Hilbert-Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations
Journal article, 2023

Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert--Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations, which couple the regularity of the driving noise with the properties of the differential operator. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert--Schmidt embeddings of Sobolev spaces. Both non-homogenenous and homogeneous kernels are considered. Important examples of homogeneous kernels covered by the results of the paper include the class of Matérn kernels.

stochastic partial differential equations

integral operators

Hilbert-Schmidt operators

reproducing kernel Hilbert spaces

elliptic operators

Author

Mihaly Kovacs

University of Gothenburg

Chalmers, Mathematical Sciences

Budapest University of Technology and Economics

Pázmány Péter Catholic University

Annika Lang

University of Gothenburg

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Andreas Petersson

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

University of Gothenburg

Stochastic Analysis and Applications

0736-2994 (ISSN) 1532-9356 (eISSN)

Vol. 41 3 564-590

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Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis

Roots

Basic sciences

DOI

10.1080/07362994.2022.2053541

Related datasets

arXiv:2107.10104 [math.PR] [dataset] [dataset]

URI: https://arxiv.org/abs/2107.10104

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7/7/2023 6