A weighted multilevel Monte Carlo method

TL;DR

This paper introduces a weighted multilevel Monte Carlo method that generalizes existing approaches by optimizing weights across multiple levels, leading to improved efficiency especially when coarse level correlations are weak.

Contribution

It extends the MLMC framework to incorporate weights for multiple levels and derives recursive formulas for optimal weights and sample sizes, enhancing efficiency.

Findings

Significant efficiency gains over standard MLMC in various numerical tests.

The weighted approach is especially beneficial when coarse level approximations are poorly correlated.

Asymptotic complexity remains unchanged, but practical performance improves.

Abstract

The Multilevel Monte Carlo (MLMC) method has been applied successfully in a wide range of settings since its first introduction by Giles (2008). When using only two levels, the method can be viewed as a kind of control-variate approach to reduce variance, as earlier proposed by Kebaier (2005). We introduce a generalization of the MLMC formulation by extending this control variate approach to any number of levels and deriving a recursive formula for computing the weights associated with the control variates and the optimal numbers of samples at the various levels. We also show how the generalisation can also be applied to the \emph{multi-index} MLMC method of Haji-Ali, Nobile, Tempone (2015), at the cost of solving a $(2^d-1)$-dimensional minimisation problem at each node when $d$ index dimensions are used. The comparative performance of the weighted MLMC method is illustrated in a range of numerical settings. While the addition of weights does not change the \emph{asymptotic} complexity of the method, the results show that significant efficiency improvements over the standard MLMC formulation are possible, particularly when the coarse level approximations are poorly correlated.