Multilevel Sequential Monte Carlo with Dimension-Independent Likelihood-Informed Proposals

TL;DR

This paper introduces a multilevel sequential Monte Carlo method with dimension-independent likelihood-informed proposals, optimized for infinite-dimensional Bayesian inverse problems, and demonstrates its efficiency on PDE and SDE examples.

Contribution

The paper develops a novel MLSMC algorithm with DILI proposals that do not require gradient evaluations, improving efficiency in high-dimensional Bayesian inference.

Findings

Achieves optimal $O(psilon^{-2})$ cost for mean-square error

Eliminates gradient computation by using empirical covariance in DILI proposals

Successfully applied to PDE-based Darcy flow and SDE path measure problems

Abstract

In this article we develop a new sequential Monte Carlo (SMC) method for multilevel (ML) Monte Carlo estimation. In particular, the method can be used to estimate expectations with respect to a target probability distribution over an infinite-dimensional and non-compact space as given, for example, by a Bayesian inverse problem with Gaussian random field prior. Under suitable assumptions the MLSMC method has the optimal $O(\epsilon^{-2})$ bound on the cost to obtain a mean-square error of $O(\epsilon^2)$. The algorithm is accelerated by dimension-independent likelihood-informed (DILI) proposals designed for Gaussian priors, leveraging a novel variation which uses empirical sample covariance information in lieu of Hessian information, hence eliminating the requirement for gradient evaluations. The efficiency of the algorithm is illustrated on two examples: inversion of noisy pressure measurements in a PDE model of Darcy flow to recover the posterior distribution of the permeability field, and inversion of noisy measurements of the solution of an SDE to recover the posterior path measure.