AIS - Inverse Problems (2022)

Venue: TIFR, Hybrid Mode

Dates: 4 Jul 2022 to 23 Jul 2022

Convener(s)

An inverse problem may be described as a problem of determination of model parameters of an object based on some measured or observed data from the exterior of the object. This requires sending probes (for example, X-rays,pressure waves, sound waves, electric current etc.), measuring the response at the boundary, and analyzing it to determine the object parameters. Solving an inverse problem involves the study of whether unique recovery is possible and if so, whether explicit inversion formulas can be given. Finally, one is interested in numerical simulation of these inversion formulas for potential real-world applications. In many situations, inverse problems are severely ill-posed, making the determination of coefficients a very challenging problem both from an analytical and numerical point of view.
Inverse problems arise in several applications, including, medical imaging, seismic imaging, radar imaging, remote sensing, sonar, signal processing, non-destructive material testing, astronomy, oceanography to name a few. A mathematical formulation of an inverse problem frequently leads to the determination of coefficients of a linear or non-linear partial differential equation (PDE) from boundary data or in the inversion of certain linear or non-linear integral transforms. For example, in X-ray imaging, we are interested in recovering a function whose line integrals are known. In thermoacoustic tomography, image reconstruction involves inversion of a spherical transform, that is, one has to recover a function knowing its integrals along a family of spheres. Depending on the imaging modality, one is interested in the inversion of these and other integral transforms. Electrical impedance tomography or seismic imaging are some imaging problems where recovery of coefficients of a PDE is involved from boundary measurements.

In this Advanced Instructional School, we plan to give an introduction to the main theoretical tools involved in the study of inverse problems. The topics we plan to cover are:

1. Introduction to Calderón inverse problem
2. Integral geometry in inverse problems
3. Unique continuation principle and Carleman estimates
4. Inverse problems for linear hyperbolic PDEs
5. An introduction to the mathematical theory of obstacle scattering
6. Microlocal analysis and inverse problems