Abstract: Model-based projections of complex systems will play a central role in prediction and decision-making, e.g., anticipate ice sheet contribution to sea level rise. However, models are typically subject to uncertainties stemming from uncertain inputs in the model such as coefficient fields, source terms, etc. Refining our understanding of such models requires solving a (Bayesian) inverse problem that integrates measurement data and the model to estimate and quantify the uncertainties in the parameters. Such problems face several challenges, e.g., high dimensionality of the inversion parameters and expensive to evaluate models. In this talk, I will focus on identifying and exploiting low-dimensional structure in such inverse problems stemming for example from local sensitivity of the data with respect to parameters, diffusive models, sparse data, etc. The key idea is to look at the Hessian operator arising in these inverse problems, which often exhibits a high (global) rank but a lower off-diagonal rank structure and build approximations of this operator. These approximations then can be used as preconditioners for the Newton-Krylov systems encountered when computing the maximum a posteriori (MAP) point or to build high-quality proposals for Markov chain Monte Carlo algorithms to explore the full posterior of the Bayesian inverse problem. Combining an efficient matrix-free point spread function (PSF) method for approximating operators with fast hierarchical (H-)matrix methods, we show that Hessian information can be obtained using only a small number of operator applications.
Reference: https://epubs.siam.org/doi/abs/10.1137/23M1584745
Co-authors: Nick Alger (UT Austin), Tucker Hartland (LLNL), and Omar Ghattas (UT Austin)