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We present a randomized maximum a posteriori (rMAP) method for generatingapproximate samples of posteriors in high dimensional Bayesian inverse problemsgoverned by large-scale forward problems. We derive the rMAP approach by: 1)casting the problem of computing the MAP point as a stochastic optimizationproblem; 2) interchanging optimization and expectation; and 3) approximatingthe expectation with a Monte Carlo method. For a specific randomized data andprior mean, rMAP reduces to the maximum likelihood approach (RML). It can alsobe viewed as an iterative stochastic Newton method. An analysis of theconvergence of the rMAP samples is carried out for both linear and nonlinearinverse problems. Each rMAP sample requires solution of a PDE-constrainedoptimization problem; to solve these problems, we employ a state-of-the-arttrust region inexact Newton conjugate gradient method with sensitivity-basedwarm starts. An approximate Metropolization approach is presented to reduce thebias in rMAP samples. Various numerical methods will be presented todemonstrate the potential of the rMAP approach in posterior sampling ofnonlinear Bayesian inverse problems in high dimensions.

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