Special Session 169: Inverse problems arising in partial differential equations and mathematical physics

Asymptotic behavior of the Becker-D{\\\\o}ring equations with fixed monomer concentration

Inmaculada Benitez Berral Universidad de Granada Spain Co-Author(s):    Prasanta Kumar Barik, Jos{\\\\`e} A. Ca{\\\\~n}izo

Abstract: The Becker-D\{o}ring equations, introduced by Becker and D\{o}ring (1935), are a classical model for coagulation and fragmentation processes, describing how clusters of particles grow or break apart over time. They arise in a wide range of applications, including phase transitions, condensation phenomena, and aggregation processes in biological systems. While significant research has focused on the nonlinear case, including the study of metastable states by Penrose (1989) and the exponential convergence results in the subcritical case by Ca\~{n}izo & Lods (2013), in this talk, we focus on the linear Becker-D\{o}ring equations, corresponding to the case where the concentration of monomers is kept constant. In this setting, the evolution of the cluster densities $c_i(t)$ is governed by \begin{equation*} \frac{d}{dt}c_i(t) = W_{i-1}-W_i, \quad \text{for all } i \geq 2, \end{equation*} where the fluxes $W_i$ describe the balance between coagulation and fragmentation mechanisms. Although the total mass of the system is not conserved in this linear case, the equations admit explicit equilibrium solutions. We study the long-time behavior of solutions and show that they converge exponentially fast towards equilibrium under natural assumptions on the model coefficients. The analysis combines ideas from entropy methods (Kreer (1993)) and spectral theory, leading to quantitative estimates of the convergence rate in both subcritical and supercritical regimes.

On the Determination of an Anisotropic Conductivity from the Neumann-to-Dirichlet Map in a Semilinear Elliptic Equation

Elena Beretta New York University Abu Dhabi United Arab Emirates Co-Author(s):    Elisa Francini Dario Pierotti Eva Sincich

Abstract: We study the inverse boundary value problem of detecting a non-uniform conductivity motivated by pacing-guided ablation in cardiac electrophysiology. At the stationary level, the transmembrane potential $u$ in a region \(\Omega\subset\mathbb{R}^3\) of cardiac tissue satisfies \[ -\nabla\!\cdot(\gamma\nabla u)+\alpha u^3=0 \quad \text{in }\Omega,\qquad \gamma\nabla u\cdot\nu=g \quad \text{on }\partial\Omega, \] where $\gamma$ is an anisotropic conductivity tensor and $\alpha$ a nonlinear ionic response coefficient. The Neumann data $g$ represent pacing currents, and the boundary values $u|_{\partial\Omega}$ correspond to invasive voltage measurements. Ischemic regions are modeled by a subdomain $D\subset\Omega$ where $\gamma$ is piecewise constant. We address the inverse problem of determining $\gamma$ from the Neumann-to-Dirichlet (NtD) map, assuming that $\alpha$ and $D$ are known. To our knowledge, uniqueness in the case of NtD data with anisotropic conductivities in this nonlinear setting has not been analyzed in previous work. Using a first-order linearization around a nontrivial pacing current, we prove uniqueness for $\gamma$.

Context aware iterative graph Laplacian for industrial imaging

Florian Bossmann Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Davide Bianchi, Wenlong Wang

Abstract: Industrial imaging often suffers from highly noised and undersampled data. As such, many state-of-the-art reconstruction algorithms are not applicable and practitioners rely on simple heuristic methods often paired with a simplified physical model. To improve the reconstruction quality, iterative methods are a favorable choice. One such method is the iterative graph Laplacian, which iteratively solves $$x_n=\arg\min\limits_{x}\|Ax-b\|_2+\lambda\|\Delta_{x_{n-1}}x\|_1$$ where $\Delta_{x_{n-1}}$ is a graph Laplacian matrix based on the previous solution $x_{n-1}$. The initial guess $x_0$ is obtained using any chosen reconstruction algorithm. This method has proven to outperform other iterative approaches since the data adaptive graph Laplacian $\Delta_{x_{n-1}}$ imitates the structure of the given solution. It can differentiate between desired and undesired features within the data. However, the above approach requires a decent starting guess $x_0$ with structural features close to the real solution. This is not always the case in industrial imaging. We present an updated version of the iterative graph Laplacian by adjusting the construction of $\Delta_{x_{n-1}}$. The new matrix is context-aware and responds to possible artifacts or anomalies in the given solution $x_{n-1}$. We use a null space projection to counter any undesired edges within the solution as well as a directional $\ell_1/\ell_2$ weight to detect blurred edges. We demonstrate the method on non-destructive testing and seismic exploration examples.

Optimization-free coefficient identification via matched asymptotics in periodic advection-diffusion-reaction problems

DMITRII CHAIKOVSKII Shenzhen MSU-BIT University Peoples Rep of China Co-Author(s):    Dmitrii Chaikovskii, Ye Zhang, Aleksei Liubavin

Abstract: We address forward analysis and inverse coefficient recovery for reaction-diffusion-advection models with time-periodic forcing in the singularly perturbed setting where diffusion is small and solutions develop a sharp moving front. Using matched asymptotic expansions, we construct a uniformly valid periodic approximation that resolves both the outer solution away from the front and the inner transition layer governing the front dynamics. The approximation yields explicit formulas for the leading-order profile and for the front location, and we provide rigorous error bounds. Building on this asymptotic surrogate, we develop two fast identification strategies: one for spatially heterogeneous coefficients and one for time-varying coefficients. In both cases, the inverse problem is reduced to a regularized least-squares fit involving readily measurable features (e.g., local gradients and front motion), eliminating iterative PDE-constrained optimization and significantly reducing computational cost. We prove stability and obtain convergence rates. Computational experiments confirm that the method recovers unknown coefficients accurately and robustly across a range of parameter regimes and noise levels, making it suitable for rapid calibration in environmental, biological, and engineering applications.

Large-Scale Model-Based 3D Image Reconstruction for Raster-Scan Optoacoustic Mesoscopy

Lena Dunst German Electron Synchrotron (DESY), Helmholtz Imaging Germany Co-Author(s):    Martin Burger

Abstract: Optoacoustic mesoscopy is a biomedical imaging technique that provides images of the microvasculature of the skin. The skin tissue is illuminated with a short laser light pulse. Due to the optoacoustic effect this results in the generation of pressure waves, which leads to the inverse problem of reconstructing an image, representing the absorption properties of the tissue, from ultrasound measurements. We focus on a spherically focused ultrasound transducer performing a raster-scan on the skin surface and use a model-based image reconstruction approach. It consists of computing the forward model matrix describing the measurement process and using variational regularization to solve an optimization problem incorporating the forward model. A main challenge of the model-based reconstruction approach for 3D images is the excessive memory demand and the computation time. We reduce both by exploiting the symmetry of the ultrasound transducer and using a stochastic proximal gradient algorithm for iterative image reconstruction. In each iteration we randomly choose a certain number of model matrix rows and corresponding signal values for which we perform a gradient descent step on the data fidelity term and a proximal step on the regularization term. Furthermore, we use Langevin sampling to enable Bayesian uncertainty quantification.

Characterizing the detailed balance property by means of measurements in linear chemical systems

Eugenia Franco University of Bonn Germany Co-Author(s):    B. Kepka, J.J.L. Velazquez

Abstract: The question that we address during this talk is whether it is possible to determine if a linear chemical system satisfies the detailed balance property using measurements. The property of detailed balance is a property of the chemical rates of chemical systems and it has an important physical meaning: active systems are modeled by equations that do not satisfy the detailed balance property, while passive systems are modeled by equations that satisfy the detailed balance property. We assume that the chemical rates of the reactions in systems are not known and we assume that we can access information on the system via some measurements. A measurement $R_{ij}(t)$ is the concentration of the substance i at time $t>0$ after the injection of a substance j at time $t=0$. We obtain a condition involving reciprocal measurements (i.e. $R_{ij}(t)$ and $R_{ji}(t)$) that is necessary, but not sufficient for the detailed balance condition to hold in the network. Moreover, we prove that this necessary condition is also sufficient if a topological condition is satisfied, as well as a stability property that guarantees that the chemical rates are not fine tuned.

Qualitative Experimental Design: an application in math biology

Kathrin Hellmuth California Institute of Technology USA Co-Author(s):    Christian Klingenberg, Qin Li, Min Tang

Abstract: Experimental design determines the quality of available data and often has a strong effect on the well- or ill-posedness of an inverse problem. The classical optimal experimental design methodology rephrases the task of finding a design that yields informative data as an optimization problem that minimizes uncertainty of the reconstruction. In this talk, we want to extend this methodology to what we call qualitative experimental design. By relaxing the optimality to a sufficiency equirement, we concentrate on finding suitable designs that yield data which is sensitive w.r.t. the unknown parameter and thus allows its inference - at least locally. This opens a new methodological toolbox, and we lay out two strategies: \begin{itemize} \item some theoretical well-posedness proofs are constructive and rely on the design of suitable measurements. Possibly after discretization, these constructions can be translated to the practical setting in a relaxation-of-theory. \item a more generally applicable approach samples designs according to a sensitivity based distribution derived from matrix sketching. This method shall be investigated on an easy toy system, the potential reconstruction problem related to the stationary Schroedinger equation. \end{itemize} We shall demonstrate both approaches on an inverse problem from kinetic PDEs related to bacterial chemotaxis.

Inverse Source Problems for Navier-Stokes Equations

Khonatbek Khompysh Al-Farabi Kazakh National University Kazakhstan Co-Author(s):

Abstract: This paper talk I will discuss on unique solvability of some inverse source problems for the Navier-Stokes equations. The inverse problem consists in recovering a time-dependent source coefficient (intensity of external forces) from additional information on the velocity field given in an integral form over a spatial domain. We investigate this problem in three settings: the linear case, the nonlinear case, and a special case of the right-hand side. For the linear problem, we establish global-in-time existence, uniqueness, and stability results for both weak and strong solutions under suitable assumptions on the data. For the nonlinear problem, we consider this in two- and three-dimensional cases and prove the local-in-time (for sufficiently small data) existence and uniqueness of strong solutions. In addition, we analyze a special case of the right-hand side that allows to improve these results, in particular, in the two-dimensional case we prove the global-in-time unique solvability of the nonlinear problem.

Inverse problems for a nonlinear dynamical Sch\odinger operator with magnetic potential

Boya Liu North Dakota State University USA Co-Author(s):    Mandeep Kumar, Manmohan Vashith

Abstract: In this talk we discuss two inverse problems concerning a nonlinear dynamical Schr\odinger operator with magnetic potential. We show that the Dirichlet-to-Neumann map determines the nonlinear coefficients uniquely. We shall consider both the full data and the partial data setting. In particular, with the assumption that the coefficients are known near the boundary, measurements are made on arbitrarily small sets of the lateral boundary of a space-time.

On the Robustness of Adaptive Eigenspace Inversion for Inverse Problems

Nasrin Nikbakht University Of Auckland New Zealand Co-Author(s):    Nasrin Nikbakht, Dr Marie Graff, Dr Melissa Tacy

Abstract: This talk addresses the robustness of the Adaptive Eigenspace Inversion (AEI) method for inverse problems arising in partial differential equations. Such problems frequently involve the reconstruction of unknown parameters from noisy measurements, where stability and reliability are critical challenges. A revised framework is proposed by re-examining two key regularisation mechanisms. The number of eigenfunctions is selected using a sensitivity-based criterion, ensuring that only the most influential modes contribute to the reconstruction process. In parallel, the diffusivity cancellation parameter is analysed and shown to play a central role in stabilising the method, leading to the identification of an effective choice. Morozov`s discrepancy principle is further incorporated to guide the adaptation process in the presence of data uncertainty. The resulting approach, termed adaptive-$\varepsilon$ ASI, demonstrates improved stability and efficiency. Numerical experiments highlight its advantages over the original AEI method and standard techniques such as Tikhonov regularisation.

Stability of inverse problems arising from wave equations of Magnetic Schrodinger operators

Hadrian Quan University of California Santa Cruz USA Co-Author(s):    Boya Liu, Teemu Saksala, Lili Yan

Abstract: I will present recent work regarding Holder stability estimates for two inverse problems arising from the wave equation associated to a Magnetic Schrodinger operator on a simple Riemannian manifold. The first such inverse problem is the question of stably recovering a non-negative electric potential, and the solenoidal part of a magnetic potential from measured Neumann boundary observations; the second problem similarly studies the question of stably recovering such potentials from the boundary spectral data of the associated Magnetic Schrodinger operator. I will discuss the connection between these two problems, and the technical issues introduced by the magnetic potential.

Regularization and discretization in the reconstruction for elliptic inverse problems

Luca Rondi Universit\`a degli Studi di Pavia Italy Co-Author(s):

Abstract: We consider elliptic inverse problems such as the electrical impedance tomography or the determination of inhomogeneities by scattering measurements. We employ a variational approach for the reconstruction. Due to severe ill-posedness, it is needed to add to our minimization problem a regularization, for example of Tikhonov-type. The discretization used for the numerical implementation may also be another important cause of instability, which is often overlooked. We investigate how to handle simultaneously the regularization and the discretization so that the solution to the corresponding regularized and discrete minimization problem is a good approximation of the solution to the inverse problem.

Lipschitz Stability for Polyhedral Elastic Inclusions from Partial Data

Eva Sincich University of Trieste Italy Co-Author(s):    Andrea Aspri, Elena Beretta, Elisa Francini, Antonino Morassi, Edi Rosset, Sergio Vessella

Abstract: We deal with the inverse problem of determining a polyhedral inclusion compactly contained in an elastic body from boundary measurements of traction and displacement taken on an open portion of the boundary. Both the inclusion and the body are made of homogeneous isotropic material. Under suitable assumptions on the geometry of the unknown inclusion, we prove a constructive Lipschitz stability estimate from the local Dirichlet-to-Neumann map. The main tools of our approach are quantitative estimates of unique continuation, the construction of singular solutions to the Lam\`{e} system, a boundary formula for the Gateaux derivative of the Dirichlet-to-Neumann map with respect to a deformation homotopy, and the determination of a lower bound of this derivative using, among other tools, the Rongved biphase fundamental solution for the Lam\`{e} system.

A Calderon Problem for the Dirac Operator

Carlos Valero Instituto de Ciencias Matematicas (ICMAT) Spain Co-Author(s):

Abstract: We consider on a spin manifold with boundary the twisted Dirac operator $D_A$ with local boundary conditions, possibly coupled to a unitary connection $A$. For suitable values of $m$, we define an analogue of the Dirichlet-to-Neumann map corresponding to $D_A - m$, which we call the boundary conjugation (BC) map. By computing its symbol in dimensions $n \geq 3$, we show that the BC map determines the infinite-order jet of the metric and connection on the boundary in the case $m \neq 0$, when the conformal symmetry of the Dirac equation is broken. We go on to show that a real-analytic Riemannian manifold and twisted spin connection can be recovered from the BC map. Similar results hold in dimension $2$ when the auxiliary connection $A$ is absent.

Inverse scattering of the relativistic Schr\odinger operator at a fixed energy

Yiran Wang Emory University USA Co-Author(s):

Abstract: We consider the scattering of relativistic Schr\odinger operators which are perturbations of the fractional Laplacian. For poly-homogeneous potentials decaying at infinity, we show that the asymptotics of the potential can be recovered from the scattering matrix at a fixed energy. This talk is based on a joint work with G. Uhlmann.

Inverse Problems for the Burgers Equation in Degenerate Domains

Madi Yergaliyev Institute of mathematics and mathematical modeling Kazakhstan Co-Author(s):

Abstract: Let us consider a nonlinearly degenerate domain whose spatial boundaries are determined by time-dependent functions that coincide at the initial moment. In this domain, we investigate inverse problems for the Burgers equation under various combinations of unknown functions and boundary conditions. It is shown that, due to the degeneracy of the domain, the solvability of the proposed inverse problems requires additional conditions. The main objective of this work is to determine conditions under which these inverse problems are uniquely solvable.

Convergence analysis for a coefficient identification problem

Wensheng Zhang Zhang Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Huimin Huang

Abstract: In this talk, we consider the convergence rate for Tikhonov regularization of the problem of identifying the coefficient in acoustic wave equation in the frequency-domain. We assume that the imprecise measurement data in a subdomain is known with a measurement error of level. First we propose to regularize this coefficient identification problem by minimizing a special functional. Then the characters of the functional are investigated. Finally, we obtain the convergence rate for the Tikhonov regularized solution under an easily satisfied source condition.

Forward and inverse problems for the BGK model with in-flow boundary condition

Hanming Zhou University of California Santa Barbara USA Co-Author(s):    Ru-Yu Lai, Hanming Zhou

Abstract: In this talk, we will discuss the Bhatnagar-Gross-Krook (BGK) equation with in-flow boundary condition in a smooth bounded domain. First, we show that the BGK model linearized around a global Maxwellian admits a unique solution with some weighted $L^\infty$ bound if the initial data and boundary condition are small perturbation around the global Maxwellian. Second, we investigate unique reconstruction of the collision frequency of the BGK equation from the albedo operator, which maps from the in-flow density to the out-flow density. We utilize the linearization technique and highly concentrated in-flow density to infer the velocity-independent collision frequency. The talk is based on joint work with Ru-Yu Lai.