Infinite-Dimensional Inverse Problems with Finite Measurements

Explore infinite-dimensional nonlinear inverse problems with finite measurements in this 57-minute SIAM-IS Virtual Seminar Series talk. Delve into uniqueness, stability, and reconstruction techniques, focusing on scenarios where the unknown lies in or is well-approximated by finite-dimensional subspaces or submanifolds. Discover the interplay between applied harmonic analysis, sampling theory, compressed sensing, machine learning, and inverse problem theory for partial differential equations. Examine practical examples, including the Calderón problem and scattering. Learn about Lipschitz stability with linear subspaces, modeling finite measurements, and compressed sensing for inverse problems in PDEs. Investigate low-dimensional manifolds, Hölder-Lipschitz stability, and discrete and continuous generator structures. Gain insights into sufficient conditions for injectivity and the main result on Lipschitz stability with finite measurements.

Intro
The effect of instability
A general model for inverse problems
Obtaining stability with structured signals
Lipschitz stability with linear subspaces
Modeling the finite measurements
Example the Calderon problem
Example 2 inverse scattering
Classical compressed sensing
Compressed sensing for inverse problems in PDE
Low-dimensional manifolds
Holder-Lipschitz stability from an infinite number of measurements
Discrete and Continuous Generator structure in 1D
Sufficient conditions for injectivity
Lipschitz stability with finite measurements: main result