# Advanced Multilevel Monte Carlo Methods

## TL;DR

This paper reviews advanced Multilevel Monte Carlo techniques, focusing on their application with Markov chain and sequential Monte Carlo methods when exact sampling is not feasible, highlighting strategies to reduce computational costs.

## Contribution

It introduces strategies for applying MLMC to MCMC and SMC methods without requiring exact sampling, expanding MLMC's applicability.

## Key findings

- Strategies for MLMC in MCMC and SMC without exact sampling
- Reduction in computational cost for biased expectation estimation
- Enhanced applicability of MLMC in complex probabilistic models

## Abstract

This article reviews the application of advanced Monte Carlo techniques in the context of Multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations which can be biased in some sense, for instance, by using the discretization of a associated probability law. The MLMC approach works with a hierarchy of biased approximations which become progressively more accurate and more expensive. Using a telescoping representation of the most accurate approximation, the method is able to reduce the computational cost for a given level of error versus i.i.d. sampling from this latter approximation. All of these ideas originated for cases where exact sampling from couples in the hierarchy is possible. This article considers the case where such exact sampling is not currently possible. We consider Markov chain Monte Carlo and sequential Monte Carlo methods which have been introduced in the literature and we describe different strategies which facilitate the application of MLMC within these methods.