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

Abstract

General elliptic equations with spatially discontinuous diffusion coefficients may be used as a simplified model for subsurface flow in heterogeneous or fractured porous media. In such a model, data sparsity and measurement errors are often taken into account by a randomization of the diffusion coefficient of the elliptic equation which reveals the necessity of the construction of flexible, spatially discontinuous random fields. Subordinated Gaussian random fields are random functions on higher dimensional parameter domains with discontinuous sample paths and great distributional flexibility. In the present work, we consider a random elliptic partial differential equation (PDE) where the discontinuous subordinated Gaussian random fields occur in the diffusion coefficient. Problem specific multilevel Monte Carlo (MLMC) Finite Element methods are constructed to approximate the mean of the solution to the random elliptic PDE. We prove a-priori convergence of a standard MLMC estimator and a modified MLMC—control variate estimator and validate our results in various numerical examples.

On properties and applications of Gaussian subordinated Lévy fields

“…This motivates the investigation of a general elliptic PDE where the GSLF occurs in the diffusion coefficient. We introduce the considered elliptic PDE in Subsection 8.1 following [31] and [33]. Spatial discretization methods are discussed in Subsection 8.2 and we conclude this section with numerical experiments in Subsection 8.3.…”

“…Next, we shortly introduce the pathwise weak solution of problem (8.1) -(8.3) following [33] and [31]. We denote by H 1 (D) the Sobolev space and by T the trace operator [16]).…”