A PDE-Constrained Optimization Approach to Uncertainty in Inverse Problems with Applications to Inverse Scattering

Abstract

Abstract : This project addresses the statistical inverse problem of reconstruction of an uncertain shape of a scatterer or properties of a medium from noisy observations of scattered wavefields. The Bayesian solution of this inverse problem yields a posterior pdf; requiring the solution of the forward wave equation to evaluate the density for any point in parameter space. The standard approach is to sample this pdf via an MCMC method and then compute statistics of the samples. However, standard MCMC methods view the underlying parameter-to-observable map as a black box, and thus do not exploit its structure, hence becoming prohibitive for high dimensional parameter spaces and expensive simulations. We have developed a Langevin-accelerated MCMC method for sampling high-dimensional PDE-based probability densities. The method builds on previous work in Langevin dynamics, which uses gradient information to guide the sampling in useful directions, improving convergence rates. We have extended the Langevin idea to exploit local Hessian information, leading to a stochastic version of Newton's method. We have also begun to analyze the spectral structure of the Hessian for inverse scattering problems. Applications to model inverse medium scattering problems indicate several orders of magnitude improvement over a reference black-box MCMC method.