We will give a brief introduction to inverse problems and their applications.

Inverse problems is that branch of mathematics usually devoted to study physical applications governed by an incomplete mathematical model. The missing components of this latest are rather inaccessible/unknown entities whose identification leads in practice to a better understanding of the ongoing physical phenomena. For this purpose, identifiability is a sensitive study point that aims to determine what kind of local observations and where should they be taken in order to ensure uniqueness of the sought mathematical components. In this talk, we feature a constructive approach based on developing appropriate adjoint functions that leads to define reciprocity gaps fulfilled by the unknown missing components. Then, appending these functions by adequate boundary conditions enables to fix the identifiability issue as well as to establish a constructive identification method for some inverse problems. We present the exploration of this approach to three physical applications.

Subdiffusion is one transport mechanism found in nature, in which the mean squared particle displacement grows only sublinearly with the time. Mathematically these processes are described a diffusion type model involving a Caputo fractional derivative in time, which is nonlocal operator in time and can capture the memory phenomenon. In this talk, we discuss two inverse problems in subdiffusion, i.e., backward problem and point source identification, and illustrate the beneficial effect of the memory phenomenon for unique recovery. We shall discuss the uniqueness issue and numerical reconstructions.

TBA

In the talk we revisit the problem of unique continuation of a finite sum of eigenfunctions of an elliptic operator in a divergence form from an open set to the whole domain. We show that the problem is of hyperbolic type for a single eigenfunction, and of elliptic type for sums that have two and more eigenfunctions. This inverse problem can be found in many engineering applications and is also related to control theory for the heat equation.

We deal with the Data Completion Poisson problem, also called the Cauchy’s problem. The special point is that: on a portion of the boundary, Neumann and Dirichlet conditions are given, while the complementary part, no conditions, no data are available. We start by explaining how to obtain an appropriate variational formulation of the problem. Its main properties are state and in particular its severe ill posedeness is pointed out. A Lavrentiev regularizing process is proposed and analyzed. Next, we show how to use a domain extension to improve the computational performance of the Lavrentiev method. We therefore switch to the finite element approximations, announce the main estimates and provide some comments. This is the most recent part of the research on the subject (20021-2023). To our knowledge this is the first complete convergence analysis of the full discrete-regularized solution of the data completion problem. The proofs are technical and tedious and are not given in the talk.

An acoustic medium occupying a compact domain with non-constant sound speed, is probed by an impulsive plane wave, and the far-field response is measured in all directions for all frequencies. A longstanding open problem, called the fixed angle scattering inverse problem, is the recovery of the sound speed from this far-field response. In some situations, the acoustic properties of the medium are modeled by a Lorentzian metric and then the goal is the recovery of this metric from the far field measurements corresponding to a finite number of incoming plane waves. We consider a time domain, near field version of this problem and show that natural fixed angle measurements distinguish between a constant velocity (the Minkowski metric) medium and a non-constant velocity (a general Lorentzian metric) medium. The talk is based on a joint work with Rakesh (Delaware) and Mikko Salo (Jyväskylä).