Multilevel Monte Carlo FEM for Elliptic PDEs with Besov Random Tree Priors

TL;DR

This paper introduces a multilevel Monte Carlo finite element method for elliptic PDEs with Besov-tree priors, enabling efficient modeling of fractal phenomena with rigorous convergence and complexity analysis.

Contribution

It develops a novel MLMC-FEM algorithm for elliptic PDEs with Besov-tree random coefficients, including detailed convergence and complexity analysis.

Findings

Optimal convergence rates are established for the method.

Complexity estimates demonstrate efficiency for fractal coefficient modeling.

The approach handles nonuniform bounds and low regularity of coefficients.

Abstract

We develop a multilevel Monte Carlo (MLMC)-FEM algorithm for linear, elliptic diffusion problems in polytopal domain $\mathcal D\subset \mathbb R^d$, with Besov-tree random coefficients. This is to say that the logarithms of the diffusion coefficients are sampled from so-called Besov-tree priors, which have recently been proposed to model data for fractal phenomena in science and engineering. Numerical analysis of the fully discrete FEM for the elliptic PDE includes quadrature approximation and must account for a) nonuniform pathwise upper and lower coefficient bounds, and for b) low path-regularity of the Besov-tree coefficients. Admissible non-parametric random coefficients correspond to random functions exhibiting singularities on random fractals with tunable fractal dimension, but involve no a-priori specification of the fractal geometry of singular supports of sample paths. Optimal complexity and convergence rate estimates for quantities of interest and for their second moments are proved. A convergence analysis for MLMC-FEM is performed which yields choices of the algorithmic steering parameters for efficient implementation. A complexity (``error vs work'') analysis of the MLMC-FEM approximations is provided.