unique solvability of (4.5) for all z ∈ Σ(Γ, r(δ)) follows by Lax-Milgram.Choosing v = uM (z, ·) in (7.31), we find with (7.34) that

a∗min‖uM (z, ·)‖2H1

0≤ |B(z;uM (z, ·), uM (z, ·))| = |(f, uM (z, ·))| ≤ ‖f‖H−1‖uM (z, ·)‖H1

0,

from where it follows that

∀z ∈ Σ(Γ, r(δ)) : ‖uM (z, ·)‖H10≤ ‖f‖H−1

a∗min

(7.35)

and hence in particular that uM ∈ L2ρ(Γ;H1

0 (D)). We now establish analyticity withrespect to zm by a power series argument. The proof can either be done analogous to theone for Proposition 3.3.3 or, due to the linear dependence of aM on ym, also more directly.For reasons of completeness we will consider the latter option. Hence, we consider theparametric problem (7.30) for parameters (z∗m, ym) ∈ Σ(Γ∗

m, r∗m(δ))×Γm. Differentiating

(7.30) with respect to the real-valued parameter ym, we obtain that for any ν ∈ N, z∗m ∈Σ(Γ∗

m, r∗m(δ)) and every v ∈ H1

0 (D), the partial derivative ∂νymuM ((z∗m, ym); ·) ∈ H1

0 (D)solves the problem:

B((z∗m, ym); ∂νymu, v) = −ν

∫

D

√

λmϕm(x)∇(∂ν−1ym

uM ) · ∇v dx ∀v ∈ H10 (D).

Choosing here v = ∂νymuM (z∗m, ym, ·) ∈ H1

0 (D) gives

a∗min‖∂νymuM (z∗m, ym, ·)‖2

H10≤ |B((z∗m, ym); ∂νym

uM , ∂νymuM )|

= ν

∣∣∣∣

∫

D

√

λmϕm(x)∇(∂ν−1ym

uM (z∗m, ym,x)

×∇∂νymuM (z∗m, ym,x) dx

∣∣∣∣

70


≤ ν√

λm‖ϕm‖L∞(D)‖∂ν−1ym

uM (z∗m, ym, ·)‖H10

× ‖∂νymuM (z∗m, ym, ·)‖H1

0,

whence we obtain, for any z∗m ∈ Σ(Γ∗m, r

∗m(δ)) and any m, ν ∈ N,

‖∂νymuM (z∗m, ym, ·)‖H1

0≤ ν

bma∗min

‖∂ν−1ym

uM (z∗m, ym, ·)‖H10.

Iterating this estimate gives with (7.35)

‖∂νymuM (z∗m, ym, ·)‖H1

0≤ ν!

(bma∗min

)ν

‖uM (z∗m, ym, ·)‖H10

≤ ν!

a∗min

(bma∗min

)ν

‖f‖H−1 . (7.36)

Now define, for arbitrary m ∈ N and arbitrary, but fixed ym ∈ Γm, the formal powerseries Σ(Γm, rm(δ)) ∋ zm → uM (z∗m, zm, ·) ∈ H1

0 (D) by

uM (z∗m, zm, ·) =

∞∑

k=0

(zm − ym)k

k!∂kym

uM (z∗m, ym, ·). (7.37)

To estimate its convergence radius, we use (7.36):

‖uM (z∗m, zm, ·)‖H10≤

∞∑

k=0

|zm − ym|kk!

‖∂kymuM (z∗m, ym, ·)‖H1

0

≤ ‖f‖H−1

a∗min

∞∑

k=0

(

bm|zm − ym|a∗min

)k

.

This infinite sum, and hence the series (7.37), converges in H10 (D) if

bm|zm − ym|a∗min

< 1.

Since ym ∈ Γm was arbitrary, the series (7.37) converges for any ym ∈ Γm = [−1, 1] and,by a continuation argument, for any zm ∈ C satisfying dist(zm,Γm) < a∗min/bm. Thiscompletes the proof.

We now prove an analogous result in the case of lognormal coefficients (7.28).

71


Lemma 7.3.2. Let aM be given as in (7.28). Define the vector of radii r(δ)by

rm(δ) :=πτm

2C(δ)m1+δ, where 0 < τm <

1

2√λm‖ϕm‖L∞(D)

(7.38)

and

C(δ) :=∑

m≥1

1

m1+δ.

Then, for any δ > 0, (4.5) admits a unique solution uM ∈ L2G(Σ(Γ, r(δ)),H1

0 (D)), whereG is the standard Gauss measure on R as constructed in Section 2.2.4. Moreover, thesolution uM (z∗m, zm, ·) : Σ(Γ, r(δ)) → H1

0 (D) is, for fixed coordinates z∗m ∈ Σ(Γ∗m, r

∗m(δ)),

a H10 (D)-valued analytic function of zm ∈ Σ(Γm, rm(δ))

Proof. In the case of unbounded domains, we have to show that the real part Re aM of therandom field given in (7.28) can be bounded from below by some function a∗(z) > 0. Theexistence and uniqueness of a solution to (7.31) in L2

G(Σ(Γ, r(δ)),H10 (D)) then follows

essentially as in Remark 3.3.1. Denoting zm = ym + iwm we have

aM (z,x) = exp

Eloga (x) +

∑

m≥1

√

λmϕm(x)zm

= exp

Eloga (x) +

∑

m≥1

√

λmϕm(x)ymei

P

m≥1

√λmϕm(x)wm

.

It follows

Re aM (z,x) = exp

Eloga (x) +

∑

m≥1

√

λmϕm(x)ym cos

∑

m≥1

√

λmϕm(x)wm

.

Hence, to ensure that the real part of aM is positive, we require∣∣∣∣∣∣

∑

m≥1

√

λmϕm(x)wm

∣∣∣∣∣∣

≤ |bmrm(δ)| < π

2

which follows from (7.38). A lower bound for ReaM is then given, as in (3.22), by

a∗(z) = exp

min Eloga (x) −

∑

m≥1

‖ψm‖L∞(D)|Re zm|

,

where zm = ym + iwm, |wm| ≤ rm(δ) and existence and uniqueness follow by an appli-cation of the Lax-Milgram Lemma as in Remark 3.3.1. The proof of analyticity then isanalogous to the one given for Proposition 3.3.3.

72


Now we can prove analyticity of Σ(Γ, r(δ)) ∋ z → u(z, ·) ∈ H10 (D).

Theorem 7.3.3. Given the assumptions of Lemma 7.3.1 and 7.3.2, respectively, for anyM ∈ N, the solution uM to (4.5) is, as a function of z, analytic in the complex domainΣ(Γ, r(δ)) with r(δ) defined as in (7.32) or (7.38), respectively.

Proof. This is a direct consequence of Hartogs’ Theorem, see e.g. [32], Theorem 2.2.8,stating that coordinate-wise analyticity implies global analyticity.

This enables us in the following sections to find bounds on the Legendre and Hermite‘coefficients’ uν ∈ H1

0 (D) due to Cauchy’s integral formula for analytic functions ofseveral variables ([32], Theorem 2.2.1).

7.3.2 Bounds on the Legendre coefficients

Lemma 7.3.4. If uM solves the parametric deterministic problem (4.5) and is expandedin a Legendre chaos series as in (7.8) with Pν = Lν, then, for any δ1 > 0, the Legendrecoefficients satisfy

‖uν‖H10 (D) ≤ C

∏

m∈supp(ν)

2(2νm + 1)

η(δ1)−ν‖uM‖L∞(Σ(Γ,r);H10 (D)), (7.39)

where C > 0 is independent of M,ν with η(δ1) = (η1(δ1), . . . , ηM (δ1)) ∈ RM and

ηm(δ) := rm(δ) +√

1 + rm(δ)2 (7.40)

and rm(δ) as in (7.32).

Proof. Note that for notational convenience we do not indicate in the following thedependence of η on δ1. The H1

0 (D)-valued Legendre ‘coefficients’ are given by

uν =

(M∏

m=1

2νm + 1

)∫

ΓuM (y)Lν(y)ρ(y) dy, (7.41)

where the integral is a Bochner integral w.r.t. the probability measure dρ(y) = ρ(y)dyand the equality has to be read in H1

0 (D).

In the following, to keep the results general, we will denote by W(D) an arbitraryBanach space. Moreover, we will use the abbreviations S = supp(ν) ⊂ 1, . . . ,Mand S = 1, . . . ,M \ S (omitting the dependence on ν for reasons of readability ofthe formulae below) and, for a set G ⊂ 1, . . . ,M, we will write ΓG =

∏

m∈G Γm

73


and by yG ∈ ΓG the respective extraction from y. Since, by Theorem 7.3.3, uM isanalytic in Σ(Γ, r(δ1)) for any δ1 > 0, we may use Cauchy’s integral formula alongES =

∏

m∈supp(ν) Em, where Em denotes the (Bernstein-)ellipse in Σ(Γm, rm(δ1)) withfoci at ±1 and the sum of the semiaxes

ηm = rm(δ1) +√

1 + rm(δ1)2 > 1, (7.42)

to rewrite the Legendre coefficients (7.41) as

uν =

(∏

m∈S

2νm + 1

2πi

)∫

ΓLν(yS)

∮

ES

uM (zS ,yS)

(zS − yS)1dzS ρ(y)dy

=

(∏

m∈S

2νm + 1

2πi

)∫

ΓS

∮

ES

uM (zS ,yS)

∫

ΓS

Lν(yS)

(zS − yS)1ρ(y) dyS dzSdyS (7.43)

By [58] §4.9, the innermost integral in (7.43) is a representation for the Legendre poly-nomials of the second kind,

Qν(zS) =

∫

ΓS

Lν(yS)

(zS − yS)1ρS(yS) dyS . (7.44)

Furthermore, if, for m ∈ S, we substitute zm = 12(wm+w−1

m ) (Joukowski transformation)with |wm| = ηm, the Legendre polynomials of the second kind can be expanded like (cf.[16], Lemma 12.4.6)

Qνm(1

2(wm + w−1

m )) =

∞∑

k=νm+1

qνmk

wkm

with |qνmk| ≤ π. Hence,

|Qν(zS)| ≤∏

m∈S

∞∑

k=νm+1

π

ηkm=∏

m∈Sπη−νm−1m

1 − η−1m. (7.45)

Using Bochner’s Theorem 1.3.1 together with (7.45) and (7.43), we obtain

‖uν‖W(D) =

∥∥∥∥∥

(∏

m∈S

2νm + 1

2π

)∫

ΓS

∮

ES

uM (zS ,yS)|Qν(zS)| dzS ρS(yS)dyS

∥∥∥∥∥W(D)

≤(∏

m∈S

2νm + 1

2πi

)∫

ΓS

∮

ES

‖uM (zS ,yS)‖W(D)Qν(zS) dzS ρS(yS)dyS

≤(∏

m∈S

2νm + 1

2πLen(Em)

)

‖uM (z)‖L∞(ES×ΓS ;W(D)) maxzS∈ES

|Qν |

74


≤(∏

m∈S

2νm + 1

2πLen(Em)

)∏

m∈Sπη−νm−1m

1 − η−1m

‖uM (z)‖L∞(ES×ΓS ;W(D))

≤ C

(∏

m∈S2(2νm + 1)

)

η−ν‖uM (z)‖L∞(ES×ΓS ;W(D)) (7.46)

where the last estimate holds due to the fact that Len(Em) ≤ 4ηm. The constant is givenby C =

∏Mm=1

11−η−1

m.

Remark 7.3.5. Note that the constant C is bounded independently of M , since the ηm’sare summable and hence the product remains finite even if M → ∞.

Under the assumption that the covariance Va ∈ Ht,t(D×D) with t > d/2, it follows fromCorollary 2.2.8 that for δ1 > 0 sufficiently small there exists a constant C(s, d, δ1) > 0such that

∀m ∈ N : η−1m ≤ C1(s, d, δ1)m

−(s−1−δ1) (7.47)

where s is defined in (2.26). Since, furthermore, for any δ2 > 0 there exists a constantC2(δ2) s.t. 2νm + 1 ≤ C2(δ2)η

δ2νmm , it follows from the definition of ηm together with

(7.47)

Corollary 7.3.6. Given the assumptions of Lemma 7.3.4. For any 0 < t < s− 1 thereexists a constant C3(t, d) > 0, independent of m,M, ν s.t. by defining

η−νmm :=

1 νm = 0

C3(t, d)η−(1−δ2)νmm νm > 0

(7.48)

the Legendre coefficients satisfy

‖uν‖H10 (D) ≤ C

∏

m∈supp(ν)

η−νmm (7.49)

where the constant C > 0 is as in Lemma 7.3.4 and, by Remark 7.3.5, does not dependon M,ν.

Note that it follows from (7.47), that η−νmm → 0 if either m or νm tend to infinity, hence

the product in (7.49) remains finite even if |supp(ν)| → ∞.

7.3.3 Bounds on the Hermite coefficients

Lemma 7.3.7. If uM ∈ C0χ(Γ,H

10 (D)) solves the parametric deterministic problem (4.5)

and is expanded in a Hermite chaos series as in (7.8) with Pν = Hν, then, for any δ1 > 0the Hermite coefficients satisfy

‖uν‖H10 (D) ≤ CC(α)η(δ1)

−ν‖uM‖C0χ(Σ;H1

0 ), (7.50)

75


where C > 0 is independent of M,ν, α,

C(α) =

M∏

m=1

eα2

m2

(

1 + Erf

(αm√

2

))

, (7.51)

and αm given by (3.7) withη(δ1) := r(δ) (7.52)

and rm(δ) as in (7.38).

Remark 7.3.8. From Section 3.3.3, together with (3.42) and Corollary 2.2.8, we havethat αm ∼ m−s. Hence, if s > 1

2 (entailing Va ∈ Ht,t(D × D) with t > 2d) then theproduct in (7.51) is finite, independent of M .

Proof of Lemma 7.3.7. We will use the same notations and abbreviations as in the proofof 7.3.4. The H1

0 -valued Hermite ‘coefficients’ are given by

uν =1

ν!

∫

ΓuM (y)Hν(y)ρ(y) dy, (7.53)

where ρm denotes the standard Gauss probability density function ρm(ym) = 1√2πe−y

2m/2.

By Cm ⊂ Σ(R, ηm), we denote the circle centered at the origin with radius ηm andby CS =

∏

m∈supp(ν)Cm. Using Rodrigues’ formula (6.34) together with integration byparts and Cauchy’s integral formula, we can rewrite the Hermite coefficients (7.53) as

uν =(−1)|ν|

ν!

∫

ΓuM (y)∂νρ(y) dy =

1

ν!

∫

Γ(∂νuM (y))ρ(y) dy

=

(

(−1)|ν|

(2πi)|ν|0

)∫

Γ

∮

CS

uM (zS + yS,yS)

(zS)ν+1dzSρ(y) dy. (7.54)

By Bochner’s Theorem 1.3.1 we then obtain

‖uν‖W(D) ≤‖u‖C0

χ(ΣS×ΓS ;H10 )

(2π)|ν|0

∫

Γ

∮

CS

∏

m∈S eαm|Re zm|

ην+1dzS

×M∏

m=1

1√2πe−y

2m/2+αm|ym| dy

Since supzm∈Cm|Re zm| ≤ ηm and

1√2π

∫

R

e−12y2m+αm|ym| dym = e

12α2

m

(

1 + Erf

(αm√

2

))

,

76


we obtain

‖uν‖W(D) ≤ C(α)η(δ1)−ν(∏

m∈Seαmηm

)

‖uM‖C0χ(Σ;W(D))

with C(α) as in (7.51). From Section 3.3.3 and (3.42) we have that αm ∼ τ−1m . On the

other hand, by (7.52) and (7.38) we have ηm ∼ τmm1+δ , hence

∏

m∈Seαmηm ≤ exp(c1

∑

m∈S

1

m1+δ) ≤ exp(c1C(δ)) := C

with C(δ) as in (7.33). Hence, C is independent of |ν|0 and M . This completes theproof.

Under the assumption that the covariance Va is at least in Ht,t(D × D) with t > 2d,it follows from Corollary 2.2.8 that for δ1 > 0 sufficiently small there exists a constantC(s, d, δ1) > 0 such that

∀m ∈ N : η−1m ≤ C1(s, d, δ1)m

−(s−1−δ1) (7.55)

where s is defined in (2.26).

7.3.4 τ-summability of the PC coefficients

To obtain best N -term approximation rates, suitable regularity of the function underconsideration is required [18]. This means in our case that the norms of the Legendreor Hermite coefficients uν have to be τ -summable.

Lemma 7.3.9. Let Va ∈ Ht,t(D ×D) with t > 3d. If uM solves (4.5) and is expandedin a Legendre or Hermite series as in (7.8), then for any τ > 1

s−1 , where s is given by

(2.26), then ‖uν‖H10 (D) : ν ∈ N

M0 ∈ ℓτ (NM

0 ) and there holds:

‖‖uν‖W(D) : ν ∈ NM0 ‖τ

ℓτ (NM0 )

=∑

ν∈NM0

‖uν‖τW(D) ≤ C exp

(C3‖η−1‖τ

ℓτ (NM0 )

(1 − η−τ1 )

)

(7.56)

with C = C(τ, d) > 0 independent of M , C3 as in (7.48) in the Legendre case or C3 = 1in the Hermite case and with ηm defined as in (7.40).

Proof. From (7.47) or (7.55), respectively, it follows that ‖(η−1m )m‖ℓτ <∞ if τ > 1

s−1 .

77


In the Legendre case, using (7.48) together with the summation formula for geometricseries, we obtain for any finite M <∞ that

∑

ν∈NM0

‖uν‖τH10 (D) ≤ C

M∑

m=1

∞∑

νm=0

M∏

m=1

η−νmτm

= C

M∏

m=1

(

1 +C3η

−τm

1 − η−τm

)

≤ Ce

C3

1−C3η−τ1

PMm=1 η

−τm

,

where we used 1 + x ≤ ex if x ≥ 0. Letting M → ∞ gives (7.56).

In the Hermite case, we obtain by a similar argument, but due to slightly better bounds(7.50)

∑

ν∈NM0

‖uν‖τH10 (D) ≤ C

M∑

m=1

∞∑

νm=0

M∏

m=1

η−νmτm ≤ Ce

1

1−η−τ1

PMm=1 η

−τm

.

7.3.5 Proof of Proposition 7.1.2

Recall the definitions of Π(Λ;W(D)), σN and Ar(W(D)) from (7.13)-(7.15) with W(D)denoting a Banach space, e.g. H1

0 (D) in our examples. The weak-ℓτ (N) spaces ℓwτ (N)consist of all sequences (fk)k∈N for which it holds

|(fk)k|ℓwτ (N) := supk≥1

k1/τf∗k <∞ (7.57)

where (f∗k )k denotes the decreasing rearrangement of (|fk|)k. Since NM0 is a countable

set of indices, there exists a bijection ı : N → NM0 , and we can identify

uk = uı(k), u∗k = u∗ı(k), ∀k ∈ N.

Where appropriate, we will switch between these two representations of the series ukk∈N

and uνν∈NM0

, respectively. Since ℓτ (NM0 ) ⊂ ℓwτ (NM

0 ), this enables us now to character-

ize the space Ar(W(D)): In fact, a generalization of a well-known result of nonlinearapproximation theory to Bochner spaces (see e.g. [18], Theorem 4) claims that a func-tion uM ∈ L2(Γ;W(D)) belongs to Ar(W(D)), if and only if the W(D)-norm of itspolynomial chaos coefficients uν in (7.8) form a sequence in ℓwτ (NM

0 ) for τ := (r + 12)−1.

In fact, by virtue of the bijection ı, denote by (u∗k)k∈N the decreasing rearrangement of(‖uν‖W(D))ν . Due to

|u|2Ar(W(D)) = supN≥1

N2r∑

k>N

‖u∗k‖2W(D)

78


(7.57)

≤ |(‖uν‖W(D))ν |2ℓwτ (NM0 )

supN≥1

N2r∑

k>N

k−2r−1

≤ |(‖uν‖W(D))ν |2ℓwτ (NM0 )

supN≥1

∫ ∞

N

N2r

x2r+1dx

≤ |(‖uν‖W(D))ν |2ℓwτ (NM0 )

1

2r + 1

and

(u∗2k−1)2 ≤ (u∗2k)

2 ≤ 1

k

2k∑

l=k+1

(u∗k)2 ≤ 1

kσk(u)

2 ≤ |u|2Ar(W(D))k−2r−1 (7.58)

we have that

|u|Ar(W(D)) . 2τ |(‖uν‖W(D))ν |ℓwτ (NM0 ) .

2

2r + 1|u|Ar(W(D))

Since Lemma 7.3.9 ensures that the Legendre coefficients of the solution uM belong toℓwτ (NM

0 ) for τ > (s − 1)−1, it follows that the solution uM is in Ar(W(D)) for τ =(r + 1/2)−1 > (s − 1)−1 and hence for 0 < r < s − 3/2. Proposition 7.1.2 then followsfrom the definition of Ar(W(D)) (7.15).

79


80

8 Implementation and numerical examples

In this chapter, we will discuss some of the issues regarding the implementation ofthe sparse tensor stochastic Galerkin and stochastic collocation method and providenumerical examples. First, we will summarize the stochastic Galerkin and stochasticcollocation algorithms and explain some of the algorithmic technicalities which are non-standard. In particular we will describe the heuristic algorithm to find a quasi best-N -term approximation of the solution in the random parameter space. We will thenprovide numerical examples for the sparse tensor Galerkin and collocation algorithmsand verify the theoretical results provided in Chapters 5-7.

8.1 Implementational aspects

8.1.1 Algorithms

Given the truncation level M of the Karhunen-Loeve expansion, a discretization levelL and known statistics Ea(x) (2.1) and Va(x,x

′) (2.2) of the input random field a, wepropose the following algorithms

Algorithm 8.1.1. (sGFEM algorithm) .

1. Compute eigenpairs (λm, ϕm), m = 1, ...,M of the covariance operator Va (2.5).

2. Compute the wavelet basis ΨL in physical domain D up to level L as constructedin Section 5.1.

3. Choose the stochastic discretization parameter γ as in (7.21) and determine theindex set Λγ(L) of the largest PC coefficients uν .Compute the Galerkin stiffness matrix and load vector w.r.t. the sparse tensorproduct space V Γ

L ⊗V DL .

4. Solve the resulting system of linear equations by a preconditioned CG-algorithm.

5. Calculate the required statistical data from the solution (postprocessing).

Algorithm 8.1.2. (sCFEM algorithm) .

81


1. Compute eigenpairs (λm, ϕm), m = 1, ...,M of the covariance operator Va(2.5).

2. Compute the wavelet basis ΨL up to level L as constructed in Section 5.1.

3. Choose the stochastic discretization parameter γ as in (6.29) and compute theSmolyak collocation points yj based on the ⌈γL⌉ Gauss nodes in each stochasticdimension

4. Solve (4.37) for each collocation point yj by finite element projection onto thecorresponding detail space WD

l2(4.43).

5. Calculate the required statistical data from the set of solutions (postprocessing).

8.1.2 KL-eigenpair computation

For special covariances, like exponential ones, and simple domains D, closed-form so-lutions to (2.6) can be obtained by separation of variables, [27]. For more generalcorrelation functions and arbitrary domains D, however, the KL-eigenpairs have to becomputed numerically. The eigenproblem (2.6) can be formulated in variational formas: Find (λ,ϕ) ∈ R

+ × L2(D), ϕ 6= 0 s.t. ∀v ∈ L2(D)

∫

D

∫

DVa(x,x

′)ϕ(x′)v(x) dx′dx = λ

∫

Dϕ(x)v(x) dx. (8.1)

For the numerical examples following in the next section, we used a Galerkin discretiza-tion with piecewise linear finite element shape functions. The obtained generalized ma-trix eigenvalue problem was then solved by using JDBSYM, a Jacobi Davidson methodoptimized for large symmetric eigenproblems, see [26, 25] for details.

8.1.3 Localization of quasi-best-N-term gPC coefficients

The optimal index sets Λγ(l1) of ‘active’ gPC coefficients are in general not computation-ally available. However, in the following we will present a heuristic strategy to localizeindex sets Λγ(l1) which, in numerical experiments, turn out to be quasi-optimal. Theiridentification is based on the observation, proved in Section 7.3, that the decay of theKL coefficients of the input random field determines the decay of the coefficients uν inthe expansion (7.8) of the random solution which are to be approximately computed bythe stochastic Galerkin algorithm. Precisely, by Lemma 7.3.4 and 7.3.7, respectively, wehave that

‖uν‖W(D) ≤ Cη−ν :=∏

m≥1

η−νmm

82

8.1 Implementational aspects

where η−1m . ‖ψm‖L∞m1+δ and, by Corollary 2.2.8, η−1

m . m−s+(1+δ), provided thecovariance Va is in Ht,t(D ×D) with t > 3d and s given by (2.26). Hence, the heuristicindex sets Λγ(l1) consist simply of the ⌈2γl1⌉ largest upper bounds η−ν , i.e.

Λγ(l1) := arg maxΛ⊂NM

0

|Λ|=⌈2γl1⌉

(∑

ν∈Λ

η−ν)

⊂ NN

0 , l1 = 0, 1, 2, ... (8.2)

It has been shown in [8] that the computation of the index sets Λγ(L) can be done linearin time and memory requirement, w.r.t. their size.

8.1.4 Postprocessing

Since the goal of our computations is the approximation of the mean value and possiblyhigher order moments of the solution uM , we have to compute expected values of theform E[uM ], E[u2

M ] etc. out of the PC expansion (7.8) or the set of collocated solutionsu(yj,x).

sCFEM postprocessing

For the stochastic collocation method, we obtain by means of (6.8)

E[IΓl1u]

=∑

0≤|k|≤l1(−1)l1−|k|

(M − 1

l1 − |k|

)

E

[

I(1)k1

⊗ · · · ⊗ I(M)kM

u]

,

E[IΓl1u

2]

=∑

0≤|k|≤l1(−1)l1−|k|

(M − 1

l1 − |k|

)

E

[

I(1)k1

⊗ · · · ⊗ I(M)kM

u2] (8.3)

etc., where, according to (6.4)

E[I(1)k1

⊗ · · · ⊗ I(M)kM

up] =M∑

m=1

µm(km)∑

jm=0

wju(yj,x)p

and wj =∏Mm=1 wjm denotes the product of the weights of the Gaussian quadrature

formula of order µm in each direction.

83


sGFEM postprocessing

In order to extract statistical quantities from the sGFEM solution

uM (y,x) =∑

ν∈Λγ(l1)

uν(x)Pν(y),

we use the orthogonality of the Legendre and Hermite polynomials, to obtain for themean value

E[uM ] = u0(x),

where 0 =

M times︷︸︸︷

(0, 0, . . . , 0).Since the variance can be expressed as Var[uM ] = E[u2

M ] − E[uM ]2, we obtain the for-mula

Var[uM ](x) =∑

ν∈Λγ(l1)\0ζνuν(x)2,

where ζν denotes the L2ρ(Γ)-norm of the polynomials Pν , i.e.

ζν =

∏

m≥11

2νm+1 Pν = Lνν! Pν = Hν

8.2 Numerical examples

The main components of the sparse stochastic Galerkin and sparse stochastic collo-cation algorithms are the hierarchic discretizations in space and random parameterdomain. In the following we will therefore first provide numerical examples for thesemi-discretizations in D and the random parameter space Γ, and then for the sparsestochastic Galerkin and collocation algorithms, where we will in particular compare theresults to the respective full tensor algorithm.

8.2.1 Wavelet discretization

To illustrate the convergence and the condition of the computed Galerkin stiffness ma-trix w.r.t. wavelet discretization, we consider the following, purely deterministic modelproblem

−div(ex1∇u(x1, x2)) = ex1π cos

(3π

2x2

)(

cos(πx1) −13π

4sin(πx1)

)

(8.4)

84


−1−0.5

00.5

1

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

u(x1, x2)

Figure 8.1: The solution u(x1, x2) of (8.4) obtained on level l2 = 2 of the waveletdiscretization

on D = [−1, 1]2 with zero Dirichlet boundary conditions. The stiffness matrix wascomputed by a generic finite element code, assembling the matrix w.r.t. the hat basisfunctions on the finest mesh Tl2, in conjunction with a wavelet transform. The solutionto (8.4) is u(x1, x2) = sin(πx1) cos

(3π2 x2

). The initial mesh T0 consists of 200 congruent

triangular elements. Figure 8.1 shows the solution obtained after refining the initialmesh twice, i.e. on T2.

Figure 8.2 shows the convergence rate of the wavelet discretization computed up to levell2 = 6, in agreement with (5.7), as well as a comparison of the number of CG-iterationsused on each level by a hat function and wavelet discretization, respectively.

8.2.2 Smolyak interpolation

This subsection is concerned with the Smolyak interpolation operator given by (6.7).Motivated by the zero-dimensional problem, i.e. (4.4) without dependence on the spatialvariable x (see also [17, 9]),

a(y)u(y) = 1 (8.5)

with a(y) = γ0 +∑M

m=1 γmym and γm ∈ R, we consider the problem of computing theexpectation E[u] of the solution to (8.5), i.e. the integral

I =

∫

Γ

ρ(y)dy

γ0 +∑M

m=1 γmym(8.6)

85


102

103

104

105

106

10−5

10−4

10−3

10−2

10−1

100

0 1 2 3 4 5 610

1

102

103

104

waveletshat functions

L2−errorH

01−error

N−1

N−0.5

relative approximation error number of CG-iterations

NDl2

level l2

Figure 8.2: Left: Convergence rate of the wavelet discretization of (8.4) computed up tolevel l2 = 6 w.r.t. the number of wavelets ND

l2. Right: Comparison between

the number of CG-iterations used on each level.

by a Smolyak cubature based on Legendre abscissae. Here, we choose M = 20 andfor m ≥ 1 we set γm =

√λm‖ϕm‖L∞(D) where (λm, ϕm) are the largest M eigenpairs

associated to the Gaussian covariance Va(x,x′) = e−‖x−x′‖2

, see (2.6). Furthermore weset Γm = [−1, 1] and ρ to be the product of the uniform probability densities ρm(ym) = 1

2on Γm.

The expected rate of convergence, as stated Lemma 6.2.3 and Remark 6.2.4, is stronglydependent on the smallest region of analyticity Σ(Γm, rm). Therefore, in our case, raisingthe constant γ0 enlarges the domains of analyticity, due to a∗ ∼ γ0 and (7.32), andshould provide better convergence rates. This can clearly be seen in Figure 8.3 where weconsidered various choices of γ0 and where we observe superalgebraic convergence in allcases. The convergence rates have been estimated between two consecutive collocationlevels l1, l1 + 1 under the assumption that ǫl1 :=

∣∣I − E[IΓ

l11a ]∣∣ ∼

(NΓl1

)−r, where NΓ

l1

denotes the number of collocation points at level l1 and E[IΓl1

1a ] is computed as in (8.3).

Then, the estimated rate r of the convergence is computed by

r := − log(ǫl1+1/ǫl1)

log(Nl1+1/Nl1).

As a second example for the Smolyak procedure, we consider the case of Gaussian randomvariables, hence the Smolyak algorithm is based on the zeros of Hermite polynomials.Motivated again by the zero-dimensional problem (8.5) with the lognormal coefficienta = exp(

∑Mm=1 γmym), we compute the integral

I =

∫

Γ

ρ(y)dy

ePM

m=1 γmym(8.7)

86


100

102

104

106

108

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

102

104

106

108

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

γ0 = 7

γ0 = 8

γ0 = 9

γ0 = 7

γ0 = 8

γ0 = 9

relative error of Smolyak cubature estimated algebraic convergence rate

NΓl1

NΓl1

Figure 8.3: Left: Error ǫl1 = |I−E[IΓl1

1a ]| of the Smolyak cubature computing the integral

in (8.6). Right: Estimated algebraic convergence rate r. Both w.r.t. thenumber of collocation points NΓ

l1.

where Γ = (−∞,∞)M and ρ(y) denotes the product of the probability density functionsof the standard Gaussian measure. Since the integrand is an entire function, we expectexponential convergence w.r.t the number of collocation pointsNΓ

l1, which can also clearly

be seen in Figure 8.4 for different choices of the dimension M as indicated. Under the

assumption that ǫl1 :=∣∣I − E[IΓ

l11a ]∣∣ ∼ exp

(

−r 2M+1

√

NΓl1

)

, the rate r has been estimated

by r between two consecutive collocation levels as

r := − log(ǫl1+1/ǫl1)

2M+1

√

NΓl1+1 − 2M+1

√

NΓl1

.

8.2.3 Sparse tensor stochastic collocation method

To illustrate the sparse tensor stochastic collocation method, we consider the followingexample problem

−div(a(y,x)∇u(y,x)) = f(x) D,u(y,x)|x∈∂D = 0,

P−a.e. y ∈ Γ (8.8)

where D = [−1, 1]2 and the ym are uniformly distributed on Γm = [−1, 1]. The diffusioncoefficient a is given as a truncated Karhunen-Loeve expansion

a(y,x) = aM (y,x) = Ea(x) +

M∑

m=0

√

λmϕm(x)ym (8.9)

87


100

102

104

106

108

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

100

102

104

106

108

5

10

15

20

25

30

35

40

45

50

M = 10

M = 20

M = 40

M = 10

M = 20

M = 40

relative error of Smolyak cubature estimated exponential convergence rate

NΓl1

NΓl1

Figure 8.4: Left: Error ǫl1 = |I−E[IΓl1

1a ]| of the Smolyak cubature computing the integral

in (8.7). Right: Estimated exponential convergence rate r. Both w.r.t. thenumber of collocation points NΓ

l1.

with Ea(x) = 8+sin(π(x+y)) and (λm, ϕm) being the largest M computed eigenpairs ofthe eigenvalue problem (2.6) defined by the covariance Va(x,x

′) = e−‖x−x′‖2. Finally, we

set the source term f(x) = 2ex+2y. Figure 8.5 shows the obtained solution for M = 20and on level L = 2 of the refinement.

In Figure 8.6 we compare the number of total degrees of freedom NL and the L2(D)-convergence of the mean E[u] in the sparse stochastic collocation method (4.43) with theexpected respective results obtained in the ‘full method’ (4.42) for various choices of Mas indicated. Since the Smolyak procedure exhibits a faster than algebraic convergence,we expect a rate of −1

2 of the sparse tensor collocation algorithm, coinciding with therate of the spatial approximations by wavelets, indicated in Figure 8.6.

8.2.4 Sparse tensor stochastic Galerkin method

Again we consider a problem of the form (8.8) on the unit square D = [−1, 1]2 with adiffusion coefficient a given by a Karhunen-Loeve expansion as in (8.9). To verify theconvergence result provided by Proposition 7.2.1, we choose the kernel of the Wiener

process as the underlying covariance i.e. Va(x, x′) = min(x,x′)+1

2 , where x ∈ [−1, 1]. Theeigenvalue problem (2.6) w.r.t. this kernel has the exact solution

λm =8

π2(2m− 1)2, ϕm(x) = sin

(

x+ 1√

2λm

)

,

88


−1−0.5

00.5

1

−1

−0.5

0

0.5

10

0.05

0.1

0.15

0.2

−1−0.5

00.5

1

−1

−0.5

0

0.5

10

0.2

0.4

0.6

0.8

1

x 10−4

E[u] Var[u]

Figure 8.5: Mean and variance of the solution u to the sPDE described in Section 8.2.3

0 1 2 3 4 5 6 7 810

2

104

106

108

1010

1012

1014

1016

1018

M=10 sparse

M=10 full

M=20 sparse

M=20 full

M=40 sparse

M=40 full

100

102

104

106

108

10−3

10−2

10−1

N−0.5

NL rel. error of E[u]

L NL

Figure 8.6: Numerical experiment as described in Section 8.2.3. Left: Comparison be-tween the total number of DoF NL in the sparse and full stochastic colloca-tion method w.r.t. the level L. Right: relative error of the mean E[u] w.r.t.NL.

89


hence the eigenvalues are algebraically decaying with rate 2. We then choose λm =

λ5/2m , such that

√λm‖ϕm‖L∞(D) exhibits an algebraic decay with rate s = 5

2 , which, byProposition 7.1.2, in turn implies a stochastic rate of r = 1. Furthermore, we assumeEa(x) = x1 + 5 and f ≡ 1. Using piecewise linear wavelets in space corresponds to aspatial approximation rate t

d = 12 , provided the solution is accordingly regular. Hence,

by Proposition 7.2.1, we expect a sparse tensor rate of β = 12 while as in the full tensor

case we expect a rate β = 13 , see Remark 7.2.2. Those rates are exactly retrieved by

the numerical experiment as observed in Figure 8.7. There, we plot the relative error ofthe computed solutions in the L2

ρ(Γ;H10 (D))- and L2

ρ(Γ;L2(D))-norm, respectively, theexpected convergence of the Monte Carlo method and the estimated order of convergence,computed between consecutive levels and plotted w.r.t. the total number of degrees offreedom NL. To estimate the order of convergence we assumed

‖uM‖2L2

ρ(Γ;H10 (D)) − ‖uM‖2

L2ρ(Γ;H1

0 (D)) ∼ dim(V ΓL ⊗V D

L )−2β

and‖uM‖2

L2ρ(Γ;H1

0 (D)) − ‖uM‖2L2

ρ(Γ;H10 (D)) ∼ dim(V Γ

L ⊗ V DL )−2β,

where uM is the exact solution to (4.5), uM the discrete solution to (4.25) w.r.t thesparse tensor product space V Γ

L ⊗V DL and uM the discrete solution to (4.22) w.r.t the

full tensor product space V ΓL ⊗ V D

L . The three unknown quantities in each relation, i.e.the solution uM , the rates β and β, respectively and the constant are then fitted usingdiscrete solutions on three consecutive levels.

90


101

102

103

104

105

106

10−5

10−4

10−3

10−2

10−1

100

102

103

104

105

106

0

0.1

0.2

0.3

0.4

0.5

MC L2(H1)

FTG L2(H1)

STG L2(H1)

FTG L2(L2)

STG L2(L2)

STG L2(H1)

FTG L2(H1)

rel. error eoc

NLNL

Figure 8.7: Numerical experiment as described in Section 8.2.4. Left: Convergence of thesolutions computed by a sparse tensor Galerkin (STG), full tensor Galerkin(FTG) and Monte Carlo (MC) method. Right: Corresponding estimatedorders of convergence. Both plotted versus the total number of degrees offreedom NL in the full and sparse tensor case, respectively.

91


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Curriculum Vitae

Personal details

Name Marcel Bieri

Date of birth January 7, 1979

Place of birth Basel, Switzerland

Citizenship Switzerland

Education

07/2005–09/2009 PhD studies in Mathematics at ETH Zurich

Zurich, Switzerland

09/2004 Diploma in Mathematics at ETH Zurich

10/1999–09/2004 Studies in Mathematics at ETH Zurich

Zurich, Switzerland

12/1998 Matur at Gymnasium Muttenz

08/1995–12/1998 Secondary school at Gymnasium Muttenz

Muttenz, Switzerland

99

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