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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.
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