BCAM-Severo Ochoa Course: Inverse problems

DATES: 17 January to 9 March 2023. Tuesdays and Thursdays (16 sessions)
TIME: 15:00-17:00 (total of 32 hours)
LOCATION: Maryam Mirzakhani Seminar Room at BCAM and Online

Announcement video here

Inaugural lecture:
Inverse problems: A new approach to PDE by Alberto Ruiz (UAM)

DESCRIPTION
The course is intended as an introduction to inverse problems arising in the analysis of partial differential equations. To explain what these key words mean, let us present a concrete example. In the figure 1 (see poster), we see how a ray of light is refracted when passing from the air to a plastic block. In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection is caused by the change of wave propagation speed. Since the velocity of propagation depends on the transmission medium, when the properties of the medium change the wave's speed also changes. In fact, the amount of refraction of a wave is determined by the properties of the transmission medium and the initial direction of wave propagation. This means that known the properties of the transmission medium, we can determine the change of velocity of any given wave passing through it. This question can be inverted as follows: knowing the change of velocity |when entering and leaving a medium| of any given wave, can we determine the properties of the transmission medium?

These two questions can be formulated mathematically. In the first one, the forward problem, we are given a partial differential equation where the coefficient (modelling the properties of the transmission medium) and the initial data are known, and we look for a solution (the wave propagation speed). In the second question, the inverse problem, we know the solution at the boundary of the medium for all initial data (the exit propagation speed of any wave passing through a medium), and we aim at determining the coefficient of the equation (which models the properties of the medium).

PROGRAM
The course will start recalling the basic properties of the Fourier transform together with the trace theorem. Then, we recall the boundary value problems of elliptic partial di erential equations (PDEs). We will continue with an introduction to scattering. After these prerequisites we can formulate the  rst inverse problems. At this point, the course will be focused in the so-called inverse Calder on problem. We will introduce the concept of complex geometrical optics solutions that is used to prove the uniqueness, stability and reconstruction of the Calder on problem. Finally, we will discuss the question of regularity. Our goal there will be just to understand why the problem becomes harder in less regular settings.

An overview of the contents is as follows:
1. Basics of the Fourier transform. The trace theorem.
2. Basics of boundary value problem for elliptic PDEs.
3. Introduction to scattering theory.
4. Introduction to inverse boundary value problems and inverse
1. scattering problems.
5. The inverse Calder on problem.
6. Complex geometrical optics.
7. Comments on the problem of regularity.

*Registration is free, but mandatory before 4 January 2023 To sign-up go to form link and fill the registration form.

• BCAM-SO_Course_2023_01_17.Pdf