On the effective dimension and multilevel Monte Carlo

TL;DR

This paper introduces a multilevel Monte Carlo method for high-dimensional integration that achieves lower computational complexity compared to standard Monte Carlo, especially when the function has a low truncation dimension.

Contribution

It presents a novel multilevel Monte Carlo approach with improved complexity bounds for integrating functions over high-dimensional spaces.

Findings

Multilevel Monte Carlo reduces computational time for a given variance.

The method's complexity is $O(d+ ext{ln}(d)d_{t} ext{epsilon}^{-2})$, better than standard Monte Carlo.

A lower bound of $d+d_{t} ext{epsilon}^{-2}$ is established for certain multilevel methods.

Abstract

I consider the problem of integrating a function $f$ over the $d$-dimensional unit cube. I describe a multilevel Monte Carlo method that estimates the integral with variance at most $\epsilon^{2}$ in $O(d+\ln(d)d_{t}\epsilon^{-2})$ time, for $\epsilon>0$, where $d_{t}$ is the truncation dimension of $f$. In contrast, the standard Monte Carlo method typically achieves such variance in $O(d\epsilon^{-2})$ time. A lower bound of order $d+d_{t}\epsilon^{-2}$ is described for a class of multilevel Monte Carlo methods.