Randomized algorithms for Bayesian Inverse Problems based on random Fourier features

Mentors:
Lead: Arvind K. Saibaba (Mathematics, NC State)
Collaborator: Julianne Chung (Mathematics, Emory University)

Intellectual merit and significance:
Inverse Problems use measurements to make inferences about parameters of interest. In this application, parameters refer to detailed spatial or spatiotemporal maps of quantities of interest such as atmospheric patterns of carbon dioxide and other greenhouse gases in atmospheric to-mography. The Bayesian approach to solving inverse problems can be used to produce reconstructions with quantified uncertainty. This project will adapt a technique used in data science, known as random Fourier features [1] to efficiently solve Bayesian inverse problems. Recent work by Saibaba and coauthors has shown the preliminary benefits of this approach on linear inverse problems. In this project, students will extend this approach to nonlinear inverse problems and inverse problems with piecewise constant reconstructions. This method is expected to be applicable to a wide range of
applications in materials sciences, environmental sciences, biomedical imaging, etc.

Objectives:
(i) Implementation of linear inverse problems, (ii) adapt to nonlinear inverse problems and problems with piecewise constant reconstructions, (iii) demonstrate performance on application problems, and (iv) integrate the implementations into a software package that is currently under
development.

Outcomes:
The students will submit a paper for publication and they will integrate their implementations into an existing software package.

References:

1. Cho, T., J. Chung, S. Miller, and AK. Saibaba. Computationally efficient methods for large- scale atmospheric inverse modeling. Geoscientific Model Development, 2022. 15.14: p. 5547-5565.