Section: Research Program

Inverse problems for heterogeneous systems

The area of inverse problems covers a large class of theoretical and practical issues which are important in many applications (see for instance the books of Isakov [68] or Kaltenbacher, Neubauer, and Scherzer [70] ). Roughly speaking, an inverse problem is a problem where one attempts to recover an unknown property of a given system from its response to an external probing signal. For systems described by evolution PDE, one can be interested in the reconstruction from partial measurements of the state (initial, final or current), the inputs (a source term, for instance) or the parameters of the model (a physical coefficient for example). For stationary or periodic problems (i.e. problems where the time dependence is given), one can be interested in determining from boundary data a local heterogeneity (shape of an obstacle, value of a physical coefficient describing the medium, etc.). Such inverse problems are known to be generally ill-posed and their study leads to investigate the following questions:

• Uniqueness. The question here is to know whether the measurements uniquely determine the unknown quantity to be recovered. This theoretical issue is a preliminary step in the study of any inverse problem and can be a hard task.

• Stability. When uniqueness is ensured, the question of stability, which is closely related to sensitivity, deserves special attention. Stability estimates provides an upper bound for the parameter error given some uncertainty on data. This issue is closely related to the so-called observability inequality in systems theory.

• Reconstruction. Inverse problems being usually ill-posed, one needs to develop specific reconstruction algorithms which are robust with respect to noise, disturbances and discretization. A wide class of methods is based on optimization techniques.

In this project, we investigate two classes of inverse problems, which both appear in FSIS and CWS:

1. Identification for evolution PDE.

   Driven by applications, the identification problem for infinite dimensional systems described by evolution PDE has known in the last three decades a fast and significant growth. The unknown to be recovered can be the (initial/final) state (e.g. state estimation problems [29] , [56] , [64] , [93] for the design feedback controllers), an input (for instance source inverse problems [26] , [39] , [50] ) or a parameter of the system. These -linear or non linear- problems are generally ill-posed and many regularization approaches have been developed. Among the different methods used for identification, let us mention optimization techniques ( [41] ), specific one dimensional techniques (like in [30] ) or observer-based methods as in [77] .

   In the last few years, we have developed observers to solve initial data inverse problems for a class of linear infinite dimensional systems of the form $\dot{z}\left(t\right)=Az\left(t\right)$ ($A$ denotes here the generator of a $C_0$ semigroup) from an output $y\left(t\right)=Cz\left(t\right)$ measured through a finite time interval. Let us recall that observers (or Luenberger observers [76] ) have been introduced in automatic control theory to estimate the state of a (finite dimensional) dynamical system from the knowledge of an output (and, of course, assuming that the initial state is unknown). Roughly speaking, an observer is an auxiliary dynamical system that uses as inputs the available measurements (that is the output of the original system) that converges asymptotically (in time) towards the state of the original system. Observers are very popular in the community of automatic control and have given rise to a wide literature (for more references, see for instance the book by O'Reilly [80] and more recently the one by Trinh and Fernando [96] devoted to functional observers). The generalization of observers (also called estimators or filters in the stochastic framework) to infinite dimensional systems goes back to the seventies (see for instance Bensoussan [33] or Curtain and Zwart [46] ) and the theory is definitely less developed than in the finite dimensional case. Using observers, we have proposed in [82] , [65] an iterative algorithm to reconstruct initial data from partial measurements for some evolution equations, including the wave and Schrödinger systems (and more generally for skew-adjoint generators). This algorithm also provides a new method to solve source inverse problems, in the case where the source term has a specific structure (separate variables in time-space with known time dependence). We are deepening our activities in this direction by considering more general operators or more general sources and the reconstruction of coefficients for the wave equation. In connection with this last problem, we study the stability in the determination of these coefficients. To achieve this, we use geometrical optics, which is a classical albeit powerful tool to obtain quantitative stability estimates on some inverse problems with a geometrical background, see for instance [32] , [31] .

2. Geometric inverse problems.

   We investigate some geometric inverse problems that appear naturally in many applications, like medical imaging and non destructive testing. A typical problem we have in mind is the following: given a domain $\Omega$ containing an (unknown) local heterogeneity $\omega$, we consider the boundary value problem of the form

   $\left\{\begin{array}{cc}Lu=0,\hfill & (\Omega \setminus \omega )\hfill \\ u=f,\hfill & \left(\partial \Omega \right)\hfill \\ Bu=0,\hfill & \left(\partial \omega \right)\hfill \end{array}\right.$

   where $L$ is a given partial differential operator describing the physical phenomenon under consideration (typically a second order differential operator), $B$ the (possibly unknown) operator describing the boundary condition on the boundary of the heterogeneity and $f$ the exterior source used to probe the medium. The question is then to recover the shape of $\omega$ and/or the boundary operator $B$ from some measurement $Mu$ on the outer boundary $\partial \Omega$. This setting includes in particular inverse scattering problems in acoustics and electromagnetics (in this case $\Omega$ is the whole space and the data are far field measurements) and the inverse problem of detecting solids moving in a fluid. It also includes, with slight modifications, more general situations of incomplete data (i.e. measurements on part of the outer boundary) or penetrable inhomogeneities. Our approach to tackle this type of problems is based on the derivation of a series expansion of the input-to-output map of the problem (typically the Dirchlet-to-Neumann map of the problem for the Calderón problem) in terms of the size of the obstacle.