Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs / Dinh Dũng, Van Kien Nguyen, Christoph Schwab, Jakob Zech.

Format:

Book

Author/Creator:

Dũng, Dinh, author.

Nguyen, Van Kien, author.

Schwab, Ch. (Christoph), author.

Zech, Jakob, author.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 0075-8434 volume 2334.

Lecture notes in mathematics, 0075-8434 ; volume 2334

Language:

English

Subjects (All):

Differential equations, Partial.

Gaussian processes.

Random fields.

Mathematical analysis.

Physical Description:

xv, 205 pages : illustrations ; 24 cm.

Place of Publication:

Cham, Switzerland : Springer, [2023]

Summary:

The present book develops the mathematical and numerical analysis of linear, elliptic and parabolic partial differential equations (PDEs) with coefficients whose logarithms are modelled as Gaussian random fields (GRFs), in polygonal and polyhedral physical domains. Both, forward and Bayesian inverse PDE problems subject to GRF priors are considered. Adopting a pathwise, affine-parametric representation of the GRFs, turns the random PDEs into equivalent, countably-parametric, deterministic PDEs, with nonuniform ellipticity constants. A detailed sparsity analysis of Wiener-Hermite polynomial chaos expansions of the corresponding parametric PDE solution families by analytic continuation into the complex domain is developed, in corner- and edge-weighted function spaces on the physical domain. The presented Algorithms and results are relevant for the mathematical analysis of many approximation methods for PDEs with GRF inputs, such as model order reduction, neural network and tensor-formatted surrogates of parametric solution families. They are expected to impact computational uncertainty quantification subject to GRF models of uncertainty in PDEs, and are of interest for researchers and graduate students in both, applied and computational mathematics, as well as in computational science and engineering.

Contents:

Intro

Preface

Acknowledgement

Contents

List of Symbols

List of Abbreviations

1 Introduction

1.1 An Example

1.2 Contributions

1.3 Scope of Results

1.4 Structure and Content of This Text

1.5 Notation and Conventions

2 Preliminaries

2.1 Finite Dimensional Gaussian Measures

2.1.1 Univariate Gaussian Measures

2.1.2 Multivariate Gaussian Measures

2.1.3 Hermite Polynomials

2.2 Gaussian Measures on Separable Locally Convex Spaces

2.2.1 Cylindrical Sets

2.2.2 Definition and Basic Properties of Gaussian Measures

2.3 Cameron-Martin Space

2.4 Gaussian Product Measures

2.5 Gaussian Series

2.5.1 Some Abstract Results

2.5.2 Karhunen-Loève Expansion

2.5.3 Multiresolution Representations of GRFs

2.5.4 Periodic Continuation of a Stationary GRF

2.5.5 Sampling Stationary GRFs

2.6 Finite Element Discretization

2.6.1 Function Spaces

2.6.2 Finite Element Interpolation

3 Elliptic Divergence-Form PDEs with Log-Gaussian Coefficient

3.1 Statement of the Problem and Well-Posedness

3.2 Lipschitz Continuous Dependence

3.3 Regularity of the Solution

3.4 Random Input Data

3.5 Parametric Deterministic Coefficient

3.5.1 Deterministic Countably Parametric Elliptic PDEs

3.5.2 Probabilistic Setting

3.5.3 Deterministic Complex-Parametric Elliptic PDEs

3.6 Analyticity and Sparsity

3.6.1 Parametric Holomorphy

3.6.2 Sparsity of Wiener-Hermite PC Expansion Coefficients

3.7 Parametric Hs(D)-Analyticity and Sparsity

3.7.1 Hs(D)-Analyticity

3.7.2 Sparsity of Wiener-Hermite PC Expansion Coefficients

3.8 Parametric Kondrat'ev Analyticity and Sparsity

3.8.1 Parametric Ks(D)-Analyticity.

Notes:

Includes bibliographical references (pages 199-204) and index.

ISBN:

3031383834

9783031383830

OCLC:

1385448228