Subordinated Gaussian random fields in elliptic partial differential equations

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.

On properties and applications of Gaussian subordinated Lévy fields

“…Assumptions of this type are natural and well known in different situations (see e.g. [10,33,13]). For simplicity, we consider centered GRFs in this subsection.…”

“…Assumtion 5.1 i is natural for GRFs and guarantees certain regularity properties for the paths of the GRF (see e.g. [10,33,13]). Equation (5.1) ensures that we can approximate the Lévy subordinators in an L s -sense.…”

“…Equation (5.1) ensures that we can approximate the Lévy subordinators in an L s -sense. This can be achieved under appropriate assumptions on the tails of the distribution of the Lévy processes, see [9, Assumption 3.6, Assumption 3.7, Theorem 3.21] and [33,Section 7]). There are several results in the direction of condition (5.2).…”

“…Such a construction yields an admissible approximation in the sense of Assumption 5.1 iii for many Lévy processes (see e.g. [33,Section 7] for the case of Poisson and Gamma processes).…”

“…Assumption 5.1 allows conclusions to be drawn on the spatial regularity of the GRF W . The next lemma on the approximation of the GRF W follows from the regularity properties of W (see [33,Lemma 4.4] and [13]).…”