Adaptive particle-based approximations of the Gibbs posterior for inverse problems

Abstract

In this work, we adopt a general framework based on the Gibbs posterior to update belief distributions for inverse problems governed by partial differential equations (PDEs). The Gibbs posterior formulation is a generalization of standard Bayesian inference that only relies on a loss function connecting the unknown parameters to the data. It is particularly useful when the true data generating mechanism (or noise distribution) is unknown or difficult to specify. The Gibbs posterior coincides with Bayesian updating when a true likelihood function is known and the loss function corresponds to the negative log-likelihood, yet provides subjective inference in more general settings. We employ a sequential Monte Carlo (SMC) approach to approximate the Gibbs posterior using particles. To manage the computational cost of propagating increasing numbers of particles through the loss function, we employ a recently developed local reduced basis method to build an efficient surrogate loss function that is used in the Gibbs update formula in place of the true loss. We derive error bounds for our approximation and propose an adaptive approach to construct the surrogate model in an efficient manner. We demonstrate the efficiency of our approach through several numerical examples.