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

Beschreibung

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, suchas 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.
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.
There is no similar text, at present, where sparsity forward and inverse UQ for these PDEs can be currently found About subsurface flow, linearly elastic deformations of random medium in solid mechanics, time-harmonic electromagnetics Applications to math. of computational PDE uncertainty quantification, comp. sci. and engineering, approximation theory

Autor*in

Dinh Dũng

Themen in »Analyticity and Sparsity in Uncertainty Quantification for PDEs with Gaussian Random Field Inputs«

Gaussian Measures Uncertainty Quantification Sparse-Grid Interpolation Smolyak Quadrature Finite Element Methods Parametric and Stochastic PDE Polynomial Chaos High-Dimensional Approximation Partial Differential Equations

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