A randomized maximum a posterior method for posterior sampling of high dimensional nonlinear Bayesian inverse problems

TL;DR

This paper introduces a randomized MAP method for efficiently generating approximate posterior samples in high-dimensional Bayesian inverse problems, combining stochastic optimization, Monte Carlo approximation, and PDE-constrained optimization techniques.

Contribution

The paper develops a novel randomized MAP approach that simplifies posterior sampling in high-dimensional nonlinear inverse problems, with convergence analysis and bias reduction strategies.

Findings

rMAP reduces to RML for specific data and prior

Method effectively samples posteriors in high dimensions

Numerical results demonstrate potential in nonlinear problems

Abstract

We present a randomized maximum a posteriori (rMAP) method for generating approximate samples of posteriors in high dimensional Bayesian inverse problems governed by large-scale forward problems. We derive the rMAP approach by: 1) casting the problem of computing the MAP point as a stochastic optimization problem; 2) interchanging optimization and expectation; and 3) approximating the expectation with a Monte Carlo method. For a specific randomized data and prior mean, rMAP reduces to the maximum likelihood approach (RML). It can also be viewed as an iterative stochastic Newton method. An analysis of the convergence of the rMAP samples is carried out for both linear and nonlinear inverse problems. Each rMAP sample requires solution of a PDE-constrained optimization problem; to solve these problems, we employ a state-of-the-art trust region inexact Newton conjugate gradient method with sensitivity-based warm starts. An approximate Metropolization approach is presented to reduce the bias in rMAP samples. Various numerical methods will be presented to demonstrate the potential of the rMAP approach in posterior sampling of nonlinear Bayesian inverse problems in high dimensions.