A PDE-Constrained Optimization Approach to Uncertainty Quantification in Inverse Problems, with Applications to Inverse Scattering
Abstract
This project addressed 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 probability of 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, becoming prohibitive for high dimensional parameter spaces and expensive simulations. A preconditioned Langevin-accelerated MCMC method for sampling high-dimensional PDE-based probability densities was developed. The preconditioner exploits local Hessian (of the negative log posterior) information to greatly speed up sampling, leading to a stochastic version of Newton's method. Fast Hessian approximations were developed for several inverse scattering problems. Applications to model inverse medium scattering problems indicated several orders of magnitude improvement over a reference black-box MCMC method.
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 28, 2010
- Accession Number
- ADA515611
Entities
People
- George Biros
- James S. Martin
- Jucas Wilcox
- Omar Ghattas
- Stephanie Chaillet
- Tan Bui-thanh
Organizations
- University of Texas at Austin