Mini Symposium

Imen Rassas : Stable recovery of a time dependent matrix potential for wave equation from arbitrary measurements

Affiliation : Institut Supérieur de biotechnologie de Béja, LAMSIN-ENIT. e-mail:imen.rassass@gmail.com

In this work, we study an inverse problem related to the wave equation, where the goal is to determine a time-dependent matrix potential from boundary measurements taken on an arbitrary sub-boundary of the domain. We establish a logarithmic stability estimate showing that the potential can be recovered from partial knowledge of the Dirichlet-to-Neumann map.

Yosra Soussi : Inverse problems for a Schr\" odinger equation defined in an unbounded domain

Affiliation : Tunis Business School, LAMSIN-ENIT. e-mail:yosra.soussi@enit.utm.tn

In this talk, we discuss the stability issue for the inverse problem of determining the electric potential appearing in a Schr\" odinger equation defined on an infinite cylindrical waveguide. We consider both results of stability from full and partial boundary measurements associated with the so-called Dirichlet-to-Neumann map. In the presence of the magnetic potential, a second problem is considered for which we prove that the electric potential and the magnetic field depend stably on the global and partial Dirichlet-to-Neumann maps. Our approach combines construction of complex geometric optics solutions and Carleman estimates suitably designed for our stability results stated in an unbounded domain.

Yosra Mannoubi : Identification of Unbounded Electric Potentials through Asymptotic Boundary Spectral Data

Affiliation : ESPRIT Ingénieurs, LAMSIN-ENIT. e-mail:yosra.mannoubi@enit.utm.tn

In this talk, we address an inverse spectral problem for the Dirichlet Laplacian in a bounded domain $(\Omega \subset \mathbb{R}^n$, with $(n \geq 3$. We prove that a real-valued electric potential $(q \in L^{\max(2, \frac{3n}{5})}(\Omega)$ is uniquely determined by the asymptotic behavior of the eigenpairs $(\lambda_k, \varphi_k)$, where $\lambda_k$ are the eigenvalues and the boundary data consist of the normal derivatives $\partial_\nu \varphi_k|_{\partial \Omega}$ of the corresponding eigenfunctions. Our result establishes that the high-frequency asymptotics of spectral data encodes sufficient information to recover the potential uniquely, highlighting a new uniqueness result in the context of inverse boundary spectral problems.

Zouhour Rezig : An inverse problem for a Riemannian non-stationary transport equation with time-dependent coefficients

Affiliation : Faculté des Sciences de Tunis, LAMSIN-ENIT. e-mail:zouhour.rezig@fst.utm.tn

This talk addresses the recovery of time dependent absorption and scattering coefficients in the Riemannian transport equation from the albedo operator. Given $ \mathrm{ M} \subset\mathbb{R}^n$, $n\geq 2$, a compact domain with smooth boundary, equipped with a Riemannian metric $\mathrm{ g} $, we first prove the unique determination of a time dependent absorption coefficient in a subset of the domain of interest, provided that it is known outside this subset. We then show that we can recover the coefficient in a larger region (and eventually in the entire domain) by enlarging the data set. Next we present a uniqueness result for the reconstruction of the scattering parameter based on the knowledge of the albedo operator. The proof is based on geometric optics solutions and inversion of the light ray transform on static Lorentzian manifolds, assuming that the Lorentzian manifold is a product of a time interval with a simple Riemannian manifold.

Saber Amdouni : Generalized Error Bounds for Tikhonov Regularization in Ill-Posed Problems: From Abstract Theory to Game-Theoretic Applications

Affiliation : University of Tunis El Manar, National Engineering School of Tunis.

This talk will focus on recent advances in the robust numerical treatment of linear ill-posed problems and their applications to data completion in linear elasticity. The first part investigates the classical Tikhonov regularization method for the equation $$ Ax = y $$ where $A$ is a compact, injective linear operator between Hilbert spaces. Traditionally, error estimates in Tikhonov regularization rely on strong regularity assumptions on the exact solution. In contrast, the presented work removes these assumptions and derives a generalized error estimate of the form $$ \|x^\delta_\alpha - x\| = O\left( \frac{\delta}{\sqrt{\alpha}} + \sqrt{\alpha} \right), $$ which holds even for solutions with minimal smoothness. This extension significantly broadens the scope of Tikhonov regularization for practical inverse problems with noisy data. The second part of the work presents a concrete application to the Cauchy problem for Laplace’s equation, which is a prototypical example of an ill-posed problem. We reformulate the problem using a Nash game approach, treating unknown boundary conditions as control variables in a two-player game. Each player minimizes a cost functional based on either Dirichlet or Neumann data gaps, with a coupling term that enforces compatibility. We prove the existence and uniqueness of Nash equilibria, even in the presence of noisy data, and derive error estimates that demonstrate stability with respect to the noise level and the regularization parameter. The model is implemented numerically using the finite element method. Together, this research illustrates a seamless transition from abstract regularization theory to a practical and robust computational strategy for inverse boundary problems. It highlights the power of combining functional analysis with game theory to resolve ill-posed problems in a stable and reliable manner.

The second part of the work can be seen as a particular application of these general results, where the same generalized Tikhonov error estimate is used to establish a priori error estimates in a concrete data completion problem in linear elasticity. In this setting, the problem is reformulated as a variational optimal control problem posed in an energy space and discretized by the finite element method. The analysis demonstrates that the robust error estimate applies at both continuous and discrete levels, guaranteeing convergence rates even in the presence of noise and without strong assumptions on the exact solution. In particular, the discrete approximation achieves at least an order $O(h^{k/3})$ for sufficiently regular data, where $h$ denotes the mesh size and $k$ the polynomial degree of the finite elements. Numerical experiments confirm these theoretical results.

Together, these contributions illustrate a unified and robust framework for tackling ill-posed inverse problems — from abstract linear operator equations to practical engineering applications in elasticity — using advanced Tikhonov regularization strategies without restrictive smoothness assumptions on the unknown solution.

Faten Khayat : Boundary data completion for advection-diffusion equations

Affiliation : University of Carthage, National School of Advanced Sciences and Technologies in Borj Cedria.

The advection-diffusion equations model the transport, by a fluid with velocity $V$ , of a substance whose distribution is represented by a density or concentration $C$, which may or may not vary over time. We considered these equations in two ways. First, in the steady-state regime, to complete boundary data in an aquifer in the presence of source points; and second, in the transient regime, to study the phenomenon of seawater intrusion into a coastal aquifer.

For the steady-state case, we determined the concentration and the flux on an inaccessible boundary from measurements available on another boundary, using an energy-type functional minimization method. Defining an energy type functional in this context is not straightforward, since a symmetry issue constitutes a serious obstacle.

For the unsteady case, we addressed two coupled problems: water flow and solute transport in porous media. The flow can be either towards the sea (discharge case) or from the sea into the aquifer (intrusion situation). This boundary can also be in a mixed state: partly discharging and partly intruding.

The boundary conditions for the solute transport problem vary along the sea-side boundary $\Gamma_s$: in the case of intrusion, the salt concentration is that of the seawater, and in the case of discharge, it is that of the aquifer. On the land-side boundary $\Gamma_l$, the salt concentration is that of the freshwater entering upstream of the aquifer. Since overdetermined boundary conditions may exist on $\Gamma_l$ while they are missing on $\Gamma_s$ for the flow problem, we therefore use a data completion method to solve this problem. Conversely, when seawater intrusion occurs, the opposite situation arises for the transport problem: conditions are overdetermined on $\Gamma_s$ while unknown on $\Gamma_l$. A second data completion must therefore be performed.

Nabil Hafyene : Data Completion Problem by Reproducing Kernel Hilbert Spaces.

Affiliation : Laboratory of Mathematical and Numerical Modeling for Engineering Sciences.

In this study, we address the classical ill-posed Cauchy problem for the Laplace equation:

$$ \left\{ \begin{aligned} - \Delta u &= 0 && \text{in } \Omega, \\ u &= \psi && \text{on } \Gamma_{c}, \\ \partial_{n}u &= \varphi && \text{on } \Gamma_{c}. \end{aligned} \right. $$

where $\Omega$ is a bounded regular open set of $\mathbb{R}^2$, and the boundary $\partial\Omega$ is composed of two disjoint parts of non-vanishing measure: $\Gamma_{c}$ and $\Gamma_{i}$, both of which are closed.

We assume that there exists $ s > \frac{1}{2} $ such that the problem above has a solution $ u_0 \in H^{s + \frac{3}{2}}(\Omega)$.

We reformulate this inverse problem using fictitious domain decomposition tools to obtain the ill-posed linear equation:

$$ R \lambda = g, \quad \lambda \in H^s(\Gamma_i) $$

where $R$ is a compact, injective operator from $H^s(\Gamma_i)$ into $L^2(\Gamma_i)$.

To ensure a stable reconstruction, we propose a discretized Tikhonov regularization within reproducing kernel Hilbert spaces. We select a positive definite kernel $K$ such that its native space coincides with $ H^s(\Gamma_i)$ with equivalent norms. Let $(X_m)_{m \in \mathbb{N}^*} $ be a sequence of subsets of $\Gamma_i$, where each $X_m$ consists of $m$ distinct points $ x_{1,m}, \ldots, x_{m,m} \in \Gamma_i$.

We define:

$$ \mathcal{H}_m = \text{span} \{ K(\cdot, x) \mid x \in X_m \} = \text{span} \{ K(\cdot, x_{1,m}), \ldots, K(\cdot, x_{m,m}) \} $$

We approximate the problem using the following discretized formulation: find $ f_{\alpha, m}^\delta \in \mathcal{H}_m $ such that:

$$ f_{\alpha, m}^\delta = \arg\min_{f \in \mathcal{H}_m} \left\{ \frac{1}{2} \Vert Rf - g^\delta \Vert_{L^2(\Gamma_i)}^2 + \frac{\alpha}{2} \Vert f \Vert_{\mathcal{H}}^2 \right\} $$

where $ g^\delta $ is the noisy data and $ \Vert \cdot \Vert_{\mathcal{H}} $ is the norm of the native space associated with the kernel $K$.

We establish convergence and error estimates based on the fill distance of $ X_m $ in $\Gamma_i$, defined by:

$$ h_{X_m} = \sup_{x \in \Gamma_i} \min_{1 \leq j \leq m} \| x - x_{j,m} \|_2 $$

Finally, numerical simulations are presented to illustrate the performance of our approach.

Anis Samaali : Robustness Analysis of an Energy Space Method for Solving a Data Completion Problem in Linear Elasticity.

Affiliation : National Engineering School of Tuni.

This talk investigates the robustness of an energy space approach for solving a data completion problem in linear elasticity equations. The problem is reformulated as a variational optimal control problem, with the Dirichlet control defined in the energy space. Both continuous and discrete levels incorporate Tikhonov regularization with a parameter $\alpha$. At the continuous level, we show that this regularization provides a robust solution of the proposed energy space method and derive conditions that guarantee a convergence rate of $O\left(\frac{\delta}{{ \sqrt{\alpha}} }+ \sqrt{\alpha}\right)$ for noisy regularized solutions, where $\delta$ denotes the noise level. Importantly, this result is obtained without imposing assumptions on the exact solution. At the discrete level, Tikhonov regularization is integrated with a finite element method of order $k$. This combination yields an optimal convergence rate for discrete noisy regularized solutions, achieving at least $O(h^{k/3})$ for sufficiently regular data, where $h$ represents the mesh size. Numerical experiments are included to validate the theoretical findings.

Pedro Serranho : On a Hybrid Method for Inverse Scattering Problems and Applications in Elastography

Affiliation : Universidade Aberta, Portugal

In this talk, we will start by presenting the direct and inverse scattering problems we are dealing with and recent techniques for dealing with the ill-conditioning of the method of fundamental solutions for the direct problem. Then, we will revise a hybrid method for inverse scattering problems and illustrate its extensions for the inverse transmission problem and to elastography.

Maria-Luısa Rapun : Processing poor experimental databases via topological derivative based methods

Affiliation : E.T.S.I. Aeronautica y del Espacio, Universidad Polit´ecnica de Madrid, Spain

The main goal in inverse scattering problems is to reconstruct information about unknown objects or defects from measurements of scattered acoustic or electromagnetic waves. In many practical experimental setups, only limited aperture excitations and limited aperture receptor configurations are available. In this talk we will study the performance of numerical methods based on the computation of topological derivatives and topological energies to process very reduced experimental measurements of the electromagnetic scattering produced by different objects extracted from the Fresnel database.

Silvia Barbeiro : On the design of discrete cross-diffusion filters

Affiliation : University of Coimbra, Portugal

After the pioneering work of Keler and Segel in the 1970s cross-diffusion models became very popular in biology, chemistry and physics to emulate systems with multiple species. From a mathematical point of view, cross-diffusion models are described by time-dependent partial differential equations (PDE) of diffusion or reaction-diffusion type, where the diffusive part involves a general nonlinear non-diagonal diffusion matrix. This leads to a strongly coupled system where the evolution of each dependent variable depends on itself and on the others in a way governed by the diffusion matrix. A central challenge in modeling real-world phenomena is the selection of appropriate coefficients and influence functions that govern the PDE system, as effectively solving this inverse problem is essential for accurate and reliable representation.

In this talk we will focus on nonlinear cross-diffusion systems for image filtering. We will discuss a flexible learning framework in order to optimize the parameters of the models improving the quality of the denoising process. In particular, we use a back propagation technique in order to minimize a cost function related to the quality of the denoising process while we ensure stability during the learning procedure.

Tram Thi Ngoc Nguyen : Bi-level iterative regularization for inverse problems in nonlinear evolution PDEs and astrophysics applications

Affiliation : Max Planck Institute for Solar System Research, Germany

We present the novel bi-level regularization methods for illposed inverse problems governed by nonlinear evolution PDEs. By a bi-level Landweber scheme, one alternates between upper-level parameter reconstruction and lower-level state approximation, bypassing exact PDE solves. We derive adaptive stopping rules for both levels and prove convergence, stability and acceleration of the bi-level algorithm. The bilevel approach illustrates its universality through several reaction-diffusion applications, such as the Lane-Emden equation in astrophysics, in which the nonlinear reaction law needs to be determined.

Sue Claret : Exact boundary controllability of semilinear wave equations

Affiliation : LMBP, Univ. Clermont Auvergne

In this talk, we adress the exact controllability of the semilinear wave equation $\partial_{tt} y − \Delta y + f(y) = 0$ posed over a bounded domain $\Omega$ of $\mathbb{R}^d$ with initial data in $L^2(\Omega) \times H^{-1}(\Omega)$. We focus on the existence of a Dirichlet boundary control for the equation under a growth condition at infinity on the nonlinearity $f$ of the type $r \ln^p\, r$, with $p \in [0, 3/2)$. This result is based on a Schauder fixed-point argument. Then, assuming additional assumptions on $f^\prime$, we consider the approximation of a control function. This is a joint work with Arnaud Munch and Jér\^ome Lemoine.

Tiphaine Delaunay : Solving inverse source wave problem from observability to observer design

Affiliation : LAGA, Univ. Sorbonne Paris Nord

The context of this talk is the study of inverse problems for wave propagation phenomena using observation theory. The objective is to formalize, analyze, and discretize so-called sequential strategies in data assimilation, where measurements are incorporated as they become available. The resulting system, known as an observer, stabilizes on the observed trajectory, thereby reconstructing the state and possibly some unknown parameters of the system. We focus in particular on the reconstruction of a source term appearing on the right-hand side of a wave equation, an estimation problem of intermediate complexity between state (or initial condition) estimation and parameter identification. In this context, we propose to define in a deterministic framework in infinite dimension a so-called Kalman estimator that sequentially estimates the source term to be identified. Using dynamic programming tools, we show that this sequential estimator is equivalent to the minimization of a functional. This equivalence enables us to propose convergence results under observability assumptions. These observability inequalities are established for various types of sources by combining techniques from functional analysis, multiplier methods, and Carleman estimates. In particular, they highlight the ill-posed nature of the inverse problems under consideration and allow us to quantify their degree of ill-posedness, hence to propose adapted regularizations.

Spyridon Filippas : On the stability of a hyperbolic inverse problem

Affiliation : Dpt of Maths & Statistics, Univ. Helsinki,

The Boundary Control method is one of the main techniques in the theory of inverse problems. It allows to recover the metric or the potential of a wave equation in a Riemannian manifold from its Dirichlet to Neumann map (or variants) under very general geometric assumptions. In this talk we will address the issue of obtaining stability estimates for the recovery of a potential in some specific situations. As it turns out, this problem is related to the study of the blow-up of quantities coming from control theory and unique continuation. This is based on joints works with Lauri Oksanen.

Lotfi Thabouti : Quantitative unique continuation for nonregular perturbations of the Laplacian

Affiliation : IMB, Univ. Bordeaux

In this talk, I will present recent results on the quantitative unique continuation property for solutions to elliptic equations of the form $$ \Delta u = V\, u + W_1\, \cdot\, \nabla u + div(W_2\, u) $$ in an open, connected subset of $\mathbb{R}^d$ , with $d > 3$, where the potentials $V$ , $W_1$, and $W_2$ belong to Lebesgue spaces $L^{q_0}$, $L^{q_1}$, and $L^{q_2}$ respectively, with minimal regularity assumptions: $q_0 > d/2$, $q_1 > d$, and $q_2 > d$. Our goal is to obtain explicit quantitative estimates for unique continuation in terms of the norms of these potentials. The strategy builds upon a refined version of $L^p$ Carleman estimates introduced in [Dehman-Ervedoza-Thabouti Annales Henri Lebesgue 2024], and an argument due to T. Wolff introduced in [Wolff Geom. Funct. Anal. 1992] for the proof of unique continuation for solutions of equations of the form $\Delta u - V u + W_1 u \cdot \nabla u$.

This is a joint work with Sylvain Ervedoza (Universit´e de Bordeaux \& CNRS, Bordeaux, France) and Pedro Caro (Ikerbasque \& BCAM, Bilbao, Spain).

Rahma Jerbi : Reconstruction of a piecewise constant conductivity by coupling the Kohn-Vogelius type functional with a non-overlapping DDM

Affiliation : ISI Kef, University of Jendouba

In our 2023 work [1] on a geometrical inverse problem, we developed an inversion method that combines gradient descent on the KohnVogelius cost functional with a non-overlapping domain decomposition method (DDM), namely an optimized Schwarz method, which includes the unknown curve in the domain partitioning. In this talk, we extend this approach by proving that a similar result holds even when the parameter values are also unknown. The framework is developed within the inverse electrical conductivity problem and aims to reconstruct a piecewise constant conductivity from a finite number of Cauchy pairs. We study the convergence of descent schemes that perform partial DDM steps per gradient iteration, with a particular focus on the one-shot method (i.e., one DDM step per iteration). We prove its local convergence in a simplified setting by examining the eigenvalues of the block matrix governing the coupled iterations. We finally illustrate our theoretical findings and the inversion scheme’s performance through 2D numerical experiments, comparing them with the classical gradient descent method. This is joint work with Slim Chaabane and Houssem Haddar.

Alejandro Sior : Numerical Assessment of a Multi-Step One-Shot Approach for Frequency-Domain Acoustic FWI

Affiliation : Université de Liège, Belgique

Full waveform inversion (FWI) aims to reconstruct spatially parameters of a wave propagation model by minimizing a misfit between simulated and measured wavefields. The process generally involves two main components: a numerical solution stage for the forward and adjoint wave problems (to estimate the gradient of the misfit), and an outer optimization algorithm. In large-scale scenarios, the wave problems are solved using iterative domain decomposition techniques, and local gradient-based methods are used as optimization algorithms. The one-shot methodology modifies the traditional approach by intertwining the solution and optimization steps. Instead of fully resolving the forward and adjoint systems at each optimization iteration, it uses a limited number of solver iterations, yielding an approximate gradient. This presentation explores the viability and effectiveness of such a one-shot strategy for frequency-domain FWI when an Optimized Restricted Additive Schwarz (ORAS) method is combined with a Gauss-Newton optimization strategy. The analysis focuses on computational performance, considering the number of solver iterations, gradient updates, and subdomain partitions. This is joint work with Boris Martin, and Christophe Geuzaine.

Nicolas R. Gauger : One-Shot Methods for Efficient Shape Optimization and Layerparallel Training of Deep RNNs

Affiliation : University of Kaiserslautern-Landau (RPTU), Germany

We introduce different variants of one-shot methods for the shape optimization with PDEs. The derivation of the different one-shot methods is first done for steady PDEs. In the further course of the lecture we will see how the derived one-shot approaches can also be used for unsteady cases and we furthermore apply it for the layer-parallel training of deep RNNs.

Marcella Bonazzoli : On the convergence analysis of one-shot inversion methods

Affiliation : Inria, ENSTA, Institut Polytechnique de Paris, France

We analyze the convergence of (multi-step) one-shot methods for general linear inverse problems and fixed-point iterations for the associated forward/adjoint problems. In particular, we establish sufficient conditions on the descent step for convergence, which are explicit in the number of inner iterations [2]. We provide numerical experiments to illustrate the convergence of these methods in comparison with the classical gradient descent method, where the forward and adjoint problems are solved exactly by a direct solver. We conclude with a recent numerical investigation for a variant where the one-shot iteration order is reversed so that an actual descent direction is used at each iteration. This is joint work with Emna Abida, Houssem Haddar, and Tuan Anh Vu.

Pierre Lissy : Desensitizing control for the heat equation with respect to domain variations

Affiliation : CERMICS - Ecole nationale des ponts et chaussées

In this talk, I will present some recent results obtained in collaboration with Sylvain Ervedoza and Yannick Privat on desensitizing control of the heat equation posed on a bounded domain. The question is to find a control, distributed here on a subdomain, such that a certain functional depending on the solution of the heat equation (in our case, the energy of the solution localized on another subdomain of the heat equation) is locally insensitive to a certain perturbation of the equation. Here, the main originality of our work lies in the fact that the perturbation is the domain itself, in the sense that its boundary can be subject to small variations. I will present various definitions of the desensitization problem and give some positive and negative results, giving an interpretation in terms of robustness for controllability issues.

Clément Moreau : Multi-scale analysis and reduced models for low-Reynolds swimmers

Affiliation : Laboratoire des Sciences du Numérique de Nantes - Nantes University

The topic of this presentation is a multi-timescale approach to derive reduced models of particles and swimmers in a viscous (low Reynolds-number) fluid. Over a long period of time, or from a distance, the trajectory of self-propelling bodies such as swimmers appears smooth, with their trajectories appearing almost ballistic. This long-time behaviour, however, masks more complex dynamics, such as the side-to-side snakelike motion exhibited by spermatozoa, or shape-changing microorganisms and microrobots. Many models of motion at microscopic scale, such as the celebrated Jeffery equations established in 1922, neglect, often without formal justification, these effects in favour of smoother long-term behaviours. In this talk, I will present recent results based on multi-timescale analysis and evaluating the long-term effects of high-frequency oscillations on translational and angular motion for various classical swimming models of micro-scale swimmers, with the purpose of assessing the relevance of neglecting these oscillations, and derive simplified equivalent models. I will particularly focus on Jeffery equations and subsequent generalisations. When adding a fast-timescale term in the orientational dynamics, one can show that the averaged system still follows Jeffery trajectories, with effective parameters being explicitly calculable. I will detail the multiscale method on this example, discuss its physical interpretation, and describe extensions to deformable particles and three-dimensional motion. This work was conducted with M. Dalwadi (UCL), E. Gaffney (Oxford University), K. Ishimoto (Kyoto University), and B. Walker (University of Bath).

Laurent Baratchart : Inverse Helmholtz problems : consistency and sparsity

Affiliation : Centre Inria d'Université Cote d'Azur

Inverse Hodge problems arise in quasi-static electromagnetism: geomagnetism, magnetization recovery, EEG analysis, electro-cardiography. They are governed by elliptic PDEs and, from a mathematical viewpoint, aim at reconstructing a vector field from knowledge of its support and of the gradient component of its Helmholtz-Hodge decomposition with respect to some Riemannian metric; even the simplest, Euclidean case is by no means fully understood. We will consider the case where the source term is the divergence of some unknown measure, to be recovered from knowledge of the field away from the support of that measure. After presenting regularity of solutions, we shall define notions of sparsity in this infinite dimensional context, when the support of the source is contained in a compact set of zero measure with connected complement (a so-called slender set), and discuss the case of volumic sample which is quite open. We will also touch upon some aspects of discretization.

Laetitia Giraldi : Guiding swarms of micro-swimmers through collective patterns

Affiliation : Centre Inria d'Université Cote d'Azur

Swarms of micro-swimmers offer promising applications, particularly in medicine, where they can revolutionize targeted drug delivery. However, modeling such swarms is highly challenging, as it requires accounting for hydrodynamic forces among all agents and their interactions. This complexity is further amplified in large populations, where lubrication forces dominate. This talk will explore collective behaviors, focusing on the transition from disordered phases to the formation of orientationally ordered patterns, and their influence on particle transport and flux. In confined environments without fluid flow, particles tend to accumulate near walls, causing clogs or obstructions that hinder movement, or form bands aligned with the channel. Leveraging these collective patterns, we utilize external magnetic fields to uniformly reorient swimmers and apply reinforcement learning tools to develop systematic control strategies that enhance their flux within the channel. By integrating mathematical modeling, statistical analysis, and reinforcement learning, these approaches demonstrate effective methods to guide magnetic micro-swimmers, highlighting the synergy between advanced modeling techniques and AI in addressing the complex challenges of swarm control.

Lauri Oksanen : Optimality of stabilized finite element methods for elliptic unique continuation

Affiliation : University of Helsinki, Finland

We consider finite element approximation in the context of the ill-posed elliptic unique continuation problem and introduce a notion of optimal error estimates that includes convergence with respect to a mesh parameter and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the finite element approximation space. We present a stabilized finite element method satisfying the optimal estimate and discuss a proof showing that no finite element approx- imation can converge at a better rate. The talk is based on joint work with Erik Burman and Mihai Nechita .

Janosch Preuss : Unique continuation for the wave equation: the stability landscape

Affiliation : Inria Makutu, France

We consider a unique continuation problem for the wave equation given data in a volumetric subset of the space time domain. The stability of this problem depends on the amount of additional knowledge that is available on the trace of the wave displacement on the lateral boundary of the space-time cylinder. In the absence of any data, a conditional stability estimate of H¨older-type holds in a certain subset of the space-time domain whereas full Lipschitz stability can be recovered globally given a finite dimensional space in which the trace is contained. The aim of this talk is to present a finite element method that can cope with these different regimes of stability thanks to specifically tailored stabilization terms. Combining the properties of the stabilization with conditional stability estimates then allows to prove convergence rates that reflect the stability of the continuous problem. Finally, we show numerical experiments not only to validate our theoretical results but also to explore further aspects that our currently not covered by the theory.

The talk is based on joint work with Erik Burman, Lauri Oksanen and Ziyao Zhao.

Jérémy Héleine : Finite element discretization aspects of the iterated quasi-reversibility method

Affiliation : Université Paul Sabatier IMT, France

We consider the iterated quasi-reversibility method applied to the data assimilation problem for the Laplacian. More precisely, we are particularly interested in the discretization of this method, using finite elements. The aim of our study is to determine the explicit dependence of the sequence built by the method over the different involved parameters: the number of iterations, the regularization parameter, the size of the mesh, and the amplitude of noise. Numerical experiments illustrate these relations.

This is a joint work with Jérémi Dardé.

Raphael Terrine : Convergence of a time finite difference scheme for solving an inverse problem via a mixed variational formulation : Application to tsunami reconstruction

Affiliation : Inria MEDISIM and ENSTA, POEMS, France

This paper addresses the inverse problem of reconstructing a seabed displacement responsible for a tsunami, based on surface measurements of the free surface. Starting from a linearized model derived from Euler equations, the study formulates the wave propagation problem as a first-order system within a suitable functional analytical framework. Due to the lack of coercivity in the original hyperbolic operator, a parabolic regularization is introduced to ensure well-posedness and to simplify the numerical treatment. The inverse problem is recast as a minimization problem of a quadratic cost functional combining a data fidelity term and a regularization term. We then derive the associated optimality system using an adjoint state, leading to a coupled forwardadjoint system. A time-discrete iterative scheme based on finite differences is developed, and its consistency and convergence are rigorously analyzed using a mixed space-time finite element formulation. Finally, an iterative gradient descent algorithm is proposed for practical implementation, the convergence being proved under standard assumptions. This methodology has direct applications in tsunami source reconstruction from surface wave measurements.

This is a joint work with Philippe Moireau and Laurent Bourgeois.

Larisa Beilina : Adaptive finite element method for solution of electromagnetic coefficient inverse problem

Affiliation : University of Gothenburg and Chalmers University of Technology, Sweden

We present an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and conductivity functions in time-dependent Maxwell's system using limited boundary observations of the electric field in 3D. A posteriori error estimates in the reconstructed functions, Lagrangian and Tikhonov functional are derived and used in the adaptive conjugate gradient algorithm. Numerical examples justify eciency of a posteriori error estimates on the reconstruction of malign melanoma in 3D.

Romina Gaburro : Recent advances in anisotropic inverse problems

Affiliation : University of Limerick, Ireland

In this talk we discuss the well known issue of non-uniqueness in Calderon's inverse problem and the related inverse problem for a Schrodinger type of equation in the presence of anisotropy. We present some positive answers to the question of non-uniqueness for these two anisotropic inverse problems under various a-priori assumptions that constrain the unknown leading coecient of the underlying forward model to belong to a nite dimensional space.

Mirza Karamehmedovic : Spectral properties of radiation for the Helmholtz equation with a random coefficient

Affiliation : Technical University of Denmark

We study the Helmholtz equation $[\Delta+k_0^2(1+q(x,\omega))]u=-f$ in $\mathbf{R}^2$ and $\mathbf{R}^3$, where $q(x,\omega)$ is a Gaussian random field. For an open ball $B$ that includes the support of $x\mapsto q(x,\omega)$, we investigate the source-to-measurement map $f\mapsto u|_{\partial B}$ in the low-frequency or weak-scattering regime, and discover a 'spectral leakage' phenomenon induced by the inhomogeneity. We provide a quantitative characterization of this effect, as well as a numerical validation of our estimates against a Finite Element Method solution of a representative radiation problem. Our results offer insight into the stability of inverse source reconstructions in the presence of deterministic or random media.

Akram Beni Hamad : Solving Inverse chemical problems via the Adjoint method: A case study on Uranium In-Situ Recovery

Affiliation : PSL University / Mines Paris, Centre de Géosciences

Inverse modeling provides a powerful tool for addressing uncertainties inherent in complex physical systems. In the context of chemical processes, this often involves solving large systems of nonlinear equations with poorly constrained parameters.

This work demonstrates the application of the adjoint state method for inverse modeling of complex chemical problems. While the direct chemical problem can be approached using various numerical methods, depending on species appearance/disappearance and the degree of implicitness, the adjoint problem remains independent of the chosen direct solver. This is a key advantage of the adjoint approach.

Moreover, the adjoint problem is numerically simpler than the direct problem, enabling efficient optimization within a reasonable computational timeframe. The effectiveness of this method is illustrated through case studies involving simplified carbonate systems and a redox-sensitive uranium–iron system.

Faouzi Triki : TBA

Affiliation : Université Grenoble Alpes, France

TBA.

Christian Gerhards : Inverse Problems in Micromagnetic Tomography

Affiliation : Bergakademie (Freiberg, Germany)

A typical task in paleomagnetism is the inversion of magnetic microscopy data (i.e., information on the magnetic field on a detector plane) for the underlying sources (i.e., magnetization within a rock sample). Such problems are known to be non-unique and highly instable. In micromagnetic tomography (MMT), microCT data of the rock sample is added as a further modality of information. In addition to the already available magnetic field information, this provides information on the shape, size, and location of magnetized grains within the rock sample. In this talk, we provide some overview on recent approaches to inverse magnetization problems in the context of MMT.

Douglas Hardin : Uniqueness for TV-Norm Regularized Inverse Problems with Source Term in Divergence Form

Affiliation : Vanderbilt University (Nashville USA)

We consider inverse problems for the Poisson equation with source term the divergence of an ${\mathbb R}^3$-valued measure $\boldsymbol{\mu}$; that is, the potential $\Phi$ satisfies $$ \Delta \Phi= \nabla \cdot \boldsymbol{\mu}, $$ and $\boldsymbol{\mu}$ is to be reconstructed from measurements of the field $\nabla \Phi$ on a set disjoint from the support of $\boldsymbol{\mu}$. Such problems arise in several electro-magnetic contexts in the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We investigate methods for recovering $\boldsymbol{\mu}$ by penalizing the measure theoretic total variation norm $\|\boldsymbol{\mu}\|_{TV}$. While the regularized inverse problem is not strictly convex due to the inherent non-uniqueness of this problem, we establish sufficient conditions for the uniqueness of the regularized problem. As a consequence, we established sufficient conditions for the unique recovery of $\boldsymbol{\mu}$, asymptotically when the regularization parameter and the noise tend to zero in a combined fashion, when it is uni-directional or when the magnetization is carried on a sufficiently `sparse' set. This is joint work with L. Baratchart and C. Villalobos Guillen.

Masimba Nemaire : Electrocardiographic imaging with vector-valued measures

Affiliation : IHU-Lyric (Bordeaux France)

we study the reconstruction of vector-valued measures $\mu\in[\mathcal{M}(\Omega)]^n$, $n \geq 3$, from measurements of their associated potentials, $\Phi$, recorded outside $\Omega$, where $\mu$ and $\Phi$ are linked by the stationary elliptic problem $\nabla\cdot (M\nabla \Phi) = \nabla\cdot\mu$ with $M$ a positive definite matrix which encodes the conduction anisotropy of $\Omega$. This elliptic problem appears in the bidomain formulation of electrically excitable tissue which faithfully models the propagation of electrical waves in such tissue.

This work focuses on measures $\mu$ whose support is either a $1$-rectifiable or $(n-1)$-rectifiable set as such measures capture the physical phenomenon of interest when $\mu$ has to be reconstructed from measurements of $\Phi$, in particular, in the context of electrocardiographic imaging (ECGI). We detail in this work a numerical algorithm for reconstructing such measures and we also provide a possible implementation of the algorithm. The method we present addresses a number of practical issues with existing methods for solving the problem of ECGI and electroencephalography (EEG). Further, our method explicitly takes into account $M$ hence is of interest in problems where structure determines conduction anisotropy. We give a numerical example of the implementation of the algorithm in the context of electrocardiographic imaging. We successfully reconstructed the activation wavefront within the volume of a simulated slab of cardiac tissue.

Richard Huber : Optimal discretizations of tomographic projection operators

Affiliation : Technical University of Denmark (Lingby)

Tomographic reconstruction is a classical inverse problem that is still of key importance for applications in medical imaging and beyond. Tomographic projection operators (certain line-integral operators) are employed to model the physical measurement processes in tomography, with the Radon transform as the most well-known example. While these models are described as operators between (infinite-dimensional) function spaces, the solution of the inverse problem has to be effectuated on a computer using finitely many basic computations. To that end, proper discretization of the tomographic projection operators is crucial. A precise definition of what makes a good discretization is surprisingly elusive, but one naturally wishes the discretization to be as representative of the continuous operator as possible. We propose to interpret discretizations as finite rank operators to compare them to their continuous counterparts on a more rigorous level. In particular, one can ask what the best finite rank operator (given certain constraints) is. This results in the novel `Weighted Strip Models' discretization framework that allows for an (in an $L^2$ sense) optimal discretization of tomographic projection operators. Moreover, the adjoint of a Weighted Strip Model is itself an optimal discretization of the adjoint operator, thus, eliminating the widespread need for mismatched (i.e., non-adjoint) operator pairs in the solution of tomographic inverse problems.

Teemu Tyni : X-ray tomography via nonlinear waves

Affiliation : University of Oulu, Finland

This talk concerns an inverse problem of recovering a potential function in a nonlinear wave equation. I will discuss a constructive method for reconstructing the potential in a Holder-stable manner. The method is based on a higher-order linearization approach, which leads to an X-ray transform of the potential function. I will also present numerical results demonstrating the effectiveness of the approach and discuss the main challenges involved.

The talk is based on joint works with Lassas, Liimatainen, Potenciano-Machado, and Takalahti.

Hamza Ammar : Identification of Stimuli Currents in the Stochastic Cardiac Bidomain Model

Affiliation : University of Tunis El Manar, Tunisia

In the stochastic framework of the bidomain model in cardiac electrophisiology elaborated by M. Bendahmane et al., we study the inverse problem of recovering a couple of space-time dependent random variables ${I}=({I}_i,{I}_e)$, called "stimuli currents" in the form of $I(\omega,t,x):=X(\omega)\mathcal{I}(t,x)$ for a certain class of real random variables $X$, by means of an interior measurement $v_{obs}$ of the transmembrane potential $v$ over a subdomain $\omega_0\subset\subset \Omega$ and over a certain duration. We focus on recovering $\mathcal{I}(t,x)$. To address these challenges, we adopt an optimal control approach. This inverse problem is expressed as a minimization one since we want to identify the optimal couple of currents, such that our transmembrane potential $v$ approaches the best the desired observation $v_{obs}$. We establish the existence of a minimizer $I^\star$ and extract the necessary optimality conditions via an adjoint method. Then we establish a stability result for this inverse problem. Our methodology integrates weak martingale solutions with the adjoint method and Tikhonov regularization.

The talk is based on joint work with Mostafa Bendahmane (University of Bordeaux) and Moncef Mahjoub (University of Tunis El Manar).

Amal Labidi : Sampling Methods in Inverse Scattering problem for the Magnetic Schrodinger Operator

Affiliation : ESPRIT, Tunisia

We study the inverse medium scattering problem for the magnetic Schrodinger operator, focusing on two qualitative reconstruction methods: the Linear Sampling Method (LSM) and the Factorization Method (FM). These approaches aim to recover the shape of a perturbation from scattered wave data at a fixed frequency. Numerical validations in two dimensions illustrate their effectiveness.

The talk is based on joint work with Houssen Haddar (ENSTA Paris Tech).

Nesrine Aroua : Stability for an inverse spectral problem of the biharmonic operator

Affiliation : University of Genoa, Italy

This talk is concerned with the inverse spectral problem of recovering the vector field $B$ and the electric potential $q$ from some asymptotic knowledge of the boundary spectral data of the bi-harmonic operator $\mathcal{H}_{B,q} = (-\Delta)^2 -2iB\cdot\nabla-i\textrm{div} B+q$. More precisely, in a bounded smooth domain of $\mathbb{R}^n$, with $n \geq 2$, we prove the stability estimate of the first order perturbation coefficients $(B, q)$ from some asymptotic knowledge of a subset of the Dirichlet eigenvalues and Neumann traces of the associated eigenfunctions of $\mathcal{H}_{B,q}$. This can be viewed as an extension of the famous one-dimensional Borg-Levinson theorem.

This is a joint work with Mourad Bellassoued (University of Tunis El Manar).

Nesrine Klebi : Parsimonious identification of a viscoelastic model using the mCRE framework and Lasso regularization

Affiliation : Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS - Laboratoire de Mécanique Paris-Saclay

In this work, we investigate the parsimonious identification of linear viscoelastic models within the modified Constitutive Relation Error (mCRE) framework~\cite{2024_Chamoin_ERC}. The proposed methodology leverages the robustness of mCRE for time-dependent materials and incorporates a Lasso-type regularization to promote sparsity in the identified parameters, following the rationale of the EUCLID framework introduced in~\cite{2023_Marino_EUCLID}. A generalized Maxwell model serves as the reference formulation. Using synthetic or real data, the approach enables the extraction of a minimal set of active Maxwell branches from an overparameterized initialization and successfully recovers the underlying model structure. This contribution lays the groundwork for data-driven yet physics-informed identification strategies in rheological modeling.

This is a joint work with Ludovic Chamoin (Université Paris-Saclay).

Renaud Ferrier : Sparse space-heterogeneous, physics-constrained PDE reconstruction for defect identification through low-frequency mechanics

Affiliation : Ecole des Mines de Saint-Etienne, CNRS, Laboratoire Georges Friedel

This contribution explores an application of automatic PDE identification for non-destructive testing. Defect localization is traditionally performed via the identification of the spatial distribution of a well-chosen physical property, which is assumed to be impacted by the defect in question. Defects whose impact on physical properties is unclear may thus cause the identification approach to fail. This contribution tries to leverage automatic PDE identification methods like~\cite{2016_Brunton_SINDY} to overcome this limitation. In the context of physics-constrained identification in mechanics, the equations of forces equilibrium is known and the relation between cinematic and statical variables is the only part of the PDE that requires identification.

The idea of the contribution is to identify a space-dependent PDE instead of a single space-dependent parameter. In practice, the PDE in question is defined in this work through a high-dimensional set of parameters and the method consists in identifying a scalar field for each of those parameters. As expected, the efficiency of the method relies on regularizing tricks, among which sparsity-promoting penalization is key. The method is evaluated on experimental data coming from the dynamical solicitation of a composite beam.

This is a joint work with Mohamed Larbi Kadri (CESI-LINEACT), Sylvain Drapier (Ecole des Mines de Saint-Etienne) and Pierre Gosselet (Université de Lille).

Walid Maherzi : Meta-Learning-Based Concrete Strength Prediction Using Stratified CatBoost Models and Domain-Informed Features

Affiliation : University Lille, IMT Nord Europe, University Artois, JUNIA, ULR 4515-LGCgE, Laboratory of Civil Engineering and Geo-Environment

Precise estimation of concrete compressive strength is critical for ensuring structural integrity and maintaining quality control throughout the construction process. Traditional experimental methods, while widely used, are often costly, time-consuming, and limited in scope. Recent advances in artificial intelligence (AI) and machine learning (ML) offer promising alternatives for predicting compressive strength based on mixture composition. However, challenges remain, including limited interpretability, reduced generalization across strength ranges, and insufficient integration of domain-specific knowledge.

This study introduces a comprehensive ensemble framework based on CatBoost, leveraging nonlinear meta-learning, strength-range stratification, and domain-informed feature engineering to address these challenges. Utilizing a benchmark dataset comprising 1,030 concrete mixture samples, the model incorporates eight primary input variables -- cement, slag, fly ash, water, superplasticizer, coarse aggregate, fine aggregate, and age. Advanced feature engineering grounded in materials science principles generated 24 derived features, effectively capturing nonlinear interactions and time-dependent behaviors.

The ensemble system employs stratified sub-models tailored to four predefined strength categories, augmented by boundary-region models and age-specific predictors. A CatBoost-based meta-learner dynamically integrates predictions from these specialized sub-models, ensuring robustness and adaptability across diverse mixture scenarios.

The proposed ensemble achieved good performance, yielding an R$^2$ of 0.9767, an RMSE of 2.61 MPa, and 88.3\% of predictions within a 10\% error margin. Notably, the model demonstrated substantial improvements in prediction accuracy for both extremely low and high strength ranges--areas where traditional models often underperform. Error analysis and statistical validation underscored the critical role of age and temporal modeling in capturing strength variance. Key engineered features, such as the maturity index and water-to-cement ratio, significantly enhanced both model interpretability and forecasting accuracy.

This work establishes a high-performance, interpretable, and replicable framework for predicting concrete compressive strength. It sets the stage for future research on cost-effective, domain-informed, and sustainable optimization of concrete mixtures.

This is a joint work with Sree Vidyasai Cheekuri, Ali Kheyrandish and Ahmed Senouc from University of Houston.

Hakim Naceur : TBA

Affiliation : INSA Hauts-de-France

Description intervention.

Christian Daveau : Determining of electromagnetic parameters in Maxwell's equations from partial boundary measurements

Affiliation : CY Cergy Paris Université, Institut des sciences et techniques, Laboratoire AGM

In this work, we deal with an inverse boundary value problem for the Maxwell equations with boundary data assumed known only in accessible part $\Gamma$ of the boundary. We aim to prove uniqueness results using the Dirichlet to Neumann data with measurements limited to an open part of the boundary and we seek to reconstruct the complex refractive index $n$ in the interior of a body. Further, using the impedance map restricted to $\Gamma$, we may identify locations of small volume fraction perturbations of the refractive index.

Aymen Jbalia : Stability estimate for an inverse problem of a hyperbolic heat equation from boundary measurement

Affiliation : ENSI, University of Manouba

We are concerned with an inverse problem arising in thermal imaging in a bounded domain . This inverse problem consists in the determination of the heat exchange coefficient $q(x)$ appearing in the boundary of a hyperbolic heat equation with Robin boundary condition. A double-logarithmic stability estimate is developed.

Saoussen Boujema : Asymptotic expansion for solution of Maxwell equation in domain with highly oscilating boundary

Affiliation : LIM, ENIB, University of Carthage

In this investigation, we derive high-order terms in the asymptotic expansions of the boundary perturbations for the electric field by taking into account of highly oscillating perturbations of an inhomogeneity with regular boundary. We show then that the asymptotic expansions depend on some parameter $\gamma$ that describes the depth $O(\delta^{\gamma})$ of irregularity. Our study is rigorous and is founded on Field expansion method.

Ahlem Jaouabi : Asymptotic behavior for eigenvalues and eigenfunctions associated to Stokes operator in the presence of a rapidly oscillating boundary

Affiliation : GAMA, University of Carthage

We analyze rigorously the asymptotic behaviors for perturbation of eigenvalues and eigenfunctions associated to the Stokes eigenvalue problem in the presence of a rapidly oscillating boundary. The aim is to construct asymptotic approximations, as $\delta \to 0$, of the eigenvalues and corresponding eigenfunctions for the case where the eigenvalue of the reference problem is simple or multiple. Taking advantage of small oscillations, we use the method of matching of asymptotic expansions to construct the associated leading terms.

Chayma Nssibi : Support identification for parameter variations in a PDE system via regularized methods

Affiliation : ENIT-LAMSIN, Tunisia

We study the inverse problem of recovering the spatial support of parameter variations in a system of partial differential equations (PDEs) from boundary measurements. A reconstruction method is developed based on the monotonicity properties of the Neumann-to-Dirichlet operator, which provides a theoretical foundation for stable support identification. To improve reconstruction accuracy, particularly when parameters have disjoint supports, we propose a combined regularization approach integrating monotonicity principles with Truncated Singular Value Decomposition (TSVD) regularization. This hybrid strategy enhances robustness against noise and ensures sharper support localization. Numerical experiments demonstrate the effectiveness of the proposed method, confirming its applicability in practical scenarios with varying parameter configurations.

Raja Dziri : An inverse obstacle problem for Stokes flow under Navier boundary condition

Affiliation : Faculty of sciences of Tunis, Tunisia

We study the inverse problem of detecting an obstacle immersed in a steady Stokes flow under Navier boundary conditions. Assuming a non-axisymmetric fluid-obstacle configuration and a perfect slip condition on the obstacle boundary, we establish an identifiability result. Furthermore, we address the stability of the inverse problem using shape optimization tools. In particular, we show that the Riesz operator corresponding to the shape Hessian at a critical shape is compact.

Ilias Ftouhi : Optimal placement of sensors via distance criteria

Affiliation : Friedrich-Alexander-University Erlangen-Nurnberg, Germany

In this talk, we focus on the optimal placement of a finite number of sensors inside a given region to minimize the mean/maximal distance to the points of the domain. To address this simple and natural geometric criterion of performance, based on distance functions, we propose to employ geometric analysis methods combined with a classical result of Varadhan, which provides an efficient approximation of the distance function through the solution of a simple elliptic PDE. Our approximation scheme is validated by theoretical $\Gamma$-convergence results and illustrated by relevant numerical simulations. This talk is based on previous and ongoing works in collaboration with Enrique Zuazua (FAU Erlangen, Germany).

Fatma Boumiza : Identification and control of point sources locations in unsteady Newtonian Flow

Affiliation : Analysis and Control of PDE (UR13ES64) Monastir, Tunisia

We deal with non homogeneous unsteady Stokes equations in spaces of low regularity. We scrutinize a viscous incompressible fluid under the action of a finite number of particles located inside the flow domain. Each particle exerts a point-wise force on the fluid modeled by the Dirac distribution. The main targets of this task are the solvability outcome in a very weak sense and the identification of point-sources locations through an optimization process of an established performance functional.

The regularity-lack manufactured by the source term does not enable to extract neither optimality conditions nor numerical algorithms. For the sake of evading the regularity trouble, a relaxation method is stipulated and therefrom the identification model is formulated as a topological optimization problem. Numerical results based on efficient algorithms are provided to illustrate the effectiveness of the established theory.

Fabien Caubet : Shape optimization for contact problems

LMAP, Université de Pau

We are interested in the sensitivity analysis with respect to shape of contact problems. We consider a boundary condition in the form of Tresca's law of friction, and aim to solve a non-linear, non-differentiable (shape) optimization problem without any regularization or penalization procedure in this context. More precisely, using tools of convex and nonsmooth analysis, such as the proximal operator and the notion of second-order epi- dierentiability, we show that the solution of a scalar Tresca problem admits a shape derivative, given by the solution of a Signorini-type problem. We then explicitly characterize the shape gradient of the corresponding energy functional, and design a descent method for numerical simulations. Finally, we discuss the possible extension of this type of results to the (more concrete) case of linear elasticity.

Carlos Brito-Pacheco : Shape and topology optimization of regions supporting boundary conditions

Affiliation : Equipe Commedia, INRIA Paris

This work focuses on the optimization of a region along the boundary of a fixed domain in 3 dimensional space, governed by the boundary conditions of a physical equation. Specifically, it considers the minimization of a criterion deepending on the solution of boundary value problem posed over the aforementioned region. From a theoretical standpoint, this research introduces notions of shape and topological derivatives tailored to such objective functions, along with streamlined procedures for their computation in general settings. On the numerical side, accurately representing the optimized boundary region and iteratively updating it during the optimization process necessitates a robust approach. This is achieved using a level set-based mesh evolution method designed specifically for boundary optimization problems. The methodology is demonstrated through applications to practical problems, including optimizing electric conductivity, structural mechanics, and acoustics, highlighting its versatility across various physical contexts.

Roman Moskalenko : On the asymptotic effect of Robin boundary conditions on small subsets of the boundary

Institut Fourier, Université Grenoble-Alpes

We analyze the asymptotic behavior of solutions to the Poisson equation under the perturbation of boundary conditions. Specifically, we consider the case where Neumann or Dirichlet boundary conditions are replaced by a Robin condition on a small portion of the boundary. We derive precise asymptotic formulas of the voltage potential, that characterize the influence of this localized modification. We think of such a Robin condition as the limiting eect of a thin electrode placed on the boundary. We then study, within the range of parameters in which this interpretation is valid, the uniform convergence of our asymptotic formulas for the voltage potential.

Orateur 4 : Titre présentation

Affiliation : Université/Organisation

Description intervention.

Rajae Aboulaich : Physics-Informed and Stochastic Control-Based Strategies for Robust Epidemiological Forecasting

Affiliation : Mohammed V University, Rabat, Marocco

Epidemiological models are essential tools for understanding disease dynamics. They allow us to predict their spread, evaluate control strategies, and ultimately make informed public health decisions that can save lives. However, these models face major challenges, particularly regarding parameter uncertainty. Many parameters used in models depend on assumptions or data whose quality and availability can vary, leading to uncertainty in model predictions and recommendations.

To address these challenges, physics-based neural networks (PINNs) offer an innovative framework. By integrating the physical or epidemiological laws governing diseases, typically in the form of differential equations, directly into the neural network architecture, PINNs provide an attractive way to estimate parameters and manage the complexity of disease modeling.

Another promising and equally interesting approach is stochastic optimal control, which integrates uncertainty of parameters directly into the decision-making process. This formulation allows us to account for the random nature of the biological parameters under consideration. By combining stochastic modeling and control theory, we are able to study the uncertainties due to parameter estimation.

In this work, we aim to present the results and simulations obtained using the PINN approach and those obtained using the stochastic optimal control method. Both methodologies offer interesting ways for parameter estimation and the study of the uncertainties due to this estimation.

Michel Duprez : Models of mosquito population control strategies for fighting against arboviruse

Affiliation : Centre Inria Nancy-Grand-Est, France

In the fight against vector-borne arboviruses, an important strategy of control of epidemic consists in controlling the population of the vector, Aedes mosquitoes in this case. Among possible actions, a technique consist in releasing sterile mosquitoes to reduce the size of the population (Sterile Insect Technique). This talk is devoted to studying the issue of optimizing the dissemination protocol for each of these strategies, in order to get as close as possible to these objectives. Starting from a mathematical model describing the dynamic of a mosquitoes population, we will study the control problem and introduce the cost function standing for sterile insect technique. In a second step, we will consider a model with several patches modeling the spatial repartitions of the population. Then, we will establish some properties of these two optimal control problems. Finally, we will illustrate our results with numerical simulations.

Mohammed Elghandouri : Protecting Susceptible or Isolating Infected Individuals in SIR Models: An Optimal Control Approach

Affiliation : Centre Inria Lyon, France

There are many ways to control the spread of infectious diseases. Among these, protecting susceptible individuals and isolating infected individuals are key strategies. Hence, a critical question remains: Which approach is more effective--protecting susceptibles, isolating infecteds, or combining both-to mitigate disease transmission?

In this work, we address this question by analyzing protection and isolation strategies within the framework of a classical SIR (Susceptible - Infected - Recovered) epidemic model. We consider three scenarios: (i) combined protection and isolation, (ii) protection alone, and (iii) isolation alone.

Our analysis, grounded in Optimal Control Theory, reveals that in the short-term context of epidemic control, the combination of protection and isolation is generally the most effective and reliable strategy. However, in certain scenarios, this combination is unavailable or may not be necessary. When only one strategy can be applied, isolation tends to be more effective than protection, particularly under unfavorable conditions such as high transmission rates or low recovery rates. Nevertheless, protection alone can still be successful under specific circumstances, such as when the recovery rate is high.

Tiné Léon Matar : Estimation du taux de division cellulaire pour un modèle de croissance-fragmentation

Affiliation : Universite Claude Bernard, Lyon, France

Les équations de croissance-fragmentation apparaissent dans de nombreux contextes, allant de la division cellulaire à la polymérisation des protéines, en passant par les neurosciences, etc. L'observation directe de la dynamique temporelle étant souvent difficile, il est particulièrement intéressant de développer des méthodes théoriques et numériques permettant de récuperer les vitesses de réaction et les paramètres de l'équation à partir de l'observation indirecte de la solution. Dans le prolongement des travaux de Perthame et Zubelli (2007) et de Doumic et al. (2009) pour le cas spécifique de l'équation de division cellulaire, nous abordons ici la question générale de l'estimation du taux de fragmentation à partir de l'observation de la solution asymptotique en temps, lorsque le noyau de fragmentation et les taux de croissance sont entièrement généraux. Dans cet exposé, nous présentons à la fois des résultats theoriques et des méthodes numériques, et discutons de quelques ouvertures.

Lekbir Afraites : Inverse Shape Identification in Advection-Diffusion Equations using ADMM and Coupled Complex Boundary Methods

Affiliation : FST-Sultan Moulay Slimane University, Beni Mellal, Morocco

This talk focuses on numerically solving a shape identification problem related to advection-diffusion processes with space-dependent coefficients using shape optimization techniques. Two boundary-type cost functionals are considered, and their corresponding variations with respect to shapes are derived using the adjoint method, employing the chain rule approach. This involves firstly utilizing the material derivative of the state system and secondly using its shape derivative. Subsequently, an alternating direction method of multipliers (ADMM) combined with the Sobolev-gradient-descent algorithm is applied to stably solve the shape reconstruction problem. Numerical experiments in two and three dimensions are conducted to demonstrate the feasibility of the methods.

Julius Fergy Rabago : An Inverse Shape Optimization Approach for Localizing Subdermal Tumors and Burns

Affiliation : Science and Engineering Institute, Kanazawa University, Japan

This talk presents a method for estimating the shape and location of an embedded tumor using shape optimization techniques, specifically through the coupled complex boundary method. The inverse problem, characterized by a measured temperature profile and corresponding heat flux (e.g., from infrared thermography), is reformulated as a complex boundary value problem with a complex Robin boundary condition, thereby simplifying its over-specified nature. The geometry of the tumor is identified by optimizing an objective functional that depends on the imaginary part of the solution throughout the domain. The shape derivative of the functional is derived through shape sensitivity analysis. An iterative algorithm is developed to numerically recover the tumor shape, based on the Riesz representative of the gradient and implemented using the finite element method. Numerical examples are presented to validate the theoretical results and to demonstrate the accuracy and effectiveness of the proposed method.

Mourad Hrizi : Reconstruction of Metal-Semiconductor Interfaces using the Kohn-Vogelius formulation and the Topological Gradient method

Affiliation : LR Analysis and Control of PDE -- ISMAIK, Kairouan University, Tunisia

In this talk, we address the inverse problem of reconstructing an interior interface arising in an elliptic partial differential equation defined over a bounded domain, using boundary measurement data. This problem is motivated by applications in semiconductor transistor modeling. We propose a novel shape reconstruction approach that combines the Kohn-Vogelius formulation with the topological sensitivity method. The inverse problem is recast as a topology optimization problem, and a topological sensitivity analysis is performed on an appropriate cost functional. The unknown contact interface is then identified as a level-set of the resulting topological gradient. Finally, we present several numerical examples that demonstrate the effectiveness and robustness of the proposed method.

Mohamed Bensalah : Bayesian recovery of time-varying fractional order in an anomalous diffusion equation and applications

Affiliation : LR Analysis and Control of PDE -- ISSATS, Sousse University, Tunisia

This talk concerns the inverse problem of recovering the time-varying fractional order $\alpha(t)$ in a time-fractional diffusion equation. This study is motivated by applications in subsurface flows and shale gas extraction. The fractional order $\alpha(t)$ plays a crucial role in modeling anomalous diffusion processes, such as those observed in complex geological formations. Prior to developing the reconstruction method, the uniqueness of the solution is analyzed, ensuring that the fractional order can be uniquely determined from the available space-averaged data. Following this theoretical investigation, a Bayesian approach is proposed to recover $\alpha (t)$ from space-averaged measurements, which are practical and robust against noise. The Bayesian framework allows for both parameter estimation and uncertainty quantification, making it a powerful tool for handling ill-posed inverse problems. Numerical experiments validate the proposed approach.

Helmut Harbrecht : Shape optimization of a thermoelastic body under thermal uncertainties

Affiliation : University of Basel (Switzerland)

We consider a shape optimization problem in the framework of the thermoelasticity model under uncertainty. The uncertainty is supposed to be located in the Robin boundary condition of the heat equation. The purpose of considering this model is to account for thermal residual stresses or thermal deformations, which may hinder the mechanical properties of the final design in case of a high environmental temperature. In this situation, the presence of uncertainty in the external temperature must be taken into account to ensure the correct manufacturing and performance of the device of interest. The objective functional under consideration is based on volume minimization in the presence of an inequality constraint for a quadratic shape functional. Exemplarily, we consider the $L^2$-norm of the von Mises stress and demonstrate that the robust constraint and its derivative are completely determined by low order moments of the random input, thus computable by means of low-rank approximation. The resulting shape optimization problem is discretized by using the finite element method for the underlying partial differential equations and the level-set method to represent the sought domain. Numerical results for a model case in structural optimization are given.

This is a joint with Marc Dambrine, Giulio Gargantini, and Viacheslav Karnaev.

Mohamed Aziz Boukraa : Shear-wave speed estimation in ultrasound time-harmonic elastography.

Affiliation : University of Oslo, Norway.

The stiffness of biological tissue serves as an important indicator in medical diagnostics, for example, in the assessment of liver cirrhosis or in distinguishing between benign and malignant tumors.

In this work, we consider the inverse problem of recovering the elastic parameters of biological tissue, often reduced to the shear-wave speed in elastography, from ultrasound-measured shear-wave displacement fields. In ultrasound time-harmonic elastography, shear-waves are first induced using external vibrators, and their propagation is tracked using high-frame-rate plane wave ultrasound imaging. However, ultrasound data are particularly challenging to work with, as they are often corrupted by high levels of noise, making the inverse problem significantly more difficult to solve.

We develop an iterative inversion approach based on full-waveform inversion (FWI), where a misfit functional that compares simulated and measured ultrasound data is minimized. A total variation regularization term is also considered to promote piecewise constant solutions. We present reconstruction results on experimental phantom data and demonstrate that the method effectively handles noisy measurements and accurately recovers the shear-wave speed of the medium. We also address common assumptions made in the physical model and provide numerical reconstructions for various synthetic ground truths, including comparisons with other methods.

Hatem Zayeni : Simultaneous heat source identification and thermal field reconstruction using the fading regularization method applied to noisy experimental data

Affiliation : Université de Poitiers, Institut Pprime.

Inverse problems associated with elliptic equations or systems refer to the challenging task of determining unknown parameters, geometry, or boundary conditions. Since the leading work of Hadamard, such problems are known to be ill-posed and regularization methods are used. In this presentation, we investigate the application of the fading regularization method (FRM) to address the ill-posed data assimilation and the source identification of a thermal problem, that of a thin plate with a heating patch. This regularization approach is an iterative process and convergence of which has also been demonstrated. This method simplifies the solution of the inverse problem by transforming it into a sequence of optimization problems with an equality constraint. At each iterative step, one searches for the solution of the equilibrium equation that best approximates the experimental data and the previously optimal element.

The algorithm is numerically implemented using the meshless method of fundamental solutions (MFS). The inverse technique is first validated with synthetic data from a reference finite element solution. In a second step, the method is applied to experimental data obtained by infrared thermography.

Chaiyma Abid: Fading Regularization for the Cauchy Problem in Helmholtz Equation via Boundary Integral Methods

Affiliation : Laboratory of Mathematical and Numerical Modeling for Engineering Sciences.

In this work, we investigate the resolution of the Cauchy problem associated with the Helmholtz equation by employing the fading regularization method combined with boundary integral equations. The main objective is to reconstruct unknown boundary data on the inaccessible part of the domain, starting from overdetermined Cauchy data given on an accessible part of the boundary. To this end, we reformulate the problem into a system of boundary integral equations defined on the whole boundary of the domain. By integrating the fading regularization method, we stabilize the inherently ill-posed  problem. The method leads to the derivation of a linear system, where the regularization term progressively fades away, allowing for a balance between stability and accuracy. A rigorous theoretical analysis is carried out, ensuring the well-posedness of the regularized problem and providing convergence guarantees as the regularization parameter vanishes. Numerical experiments are then performed in a three-dimensional framework to validate the effectiveness of the proposed approach. We consider various configurations, including domains with different geometrical shapes and measurements affected by noise. The numerical results confirm the efficiency of the method in accurately reconstructing the unknown data. The proposed algorithm shows a fast convergence rate and remains stable even in the presence of noisy data. Moreover, the approach proves to be robust and adaptable to complex domain geometries, highlighting its potential for practical applications in inverse problems related to wave propagation phenomena.

Kévin Pascreau: Identification of material properties and reconstruction of boundary conditions from partial experimental displacement fields

Affiliation : Université de Poitiers, Institut Pprime.

Determining mechanical properties from full-field measurements is a major challenge in mechanics, especially when the available displacement fields are partial or noisy. In this context, we apply the fading regularization method (FRM) to solve an inverse problem aimed at identifying a homogeneous Young’s modulus from 2D experimental displacement fields obtained via Digital Image Correlation (DIC), in a combined bending-compression test. The approach also enables the reconstruction of boundary conditions along the domain edges without any prior assumptions, relying solely on internal data. The problem is formulated as a sequence of regularized optimization problems, leading to a stable estimation of the global stiffness parameter. Initial results show good robustness to noise and the ability to recover a globally coherent mechanical response. This study represents a step toward the identification of localized parameters and extension to heterogeneous materials or non-destructive testing applications.

Zakaria Belhachmi : Motion analysis with the optical flow for constant and varying illumination: an optimal transport point of view

Affiliation : Dept. Math., IRIMAS, University of Upper Alsace

In computer vision, motion analysis involves modeling the movement of objects within a scene. A central question in this domain is optical flow estimation, which is the determination of a vector field defining the motion of 'moving parts' from one image to another. The widely dominating approach is a pixel-wise transport of the intensity map along trajectories driven by the optical flow. This approach, based on the assumption of constancy of the intensity in the motion, leads to an ill-posed problem and gives a large panel of models that differ by the regularization strategy chosen. When there are illumination changes in the scene, this constancy assumption does not hold, and within the same approach, several models were proposed, for example, adding other constancy assumptions on some gradient norm or simply adding a new variable to the vector field to model the change in the illumination.

In this talk, we discuss a new approach based on optimal transport theory, which completely changes the constancy assumption from the pixel-wise point of view to a mass transport of densities (small volumes of pixels) from one location to another. The constancy assumption becomes a "compressibility constraint" in the new approach, that is, a volume preserving transport when there is no illumination change, whereas the illumination change is given by the violation of such volume preservation. The new model is a well-posed problem, gives an elaborated sense to the illumination changes, and very satisfactory numerical results, but at a higher cost than classical models, as it does not yet take advantage of the algorithmic progress in the numerical mass transport methods.

Maher Moakher : Geometric similarity measure for multimodal image registration

Affiliation : LAMSIN, ENIT, University of Tunis El Manar

We employ deformable models for the registration of multimodal images. To construct a robust and efficient similarity measure between the images to be registered, we extract geometric features from the images, such as edges and thin structures. For this purpose, we utilize the Blake-Zisserman energy, which is particularly effective for detecting discontinuities at multiple scales-both first-and second-order. A theoretical analysis of the proposed model will be provided.

For the numerical solution, we adopt a gradient descent method, iteratively solving the corresponding Euler-Lagrange equation. We present and discuss numerical results that demonstrate the effectiveness of the proposed model.

This is a joint work with Mohamed Lajili, Anis Theljani and Badreddine Rjaibi.

Bochra Mejri : Vertex characterization via second-order topological derivatives

Affiliation : Johann Radon Institute for Computational and Applied Mathematics

This talk focuses on identifying vertex characteristics in 2D images using topological asymptotic analysis. Vertex characteristics include both the location and the type of the vertex, with the latter defined by the number of lines forming it and the corresponding angles. This problem is crucial for computer vision tasks, such as distinguishing between fore- and background objects in 3D scenes. We compute the second-order topological derivative of a Mumford-Shah type functional with respect to inclusion shapes representing various vertex types. This derivative assigns a likelihood to each pixel that a particular vertex type appears there. Numerical tests demonstrate the effectiveness of the proposed approach.

This is a joint work with Peter Gangl and Otmar Scherzer.

Orateur 4 : Titre présentation

Affiliation : Université/Organisation

Description intervention.