Analyticity and Sparsity in Uncertainty Quantification for Pdes with Gaussian Random Field Inputs

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About this book 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.

STOCHAST PARTIAL DIFFERENT EQUATION ADDITIVE GAUSSIAN NOISE

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The stochastic partial differential equations (SPDEs) arise in many applications of the probability theory. This monograph will focus on two particular (and probably the most known) equations: the stochastic heat equation and the stochastic wave equation. The focus is on the relationship between the solutions to the SPDEs and the fractional Brownian motion (and related processes). An important point of the analysis is the study of the asymptotic behavior of the p-variations of the solutions to the heat or wave equations driven by space-time Gaussian noise or by a Gaussian noise with a non-trivial correlation in space. The book is addressed to public with a reasonable background in probability theory. The idea is to keep it self-contained and avoid using of complex techniques. We also chose to insist on the basic properties of the random noise and to detail the construction of the Wiener integration with respect to them. The intention is to present the proofs complete and detailed. Sample Chapter(s) Preface Chapter 1: Gaussian processes Contents: Gaussian Processes and Sheets: Gaussian Processes Fractional and Bifractional Brownian Motion Multiparameter Gaussian Processes Isonormal Processes and Wiener Integral Stochastic Heat and Wave Equations with Additive Gaussian Noise: The Stochastic Heat Equation with Space-Time White Noise The Stochastic Heat Equation with Correlated Noise in Space The Stochastic Wave Equation with Space-Time White Noise Power Variation and Statistical Inference for Solutions to SPDEs: Variations of the Solution to the Stochastic Heat Equation Parameter Estimation for the Stochastic Heat Equation via Power Variations Power Variations and Inference for Stochastic the Wave Equation Readership: Undergraduate, graduate students and researchers working in various areas of probability theory and mathematical statistics.