Past Inverse problems seminars

Time & Location: Thursday, 29 October at 14:15, room C222 (Exactum)

Speaker: Tianyu Cai, Fudan University/ Melbourne University

Title: Single-Spike Solutions for a Schnakenberg System with Geodesic Levy Diffusion

Abstract: We study the stationary fractional Schnakenberg system on the round sphere \(\mathbb S^n\), \(n\geq2\), with diffusion governed by the full geodesic Lévy generator of order \(2\alpha\), \(0<\alpha<1\). For every sufficiently small fixed feed parameter \(b>0\), we construct positive, nonconstant, rotationally symmetric solutions concentrating at any prescribed point as \(\varepsilon\to0\). We also consider the leading behavior near the concentration point and its antipode. 

Time & Location: Thursday, 27 August at 14:15, room C222 (Exactum)

Speaker:  Catalin Carstea, National Yang Ming Chiao Tung University

Title: An anisotropic Calderón problem without a gauge, and related AI misadventures

Abstract: I will report on a recent preprint on the Calderón problem for piecewise polynomial anisotropic conductivities on partitions with many fl at faces. Depending on the maximum degree of these polynomials, the existence of a sufficient number of generically oriented flat faces on the interfaces of the partition is enough to prevent the well known diffeomorphism gauge and a uniqueness result is possible. I will present the result, report on how AI made it possible, and then describe my attempts to (auto)formalize it in Isabelle/HOL. 

Time & Location: Thursday, 30 April at 14:15, room C124 (Exactum)

Speaker:  Ilaria Trombini, University of Ferrara

Title:  Stochastic algorithm for the 3D Computed Tomography (CT) problems 

Abstract: 📄

Time & Location: Thursday, 23 April at 14:15, room C124 (Exactum)

Speaker:  Malena Espanol, Arizona State University

Title:  Variable Projection Methods for Large-Scale Separable Nonlinear Inverse Problems

Abstract: Many inverse problems in imaging and computational science lead to nonlinear least-squares formulations in which the forward model depends linearly on some parameters and nonlinearly on others. Variable projection is a powerful and efficient optimization approach that exploits this structure by eliminating the linear variables, thereby reducing dimensionality and improving the conditioning of the resulting problem. In this talk, I will provide an overview of the classical variable projection method and discuss its extension to large-scale inverse problems, with a particular focus on semi-blind deconvolution. I will present new regularization strategies and recent developments that incorporate approximate Jacobians computed using Krylov subspace techniques, enabling matrix-free implementations and improved scalability.

Time & Location: Thursday, 19 March at 14:15, room D122 (Exactum)

Speaker:  Gesa Sarnighausen, University of Göttingen

Title: Solving dynamic inverse problems with classical and data-driven methods

Abstract: We investigate time-dependent inverse problems which arise when one aims to recover a function from given observations where the function or the data depend on

time.

In this talk, we present two different approaches:

1. classical approach using Lebesgue-Bochner spaces in the function space setting

2. data-driven approach that incorporates the principle of causality
   
   

In the classical case, we first investigate geometrical properties of Lebesgue-Bochner spaces to implement Tikhononv regularization in these spaces using different regularities for time and space.

Then, we develop a regularization algorithm to solve a variational minimization problem that penalizes the time-derivative in a Lebesgue-Bochner space.

In the data-driven case, we aim to give a proof-of-concept on including the causality principle into the solution of linear time-dependent inverse problems by combining a transformer based approach with classical variational regularization which we call Transformer causality regularization (TCR).

We test all methods using the example of dynamic computerized tomography.

Time & Location: Thursday, 26 February at 14:15, room CK111 (Exactum)

Speaker:  Samuel Agenorwoth, LUT University

Title:  Uncertainty and Computation in Acoustic Imaging of Pipes. 

Abstract: In this work we compare two inverse methods for reconstructing the cross-sectional area of a one-dimensional pipe from boundary measurements, and we assess their behaviour statistically under data generated by each method’s forward model.

The first algorithm is the Sondhi Gopinath (SG) approach, originally developed for vocal-tract shape determination and adapted here to pipe area. The second algorithm is the boundary control method of Korpela Lassas Oksanen (KLO), originally proposed for wave-speed recovery and adapted here to reconstruct the pipe area. Our comparison is based on repeated experiments with randomly generated area profiles, under two complementary forward data-generation settings, with both SG and KLO algorithms applied as inverse solvers in each case. We also investigate sensitivity to measurement noise by varying the noise level. 

We quantify reconstruction accuracy using relative errors, report summary statistics, and assess paired differences using a paired t-test and the Wilcoxon signed-rank test. Finally, we complement these numerical results with graphical comparisons.

Time & Location: Thursday, 19 February at 14:15, room CK111 (Exactum)

Speaker:  Hiroshi Isozaki, University of Tsukuba

Title:  Quantum walk and velocity operator.

Abstract: We study basic properties of quantum walk taking into account of formalism in classical mechanics.

We introduce  the velocity density by the Legedre transformation and represent it in terms of Legendre's elliptic integral of 2nd kind. To make the explanation easy, we pick up only basic examples, however, the idea can be applied to general cases.

Time & Location: Thursday, 12 February at 14:15, room CK111 (Exactum)

Speaker:  Chen Xi, Fudan University

Title:  Inverse scattering by diffracted waves

Abstract: Beyond reflection and refraction, another form of wave deviation is defined as diffraction. Notably, when incident waves strike a corner, diffracted waves emanate from the corner tip and propagate in all directions. We propose a novel framework for detecting rigid cornered obstacles using measured diffracted wave data. This methodology offers two key advantages: first, the receiver's size and location can be arbitrary; second, the numerical inversion process does not require numerical data from the associated wave equations. 

Time & Location: Monday, 12 January, at 14:15, room B222 (Exactum) 

Speaker:  Jonas Latz, the University of Manchester 

Title:  Losing momentum in continuous-time stochastic optimisation 

Abstract: The training of modern machine learning models often consists in solving high-dimensional non-convex optimisation problems that are subject to large-scale data. In this context, momentum-based stochastic optimisation algorithms have become particularly widespread. The stochasticity arises from data subsampling which reduces computational cost. Both, momentum and stochasticity help the algorithm to converge globally. In this work, we propose and analyse a continuous-time model for stochastic gradient descent with momentum. This model is a piecewise-deterministic Markov process that represents the optimiser by an underdamped dynamical system and the data subsampling through a stochastic switching. We investigate longtime limits, the subsampling-to-no-subsampling limit, and the momentum-to-no-momentum limit. We are particularly interested in the case of reducing the momentum over time. Under convexity assumptions, we show convergence of our dynamical system to the global minimiser when reducing momentum over time and letting the subsampling rate go to infinity. We then propose a stable, symplectic discretisation scheme to construct an algorithm from our continuous-time dynamical system. In experiments, we study our scheme in convex and non-convex test problems. Additionally, we train a convolutional neural network in an image classification problem. Our algorithm attains competitive results compared to stochastic gradient descent with momentum.

Joint work with Kexin Jin, Chenguang Liu, and Alessandro Scagliotti. 

Associated paper: Jin et al. 2025: Journal of Machine Learning Research 26(148):1-55 (https://jmlr.org/papers/v26/23-1396.html).

Time & Location: Wednesday, 3 December at 14:15, room C222 (Exactum)

Speaker: Lisa Schätzle, Aalto University 

Title: Far field operator splitting for inhomogeneous medium scattering

Abstract: We consider scattering of time-harmonic acoustic waves by two compactly supported inhomogeneous objects, called scatterers, in a homogeneous background medium. These scatterers are illuminated by incident plane waves along all possible illumination directions and, each time, the resulting scattered waves are detected far away along all possible observation directions. This data can be described by the far field operator, which uniquely determines the scatterers in inverse medium scattering. Assuming a priori knowledge on the position of the stronger scattering scatterer, we consider the inverse problem of reconstructing the far field operator of the stronger scattering scatterer from knowledge of the far field operator of both scatterers. We refer to this problem as far field operator splitting, which particularly involves removing of certain multiple scattering effects. Provided the given data is exact this

is a uniquely solvable but severely ill-posed problem. It turns out that the far field operator component of the stronger scattering scatterer can be approximated by a sparse operator with respect to a known modulated Fourier basis. The remaining far field operator component is approximately of low rank but, however, it is no longer sparse with respect to the modulated Fourier basis. This observation motivates to approximate the solution of the original splitting problem by solving a coupled nuclear norm and $\ell^1 \times \ell^1$ norm minimization problem. Using techniques from the field of compressed sensing, we present a stability analysis for this problem formulation that requires explicit assumptions on the wave number and on the positions and sizes of the two scatterers. Furthermore, we verify our theoretical predictions by numerical examples.

Time & Location: Monday, 17 November at 14:15, room C222 (Exactum)

Speaker: Janne Nurminen, University of Jyväskylä

Title: An inverse problem for the prescribed mean curvature equation

Abstract: In this talk I will formulate an inverse source problem for the prescribed mean curvature equation (PMC) \div(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}=H

The question is if from boundary measurements one can determine the mean curvature $H$. The talk is based on joint work with Tony Liimatainen (https://arxiv.org/abs/2509.22078) and we show that it is indeed possible to recover $H$. The proof relies on the higher order linearization method and asymptotical analysis in an integral identity using complex geometric optics solutions from the work of Guillarmou and Tzou in 2011.

Time & Location: Monday, 20 October at 14:15, room C222 (Exactum)

Speaker: John C. Schotland, Yale University 

Title: Inverse scattering with entangled photons

Abstract: Quantum optics, the quantum theory of light-matter interactions, is primarily concerned with systems consisting of a small number of atoms. In this talk I will describe recent work on many-body problems and relations to kinetic theory. As an application, I will discuss related inverse scattering problems.

Time & Location: Thursday, 9 October at 14:15, room C222 (Exactum)

Speaker: Jhon Alexander Rodríguez Murillo, Pontificia Universidad Católica de Chile.

Title: Unique Continuation and Stability for a Moore-Gibson-Thompson Equation

Abstract: We study a Moore-Gibson-Thompson (MGT) equation with variable coefficients, which models acoustic wave propagation in thermoviscous media through a third-order-in-time evolution law. Following the ideas of Bukhgeim, Dyatlov, and Uhlmann (2013), we establish a unique continuation property by deriving Carleman estimates adapted to the MGT operator. This result extends existing continuation principles for hyperbolic equations with memory to a higher-order framework. Motivated by the geometric microlocal approach of Stefanov and Uhlmann (2009), we also construct a parametrix for the MGT equation and obtain a stability estimate for an associated inverse problem. The analysis clarifies the propagation of singularities and provides a precise description of how internal coefficients can be stably recovered from boundary data in the presence of memory effects.

Time & Location: Thursday 18 September, 14:15, room C222 (Exactum)

Speaker: Dr. Marvin Knöller, University of Helsinki

Title: A Computational Method for the Inverse Robin Problem with Convergence Rate

Abstract: In this talk we consider the inverse Robin problem, which is the determination of the Robin parameter $a$ appearing in an elliptic partial differential equation's boundary condition. Let $\Omega \subset \mathbb{R}^2$ be bounded and sufficiently regular. Suppose that we know the solution to the Robin problem

\begin{align*}

\Delta u \, &= \, f \quad \text{in } \Omega\, , \\

\partial_\nu u + a u \, &= \, g \quad \text{on } \partial \Omega

\end{align*}

only on a small subdomain $\omega \subset \Omega$ as well as the right hand sides $f \in L^2(\Omega)$ and $g\in H^{1/2}(\partial \Omega)$. Under the main assumption that the Robin parameter lies in a finite dimensional space of continuously differentiable functions, we establish a Newton-type algorithm for its reconstruction. This reconstruction algorithm requires first order piecewise continuous finite elements only. We show that the algorithm converges quadratically in the finite element's mesh size to the unknown Robin parameter. Moreover, we establish a perturbation analysis. We study several numerical examples that highlight the efficacy of our approach. Furthermore, we verify our theoretical convergence rates numerically.

This talk is based on joint work with Erik Burman (University College London) and Lauri Oksanen (University of Helsinki).

Time & Location: Thursday, 11 September at 14:15, room C222 (Exactum)

Speaker: Prof. Rakesh

Title: The fixed angle inverse scattering problem for Riemannian metrics

Abstract: Wave propagation in an inhomogeneous acoustic medium may be modeled, for example, by the wave operators $\Box + q$, $\rho \pa_t^2 - \Delta$ or $\pa_t^2 - \Delta_g$,  for a real smooth function $q(x)$, a positive function $\rho(x)$, or a Riemannian metric $g(x)$ on $\R^n$, with $q, \rho -1, g - I$ supported in a ball. The medium is probed by plane waves coming from a {\bf finite} number (dimension dependent) of directions, and the resultant time dependent waves are measured on the boundary of the ball. We describe our partial results about the recovery of $q,\rho,g$ from these boundary measurements. These are long standing formally determined open problems. These results were obtained in collaboration with Lauri Oksanen and Mikko Salo.

Time & Location: Friday, 29 August, 11:00, room C222 (Exactum)

Speaker: Prof. Catalin Carstea, National Yang Ming Chiao Tung University

Slides: 📄

Title: Uniqueness in the inverse boundary value problem for the weighted p-Laplacian (in the plane)

Abstract: In this talk I will consider  the weighted p-Laplace equation $\nabla\cdot(\gamma|\nabla u|^{p-2}\nabla u)=0$, in a bounded domain. I will present a uniqueness result for the inverse boundary value problem in the plane. Time permitting, I will also mention a higher dimension under rather strong apriori assumptions on $\gamma$. The talk is based on work with Ali Feizmohammadi.

Time & Location: Monday 25 August, 14:15, room C222 (Exactum)

Speaker: Meri Laurikainen, University of Helsinki 

Slides: 📄

Title: Reconstructing the metric of a Riemannian manifold from relative distance measurements

Abstract: Relative distance measurements are qualitative statements of the form "A is closer to B than C" or "A and B are closer to each other than either is to C". In this presentation, I will show how to reconstruct the metric of a complete Riemannian manifold up to a constant multiple from such measurements. 

Time & Location: Tuesday 12 August, 14:15, room C222 (Exactum)

Speaker: Prof. Sergei Avdonin, University of Alaska Fairbanks 

Slides: 📄

Title: Control and inverse problems for the wave and Schrödinger equations on metric graphs

Abstract: Under differential equation networks, known also as quantum graphs, we understand differential equations on metric graphs coupled by certain vertex matching conditions. These models play a fundamental role in many problems of science and engineering. Interest in control and inverse theory for differential equation networks is motivated by applications to important problems of classical and quantum physics, chemistry, biology, and engineering. For trees, i.e. graphs without cycles, various types of control and inverse problems were studied in the literature, but very little was done for graphs with cycles. In this talk we describe our recent controllability and identifiability results for the wave and Schrödinger equations on compact metric graphs. Our approach to inverse problems is based on the boundary control method, which uses deep connections between controllability / observability and identifiability of dynamical systems.

Time & Location: Tuesday 17 June, 14:15, room C222 (Exactum)

Speaker: Ruirui Wu, University of Washington

Slides: 📄

Title: Calderón problem for the quasilinear conductivity equation in dimension 2 

Abstract: In this talk, we will prove a uniqueness result for the Calderón problem for the quasilinear conductivity equation on a bounded domain in R^2. The proof of the result is based on the higher order linearization method. In contrast to the higher-dimensional case, the proof involves delicate analysis of the correction terms of Bukhgeim type complex geometric solutions (CGOs), which have only limited decay. To prove our results, we construct suitable families of CGOs whose phase functions may or may not have critical points. We also combine stationary phase analysis with L^p estimates for the correction terms of the CGOs. This is a joint work with Tony Liimatainen at the University of Helsinki. 

Time & Location: Thursday 5 June, 14:15, room C222 (Exactum)

Speaker: Yuchao Yi, University of California San Diego

Slides: 📄

Title: Riemannian and Lorentzian Calderón problem under magnetic perturbation 

Abstract: In this talk, I will discuss the Calderón Problem with magnetic (electromagnetic for Lorentzian) perturbations in both the Riemannian and Lorentzian settings. For both of them, we consider the Laplace-Beltrami operator with a smooth magnetic 1-form, and ask what information about the metric can be recovered when a family of Dirichlet-to-Neumann maps are given for different magnetic 1-forms. The approaches are different for the two settings: for Riemannian, we will utilize the rigidity of elliptic equations to uniquely determine the metric without gauge equivalence; for Lorentzian, we rely on the dispersive nature of hyperbolic equations and explicitly construct the trajectory of light-like geodesics with generic perturbations.  

Time & Location: Wednesday 28 May, 14:15, room C222 (Exactum)

Speaker: Siqin Zheng, Tsinghua University (China)

Slides: 📄

Title: Hybrid Inverse Problems: Magneto-Acousto-Electric Tomography as an example 

Abstract: Coupled physics imaging has emerged as a powerful paradigm for enhancing resolution beyond the limitations of classical methods like electrical impedance tomography (EIT). These approaches give rise to hybrid inverse problems, where reconstruction typically involves decoupling the multi-physics interaction inherent to the imaging process. As an example, I will present the inversion in magneto-acousto-electric tomography (MAET) introduced by Kunyansky, and our recent results on the stability analysis of the final inversion step. This step involves recovering the isotropic conductivity distribution from internal current density data, generated by solving the electrostatic equation under well-chosen boundary conditions. Employing complex geometrical optics (CGO) solutions, we establish local Lipschitz stability for this inverse problem in both full and partial data cases. 

Time & Location: Thursday 24 April, 14:15, room B222 (Exactum)

Speaker: Dr. Marvin Knöller, University of Helsinki

Talk: The talk has been recorded. If you are interested in the recording, please contact the speaker. 

Title: Shape reconstructions from time-dependent electric near field data

Abstract: Electromagnetic wave propagation is governed by Maxwell’s equations. In scattering theory, an electromagnetic wave impinges on the surface of a scattering object, induces a current on it and produces a scattered wave. We understand the inverse scattering problem as the determination or reconstruction of the scattering object from knowledge of the incident wave and observations of the scattered wave. In this talk we discuss the reconstruction of scatterers based on a temporal domain derivative, which is applicable to perfectly conducting obstacles in free space. We first provide a short overview on Maxwell’s equations and give an introduction to the shape reconstruction problem. Subsequently we explore the derivation of the temporal domain derivative by proceeding through the Laplace domain. Applying the convolution quadrature method to the temporal domain derivative yields a convergent semi-discretization in time. Combining this with a Galerkin discretization featuring Rao–Wilton–Glisson basis functions yields a fully discrete system. We present several three-dimensional shape reconstructions and highlight algorithmic aspects, implementation difficulties and survey on possible extensions to both theory and practice.

Time & Location: Wednesday 9 April, 14:15, room C222 (Exactum)

Speaker: Benjamin Florentin, Université de Lorraine

Slides: 📄

Title: Can one hear the shape of a conformal Steklov drum? 

Abstract: Introduced at the beginning of the 20th century, the Steklov eigenvalue problem has attracted growing interest in spectral geometry in recent decades and remains a major research topic in this field. In this talk, we will focus on the spectral inverse problem consisting in recovering the metric of a compact Riemannian manifold with boundary from knowledge of its Steklov spectrum, or equivalently the spectrum of its Dirichlet-to-Neumann map. In other words, can one hear the shape of a "Steklov drum"?

When the boundary is Anosov with simple length spectrum, the study of singularities in the trace of the wave operator makes it possible to exhibit certain spectral invariants via the Duistermaat-Guillemin trace formula and we will discuss how these invariants can be exploited and naturally combined with the injectivity of the geodesic X-ray transform to attack the problem. Some recent results obtained in the class of conformal metrics will be presented and it will be explained how generically it is possible to “hear the Taylor series at the boundary” of a conformal Steklov drum. 

Time & Location: Thursday 20 March, 14:15, room B222 (Exactum)

Speaker: Prof. Hiroshi Isozaki, University of Tsukuba (Japan)

Slides: 📄

Title: Inverse scattering on non-compact manifolds with general metric

Abstract: We study the spectral theory and inverse problem associated with Laplacians on a class of non-compact Riemannian manifolds with several ends. By a unified approach to  manifolds with ends of volume growth ranging over from exponential, asymptotically Euclidean, polynomially slow order, and to exponentially decaying cusps, we prove that the whole manifold is reconstructed from the knowledge of the S-matrix of one end for all energies. The key step for the proof is the Rellich-Vekua theorem for the uniqueness of solutions to the Helmholtz equation with optimal decay rate. This is a joint work with M. Lassas. 

Time & Location: Thursday 27 February, 14:15, room D122 (Exactum)

Speaker: Prof. Teemu Saksala, North Carolina State University

Slides: 📄

Title: Geometric inverse problems arising from hyperbolic PDEs

Abstract: In this talk I will survey the classical Boundary Control method, originally developed by Belishev and Kurylev, which can be used to reduce an inverse problem for a hyperbolic equation, on a complete Riemannian manifold, to a purely geometric problem involving the so-called travel time data. For each point in the manifold the travel time data contains the distance function from this point to any point in a fixed a priori known closed observation set. If the Riemannian manifold is closed then the observation set is a closure of an open and bounded set, and in the case of a manifold with boundary the observation set is an open subset of the boundary. We will survey many known uniqueness and stability results related to the travel time data. 

Time & Location: Thursday 30 January, 14:15, room B321 (Exactum)

Speaker: Ruochong Zhang, Fudan University

Title: Stable inversion of potential in nonlinear wave equations with cubic nonlinearity

Slides: 📄

Abstract: In this talk we investigate inverse potential problems of wave equations with cubic nonlinearity. We develop a methodology for establishing stability estimates for inversion of lower order coefficients. The new ingredients of our approach include trilinear approximations of nonlinear response operators, symbol estimates of distorted plane waves, and lower order symbol calculus. This is a joint work with Professors Xi Chen and Shuai Lu.

Time & Location: Thursday 24 October, 14:15, room B121 (Exactum)

Speaker: Prof. Gunther Uhlmann, University of Washington

Title: The anisotropic fractional Calderón problem

Abstract: We discuss some recent progress on the anisotropic Calderón problem for the fractional Laplacian.