# Unbiased Multi-index Monte Carlo

## TL;DR

This paper presents a novel unbiased multi-index Monte Carlo method that reduces computational effort and removes bias in expectation approximations of discretized random variables, with applications in PDE solutions and Bayesian inference.

## Contribution

It introduces an unbiased multi-index Monte Carlo approach that improves efficiency and removes bias compared to traditional discretization sampling methods.

## Key findings

- Reduced computational effort for a given error level.
- Successfully applied to PDE solutions with random coefficients.
- Effective in Bayesian inference for stochastic PDEs.

## Abstract

We introduce a new class of Monte Carlo based approximations of expectations of random variables such that their laws are only available via certain discretizations. Sampling from the discretized versions of these laws can typically introduce a bias. In this paper, we show how to remove that bias, by introducing a new version of multi-index Monte Carlo (MIMC) that has the added advantage of reducing the computational effort, relative to i.i.d. sampling from the most precise discretization, for a given level of error. We cover extensions of results regarding variance and optimality criteria for the new approach. We apply the methodology to the problem of computing an unbiased mollified version of the solution of a partial differential equation with random coefficients. A second application concerns the Bayesian inference (the smoothing problem) of an infinite dimensional signal modelled by the solution of a stochastic partial differential equation that is observed on a discrete space grid and at discrete times. Both applications are complemented by numerical simulations.