"Accelerating Bayesian Computation in Remote Sensing Problems"
Abstract: The Bayesian approach to inverse problems arising in imaging spectroscopy can quantify uncertainty in retrievals and help elucidate the value of different information sources, but it can be computationally intractable in practice. In many Bayesian inverse problems, however, there exists a low-dimensional likelihood-informed subspace that describes optimal projections of the directions in parameter space that are most informed by the data. We demonstrate how to exploit this subspace in a Markov chain Monte Carlo (MCMC) retrieval algorithm using multiple levels of forward model fidelity, with the goal of making MCMC sufficiently fast for operations.
"A data driven heuristic for rapid convergence of Scheduled Relaxation Jacobi schemes"
Abstract: The Scheduled Relaxation Jacobi (SRJ) method is a viable candidate as a high performance linear solver for elliptic partial differential equations (PDEs). The method greatly improves the convergence of the original Jacobi iteration method by applying a sequence of M judiciously chosen relaxation parameters in each cycle of the algorithm. In previous work, the relaxation factors associated with each of the M steps (which characterize an SRJ scheme) were derived to be specific to the problem of interest and its discretization. In this work, we develop a class of SRJ schemes which could be applied to solve any symmetric linear system as long as the original Jacobi iterative method would converge. Furthermore, we use data to train an algorithm to select which scheme to use at each cycle of the SRJ method for rapid convergence. Specifically, the algorithm is trained using convergence data obtained from randomly applying SRJ schemes to the 1D Poisson problem. The automatic selection heuristic that is developed based on this limited data is found to provide good convergence for a wide range of problems, and avoids the need for user exploration of different schemes or knowledge of the exact eigenvalue spectrum. Lastly, an extension of this method to nonsymmetric matrices arising from the discretization of non-elliptic PDEs is also discussed.