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The integral equation method offers a powerful and efficient tool for solving linear partial differential equations by reducing the original problem to boundaryonly formulations. In this talk, we explore its application to data completion problems for the Helmholtz equation, where missing Cauchy boundary data on the inaccessible part of the domain boundary must be recovered from measurements available on the accessible part.

We begin by presenting a reconstruction method based on boundary integral representations of the solution. By exploiting the properties of the traces of the solution and its normal derivative, the inverse problem is reformulated as a linear system involving boundary integral operators. The effectiveness of the method is demonstrated through both theoretical analysis and numerical simulations in twoand three-dimensional settings, considering various domain geometries and noise levels.

In the second part of the talk, we enhance the performance of classical algorithms using boundary integral equations. Specifically, we revisit the GMRESbased data completion approach, incorporating the Steklov-Poincar´e operator expressed in terms of boundary integral operators. Furthermore, we investigate the use of fading regularization in combination with boundary integral formulations to address the ill-posed Cauchy problem for the Helmholtz equation. This leads to a reformulation of the inverse problem as a constrained optimization problem. Finally, the fading regularization method is extended to the case of Maxwell’s equations, and its performance is evaluated under various noise levels.

Key words : Inverse problem, Integral equation, Data completion.

Fields : Inverse Problem.

Many imaging methods in inverse scattering rely on the ability to superimpose scattering data so that the resulting scattered field corresponds to that of a point source. In the frequency domain, for compact inhomogeneities, this leads to solving two elliptic PDEs in a bounded region with a prescribed difference in Cauchy data. The case of zero scattered field leads to the study of the transmission eigenvalue problem and the regularity of free boundaries. In this talk, we introduce this concept for linear media and review key results on the trans- mission eigenvalue problem, its resolvent, and non-scattering phenomena [1]. We then present recent results from [2] on the scattering problem for a nonlinear medium with compact support in the second-harmonic generation. When such a medium is probed with monochromatic light beams at a frequency $\omega$, it generates additional waves at the frequency $2 \omega$. The response of the medium is governed by a system of two coupled semilinear PDEs. We explore the possibility that the generated $2 \omega$ wave remains localized within the support of the medium, effectively rendering the nonlinear interaction with the probing wave invisible to an outside observer.

References :

[1] F. Cakoni, D. Colton, and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF SIAM publication, Philadelphia, PA, 2016, Second edition 98, 2023.

[2] F. Cakoni, N. Hovsepyan, M. Lassas and M. Vogelius, On the lack of external response of a nonlinear medium in the second-harmonic generation process SIAM J. Math. Anal. 57 (2) 1370-1405 (2025).

Key words : inverse scattering, inhomogeneous media, nonlinear media, inverse scattering, qualitative approach.

Fields : Inverse Problem.

In this talk, two types of results are presented. The first one is a joint work with I. Feki concerning some optimal logarithmic estimates in the Hardy-Sobolev space $H^{k, p}$ of the unit disk of $\mathbb{R}^2$ and its applications, firstly to prove an optimal global stability result of logarithmic type of the Cauchy problem, and secondly to study the stability of some inverse problems of parameter identification in second order elliptic partial differential equations. As for the second one, it is about the Kohn-Vogelius method for solving some inverse problems of the identification of singular parameters by boundary measurements, which is a joint work with B. Charfi and H. Haddar.

Key words : Hardy-Sobolev space, Cauchy problem, uniqueness, stability, Identification, optimization, Kohn-Vogelius method.

Fields : Inverse Problem.

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The Monotonocity Principle (MP), stating a monotonic relationship between a material property and a proper corresponding boundary operator, is attracting great interest in the field of inverse problems, because of its fundamental role in developing real time imaging methods. Moreover, under quite general assumptions, a MP for elliptic PDEs with nonlinear coefficients has been established. This MP provided the basis for introducing a new imaging method to deal with the inverse obstacle problem, in the presence of nonlinear anomalies.

The introduction of a MP based imaging method poses a set of fundamental questions regarding the performance of the method in the presence of noise. The main contribution is focused on theoretical aspects and consists in proving that (i) the imaging method is stable and robust with respect to the noise, (ii) the reconstruction approaches monotonically to a well-defined limit, as the noise level approaches to zero, and that (iii) the limit contains the unknown set and is contained in the outer boundary of the unknown set.

Results (i) and (ii) come directly from the Monotonicity Principle, while results (iii) requires to prove the so-called Converse of the Monotonicity Principle, a theoretical results of fundamental relevance to evaluate the ideal (noise-free) performances of the imaging method.

The results are provided in a quite general setting for Calder´on problem, and proved for three wide classes where the nonlinearity of the anomaly can be either bounded from infinity and zero, or bounded from zero only, or bounded by infinity only. These classes of constitutive relationships cover the wide majority of cases encountered in applications.

Key words : Inverse obstacle problem; Nonlinear material; Monotonicity Principle.

Fields : Inverse Problem.

Electrical impedance tomography (EIT) is a non-invasive imaging technique designed to reconstruct the distribution of electrical conductivity within a domain by applying electrical currents at the edge of the domain to electrodes and measuring the resulting voltages.

The direct EIT problem consists of determining the electric potential in the domain for a given conductivity distribution and for a set of currents injected through the electrodes, which results in the following system of PDE's:

\begin{equation}\label{eq1} \left\{ \begin{array}{lllllllllll} div(\sigma\, \nabla u) &=& 0 &\mbox{in}\,\, \Omega ,\\ \displaystyle\int_{E_m} \sigma\, \nabla u\, \cdot\, n\, dS &=& I_m &\mbox{on electrode}\,\, E_m ,\\ u + z_m\, \sigma\, \nabla u\, \cdot\, n &=& U_m& \mbox{on}\,\, E_m ,\\ \sigma\, \nabla u\, \cdot\, n &=& 0& \mbox{on}\,\, \delta \Omega\, \setminus\, \cup_m E_m , \end{array} \right. \end{equation}

with $I_m$ the electrical current injected in electrode $E_m$ and $z_m$ the contact impedance on electrode m. The inverse problem involves estimating the conductivities themselves from measurements taken at the body surface. Mathematically, this inverse problem, known as the Calderon problem or inverse conductivity problem, is a strongly ill-posed inverse problem. The main unknown in this inverse problem is the conductivity, but lack of information for other variables such as the geometric shape of the domain in which we aim to reconstruct the conductivity, as well as the positions of the electrodes, can influence the quality of the reconstruction. Therefore, we consider the EIT problem with an unknown moving geometry.

Solving such a problem can be computationally expensive, especially when using numerical methods that require an adapted mesh due to remeshing steps. To avoid these, we propose an Immersed Boundary Method (IBM) approach for numerically solving the complete electrode model in electrical impedance tomography. This method allows the use of a Cartesian mesh without requiring precise discretization of the boundary, which is useful in situations where the boundary is complex and/or evolving in time. We demonstrate the convergence of our method and illustrate its effectiveness in direct and inverse problems in two dimensions. The use of immersed boundary methods to solve the direct problem also allows easy parallelization of the method.

Key words : Immersed boundary methods, numerical analysis, electrical impedance tomography.

Fields : Inverse Problem.