Goal-Oriented Adaptive Modeling of Random Heterogeneous Media and Model-Based Multilevel Monte Carlo Methods

TL;DR

This paper introduces adaptive surrogate modeling techniques for two-phase random heterogeneous media, combined with model-based multilevel Monte Carlo methods to efficiently estimate quantities of interest with reduced computational cost.

Contribution

The paper develops novel adaptive surrogate modeling strategies integrated with mbMLMC to control errors and improve efficiency in stochastic PDE simulations of heterogeneous media.

Findings

Significant computational savings demonstrated in numerical experiments.

Adaptive surrogate models effectively control modeling errors.

mbMLMC accelerates convergence of Monte Carlo estimates.

Abstract

Methods for generating sequences of surrogates approximating fine scale models of two-phase random heterogeneous media are presented that are designed to adaptively control the modeling error in key quantities of interest (QoIs). For specificity, the base models considered involve stochastic partial differential equations characterizing, for example, steady-state heat conduction in random heterogeneous materials and stochastic elastostatics problems in linear elasticity. The adaptive process involves generating a sequence of surrogate models defined on a partition of the solution domain into regular subdomains and then, based on estimates of the error in the QoIs, assigning homogenized effective material properties to some subdomains and full random fine scale properties to others, to control the error so as to meet a preset tolerance. New model-based Multilevel Monte Carlo (mbMLMC) methods are presented that exploit the adaptive sequencing and are designed to reduce variances and thereby accelerate convergence of Monte Carlo sampling. Estimates of cost and mean squared error of the method are presented. The results of several numerical experiments are discussed that confirm that substantial saving in computer costs can be realized through the use of controlled surrogate models and the associated mbMLMC algorithms.