Title: Unbiased Multi-index Monte Carlo,
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.