Parallel Multiscale Gauss-Newton-Krylov Methods for Inverse Wave Propagation

One of the outstanding challenges of computational science and engineering is large-scale nonlinear parameter estimation of systems governed by partial differential equations. These are known as inverse problems, in contradistinction to the forward problems that usually characterize large-scale simulation. Inverse problems are significantly more difficult to solve than forward problems, due to ill-posedness, large dense ill-conditioned operators, multiple minima, space-time coupling, and the need to solve the forward problem repeatedly. We present a parallel algorithm for inverse problems governed by time-dependent PDEs, and scalability results for an inverse wave propagation problem of determining the material field of an acoustic medium. The difficulties mentioned above are addressed through a combination of total variation regularization, preconditioned matrix-free Gauss-Newton-Krylov iteration, algorithmic checkpointing, and multiscale continuation. We are able to solve a synthetic inverse wave propagation problem though a pelvic bone geometry involving 2.1 million inversion parameters in 3 hours on 256 processors of the Terascale Computing System at the Pittsburgh Supercomputing Center.