Inverse problems are pivotal in a variety of applications, such as target identification, non-destructive testing, and parameter estimation. Among these, the inverse scattering problem in inhomogeneous media poses significant challenges, as it seeks to estimate unknown parameters from available measurement data. To understand the mathematics of machine learning approaches for inverse scattering, we develop a low rank structure based on the generalized prolate spheroidal wave functions. This framework allows us to establish rigorous stability estimates and to develop robust numerical algorithms that are effective even in the presence of noisy and large-scale measurement data.
The presentation will conclude with a discussion of current and future work involving low rank methods for inverse problems and partial differential equations.