Multilevel Monte Carlo for Bayesian Inverse Problems

Abstract

Introduction In recent years, various methods have been developed for solving parametric operator equations, focusing on the estimation of parameters given measurements of the parametric solution, subject to a stochastic observation error model. The Bayesian approach [7] to such inverse problems for PDEs will be considered here and solved using adaptive, deterministic sparse tensor Smolyak quadrature schemes from [4, 5]. Multiple solutions of the Bayesian inverse problem based on different measurements are often averaged using a standard Monte Carlo approach. We develop a multilevel Monte Carlo method achieving an error of the same order while requiring less work [1, 2, 3].