Multilevel Monte Carlo estimators for elliptic PDEs with L\'evy-type diffusion coefficient

TL;DR

This paper develops and analyzes multilevel Monte Carlo finite element methods for solving elliptic PDEs with discontinuous Le9vy-type random diffusion coefficients, providing convergence proofs and numerical validation.

Contribution

It introduces novel MLMC estimators tailored for elliptic PDEs with Le9vy-type random coefficients, addressing discontinuities and distributional flexibility.

Findings

Proved convergence of standard MLMC estimator.

Validated effectiveness through numerical experiments.

Demonstrated applicability to complex random diffusion models.

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.