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.
Dinh Dũng is professor of applied mathematics at the Vietnam National University, Hanoi. He graduated from the Lomonosov Moscow State University in 1975 (former Soviet Union). There, he received the Ph.D. degree in Mathematics in 1979, and the Dr.Sc. degree in Mathematics in 1985. His research fields are approximation theory and numerical analysis. His recent research interests include computational uncertainty quantification for PDEs with random inputs, and high-dimensional problems: computation complexity, hyperbolic approximation, sampling recovery, numerical weighted integration, deep ReLU network approximation. He delivered plenary talks at the Third Asian Mathematical Conference, October 23-27, 2000, in Manila (Philippines), and at the Tenth Vietnam Mathematical Congress, August 08-12, 2023, Da Nang (Vietnam). He was awarded the Ta Quang Buu Prize for excellent achievement in Information and Computer Sciences in 2015.
Van Kien Nguyen is currently a lecturer in the Department of Mathematical Analysis, University of Transport and Communications, Hanoi, Vietnam. He graduated from University of Science, Vietnam National University Hanoi. He obtained his PhD in Mathematics from Friedrich Schiller University Jena, Germany. His areas of interest are function spaces, approximation theory, and numerical analysis.
Christoph Schwab is professor of mathematics at the Seminar for Applied Mathematics at ETH Zurich. His areas of research encompass Numerical Analysis of Partial Differential Equations, in particular Finite- and Boundary Element Methods, and the mathematical investigation of numerical methods for high-dimensional PDEs, with emphasis on forward and Bayesian inverse problems in numerical Uncertainty Quantification. His results are published in numerous articles in major journals in applied and computational mathematics. He was speaker at ICM2002, and PI of an ERC advanced grant.
Jakob Zech currently holds a position as Assistant Professor for Machine Learning in Scientific Computing at Heidelberg University. His academic path started at TU Wien, where he earned his Bachelor's degree in 2012. He then continued his studies at ETH Zurich, completing his Masters in 2014 and his PhD in Mathematics in 2018. His PhD research revolved around the approximation of high-dimensional parametric PDEs. Upon obtaining his PhD, he received the Early PostdocMobility fellowship from the Swiss National Science Foundation and spent a year as a Postdoc at the Massachusetts Institute of Technology in 2019. Subsequently he joined Heidelberg University in April 2020. His research interests include high-dimensional approximation, deep learning theory, statistical inference, uncertainty quantification, and numerics of PDEs, which resulted in numerous publications in top-tier academic journals.
