Subordinated Gaussian Random Fields in Elliptic Partial Differential Equations

Abstract

To model subsurface flow in uncertain heterogeneous\ 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 [11]). In this paper a new subordination approach is employed (see also [9]) 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.

Multilevel Monte Carlo estimators for elliptic PDEs with Lévy-type diffusion coefficient

“…In this subsection, we introduce the pathwise weak solution of problem (2.1) -(2.3) following [8]. We denote by H 1 (D) the Sobolev space on D equipped with the norm…”

“…Different subordinators display unique patterns in the discontinuities and have varied marginal distributions (see [7]). Existence and uniqueness of pathwise solutions to the problem was demonstrated in [8]. Spatial regularity of the solution depends on the subordinated Gaussian random field which itself depends on the subordinator.…”

“…The discontinuities in the spatial domain pose additional difficulty in the pathwise discretization. A sample-adapted approach was considered in [8], but is limited to certain subordinators. Here we investigate not only the limitations of a sample-adapted approach in multilevel sampling, but also a Control Variates ansatz as presented first in [31].…”

“…In this section, we briefly introduce the general stochastic elliptic boundary value problem. For more details on the existence, uniqueness and measurability of the solution to the considered PDE, we refer the reader to [8] and [11]. For the rest of this paper we assume that a complete probability space (Ω, F, P) is given.…”

“…The following general assumptions ensure the well-posedness of the elliptic boundary value problem (see also [8,Assumption 2.2] and [11,Assumption 2.3]).…”