Topic: Inverse Problems for Partial Differential Equations
The theory of inverse problems, which underpins a broad spectrum of contemporary scientific and technological developments, deals with situations where the ``hidden'' causes of
an observed or desired effect are to be resolved. Applications include medical imaging, seismic exploration, non-destructive testing, radar and sonar technologies, communication
theory, signal processing and machine learning. Mathematically formulated often such problems amount to reconstruct the coefficients in a differential equation from partial
information on its solution. The solution of these inverse problems lead to nonlinear and severely ill-posed equations, and as such they employ sophisticated analytical tools
from functional and complex analysis, partial differential equations and probability as well as algorithm development and computations. On the other hand the field of inverse
problems was only able to grow to maturity with the development of new methods that were unique to the field itself. An important class of inverse problems for partial differential
equations that I am interested in, is inverse scattering theory. Although the basic mathematical model of the forward scattering problem is deceptively simple, inverse scattering
continues to perplex and challenge mathematicians in diverse areas of mathematics.
The reading course on this subject will be designed as a self-contained and up-to-date discussion of the mathematical methods in inverse problems for partial differential equations
and will outline a variety of profitable directions for future research.
Some useful references:
1. A. Kirsch, An Introduction to the Mathematics Theory of Inverse Problems, Springer (2011), http://www.springer.com/us/book/9781441984739
2. F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF, SIAM Publications 88 (2016),
http://epubs.siam.org/doi/book/10.1137/1.9781611974461