To model subsurface flow in uncertain heterogeneous or fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient—also called random field—may be used. In case of a one-dimensional parameter space, Lévy processes allow for jumps and display great flexibility in the distributions used. However, in various situations (e.g. microstructure modeling), a one-dimensional parameter space is not sufficient. Classical extensions of Lévy processes on two parameter dimensions suffer from the fact that they do not allow for spatial discontinuities see for example Barth and Stein (Stoch Part Differ Equ Anal Comput 6(2):286–334, 2018). In this paper a new subordination approach is employed see also Barth and Merkle (Subordinated gaussian random fields. ArXiv e-prints, arXiv:2012.06353math.PR, 2020) to generate Lévy-type discontinuous random fields on a two-dimensional spatial parameter domain. Existence and uniqueness of a (pathwise) solution to a general elliptic partial differential equation is proved and an approximation theory for the diffusion coefficient and the corresponding solution provided. Further, numerical examples using a Monte Carlo approach on a Finite Element discretization validate our theoretical results.
%0 Journal Article
%1 merkle2022subordinated
%A Merkle, Robin
%A Barth, Andrea
%D 2022
%I Springer
%J Stochastics and Partial Differential Equations: Analysis and Computations
%K EXC2075 PN5 PN5-4 curated
%P 819-867
%R 10.1007/s40072-022-00246-w
%T Subordinated Gaussian random fields in elliptic partial differential equations
%U https://doi.org/10.1007/s40072-022-00246-w
%V 11
%X To model subsurface flow in uncertain heterogeneous or fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient—also called random field—may be used. In case of a one-dimensional parameter space, Lévy processes allow for jumps and display great flexibility in the distributions used. However, in various situations (e.g. microstructure modeling), a one-dimensional parameter space is not sufficient. Classical extensions of Lévy processes on two parameter dimensions suffer from the fact that they do not allow for spatial discontinuities see for example Barth and Stein (Stoch Part Differ Equ Anal Comput 6(2):286–334, 2018). In this paper a new subordination approach is employed see also Barth and Merkle (Subordinated gaussian random fields. ArXiv e-prints, arXiv:2012.06353math.PR, 2020) to generate Lévy-type discontinuous random fields on a two-dimensional spatial parameter domain. Existence and uniqueness of a (pathwise) solution to a general elliptic partial differential equation is proved and an approximation theory for the diffusion coefficient and the corresponding solution provided. Further, numerical examples using a Monte Carlo approach on a Finite Element discretization validate our theoretical results.
@article{merkle2022subordinated,
abstract = {To model subsurface flow in uncertain heterogeneous or fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient—also called random field—may be used. In case of a one-dimensional parameter space, Lévy processes allow for jumps and display great flexibility in the distributions used. However, in various situations (e.g. microstructure modeling), a one-dimensional parameter space is not sufficient. Classical extensions of Lévy processes on two parameter dimensions suffer from the fact that they do not allow for spatial discontinuities [see for example Barth and Stein (Stoch Part Differ Equ Anal Comput 6(2):286–334, 2018)]. In this paper a new subordination approach is employed [see also Barth and Merkle (Subordinated gaussian random fields. ArXiv e-prints, arXiv:2012.06353[math.PR], 2020)] to generate Lévy-type discontinuous random fields on a two-dimensional spatial parameter domain. Existence and uniqueness of a (pathwise) solution to a general elliptic partial differential equation is proved and an approximation theory for the diffusion coefficient and the corresponding solution provided. Further, numerical examples using a Monte Carlo approach on a Finite Element discretization validate our theoretical results.},
added-at = {2023-12-01T18:18:47.000+0100},
affiliation = {Merkle, R (Corresponding Author), Univ Stuttgart, IANS SimTech, Stuttgart, Germany.
Merkle, Robin; Barth, Andrea, Univ Stuttgart, IANS SimTech, Stuttgart, Germany.},
author = {Merkle, Robin and Barth, Andrea},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2cf8cd98ec4537fbf0433d4d24faa8864/simtech},
doi = {10.1007/s40072-022-00246-w},
interhash = {1f78395d8065f9d0b9c98329d3ddc5ad},
intrahash = {cf8cd98ec4537fbf0433d4d24faa8864},
issn = {{2194-0401} and {2194-041X}},
journal = {Stochastics and Partial Differential Equations: Analysis and Computations},
keywords = {EXC2075 PN5 PN5-4 curated},
language = {eng},
pages = {819-867},
publisher = {Springer},
pubstate = {prepublished},
research-areas = {Mathematics},
timestamp = {2023-12-06T08:55:18.000+0100},
title = {Subordinated Gaussian random fields in elliptic partial differential equations},
unique-id = {WOS:000770498600001},
url = {https://doi.org/10.1007/s40072-022-00246-w},
volume = 11,
year = 2022
}
