MLMC techniques for discontinuous functions

TL;DR

This paper reviews techniques to improve Multilevel Monte Carlo methods when estimating expectations involving discontinuous functions, focusing on recent advances like branching diffusion and adaptive sampling.

Contribution

It provides a comprehensive review of methods addressing the challenges of discontinuous functions in MLMC, highlighting recent developments and their applications.

Findings

Branching diffusion techniques can mitigate variance decay issues.

Adaptive sampling improves estimation accuracy for discontinuous functions.

Review of literature offers insights into overcoming MLMC limitations.

Abstract

The Multilevel Monte Carlo (MLMC) approach usually works well when estimating the expected value of a quantity which is a Lipschitz function of intermediate quantities, but if it is a discontinuous function it can lead to a much slower decay in the variance of the MLMC correction. This article reviews the literature on techniques which can be used to overcome this challenge in a variety of different contexts, and discusses recent developments using either a branching diffusion or adaptive sampling.