Adjoint-based Adaptive Multi-Level Monte Carlo for Differential Equations

TL;DR

This paper introduces an adaptive multi-level Monte Carlo method for differential equations that uses adjoint-based error analysis for mesh refinement and stopping criteria, improving accuracy over traditional methods.

Contribution

It develops two novel adaptive refinement strategies for MLMC that leverage adjoint problems and residuals for better error control and efficiency.

Findings

Enhanced accuracy with adaptive mesh refinement

Improved stopping criteria based on adjoint error analysis

Demonstrated efficiency gains over classical MLMC methods

Abstract

We present a multi-level Monte Carlo (MLMC) algorithm with adaptively refined meshes and accurately computed stopping-criteria utilizing adjoint-based a posteriori error analysis for differential equations. This is in contrast to classical MLMC algorithms that use either a hierarchy of uniform meshes or adaptively refined meshes based on Richardson extrapolation, and employ a stopping criteria that relies on assumptions on the convergence rate of the MLMC levels. This work develops two adaptive refinement strategies for the MLMC algorithm. These strategies are based on a decomposition of an error estimate of the MLMC bias and utilize variational analysis, adjoint problems and computable residuals.