Plenary Talk 3: October 2, Sunday 9:00am-10:00am, Click Hall, Alumni Center.
Title: The partnership of Bayesian inference and numerical analysis for the solution of inverse problems
Abstract: The numerical solution of inverse problems where the number of unknowns exceeds the available data is a notoriously difficult problem. Regularization methods designed to overcome the paucity of data penalize candidate solutions for unlikely or undesirable features. Discretization level of the underlying continuous problem can also be used to improve the accuracy of the computed solution. In this talk we show how recasting the inverse problems within the Bayesian framework makes it possible to express via a probability density function features believed to characterize the solution in a way that interfaces naturally with state of the art computational schemes. In that context, we will present an efficient computational scheme for the recovery of sparse solutions, where the sparsity is encoded in terms of hierarchical models whose parameters can be set to account for the sensitivity of the data to the solution. The computations can be organized as an inner-outer iteration scheme, where a weighted linear least squares problem is solved in the inner iteration and the outer iteration updates the scaling weights. When the least squares problems are solved approximately by the Conjugate Gradient method for least squares (CGLS) equipped with a suitable stopping rule, typically the number of CGLS iterations quickly converges to the cardinality of the support, thus providing an automatic model reduction. Computed examples will illustrate the performance of the approach in a number of applications.
Biography: Daniela Calvetti, the James Wood Williamson professor, is an applied mathematician whose work on inverse problems connects mathematical models, scientific computing, Bayesian inference and uncertainty quantification. After receiving her Laurea in Mathematics from the University of Bologna, Italy in 1980, she moved to the University of North Carolina at Chapel Hill, where she completed her PhD in 1989. After holding faculty positions at North Carolina State University, Colorado State University-Pueblo and the Stevens Institute of Technology, she moved to Case Western Reserve University in 1997, where she chaired the department from 2007 to 2013.
She has co authored three monographs, Introduction to Bayesian Scientific Computing, Computational Mathematical Modeling and Mathematics of Data Science, with two more coming soon, and approximately 150 peer reviewed papers. Her research has been supported by NSF, NIH and the Simons Foundation. She has graduated 20 PhD students and she has been in the editorial board of several scientific journals, including SIAM Journal on Matrix Analysis and its Applications, Mathematics of Computation, Inverse Problems and SIAM Review. She is currently the Program Director of the SIAM activity group on Uncertainty Quantification.
For more detail, please visit Professor Calvetti's Website: https://mathstats.case.edu/faculty/daniela-calvetti/