Geometric Methods in Inverse Problems for Partial Differential Equations

Introduction

Inverse problems are a research field at the intersection of pure and applied mathematics. The goal in inverse problems is to recover information from indirect, incomplete, or noisy observations. The problems arise in medical and seismic imaging where measurements made on the exterior of a body are used to deduce the properties of the inaccessible interior. We use mathematical methods ranging from microlocal analysis of partial differential equations and metric geometry to stochastics and computational methods to solve these problems.

Project Focus

The focus of the project is on the inverse problems for non-linear partial differential equations. We attack these problems using a recent method that we developed originally for the geometric wave equation. This method uses the non-linear interaction of waves as a beneficial tool.

Achievements

Using this method, we have been able to solve inverse problems for non-linear equations for which the corresponding problem for linear equations is still unsolved. We study the determination of a Lorentzian space-time from scattering measurements and the lens rigidity conjecture.

Methodology

We use geometric methods, originally developed for General Relativity, to analyze waves in a moving medium and to develop methods for medical imaging. By applying Riemannian geometry and our results in invisibility cloaking, we study counterexamples for non-linear inverse problems and use transformation optics to construct scatterers with exotic properties.

Solution Algorithms

We also consider solution algorithms that combine the techniques used to prove uniqueness results for inverse problems, manifold learning, and operator recurrent networks.

Applications

Applications include:

1. New virus imaging methods using electron microscopy
2. Imaging of brains

Collaboration

Practical algorithms based on the results of the research will be developed in collaboration with scientists working in medical imaging, optics, and Earth sciences.