Inverse Problems and Optimization I

Date & Time: Tuesday, September 29, 11 - 13:30
Place: tba
Chairs: Lisa Schätzle, Aada Hakula

Abstract: 

Inverse problems concern the reconstruction of unknown quantities from direct or indirect observations, and they play a central role in many areas of mathematics and applicational fields such as medical imaging, nondestructive testing or geophysical exploration. This mini-symposium brings together recent developments across several key topics, including scattering theory, optimization methods, and electrical impedance tomography. In addition to practical and computational approaches, fundamental theoretical aspects of inverse problems will also be addressed. The session features contributions primarily from early-career researchers, including PhD students and postdoctoral fellows, highlighting fresh perspectives and emerging work in the field.

Confirmed Speakers:

Speaker: Suvi Anttila (University of Oulu)
Title: X-ray imaging from nonlinear waves
Abstract: In this talk, we consider an inverse boundary value problem for a nonlinear wave equation in the plane, focusing on the recovery of an unknown potential. We present a fast, non-iterative numerical reconstruction method, based on higher-order linearization, that yields the Radon transform of the potential; this can the be inverted using standard X-ray tomography techniques to determine the potential. To mitigate the effects of noise in the boundary data, we introduce a spectral regularization procedure that stabilizes the numerical differentiation step required in the reconstruction. The talk is based on joint work with Markus Harju and Teemu Tyni.

Speaker: Khaoula El Maddah (University of Oulu)
Title: Reconstruction for an inverse scattering problem with a Kerr type nonlinearity
Abstract: We study the inverse scattering problem for the Kerr-nonlinear Helmholtz equation

\Delta u + k^2(1+q(x)|u|^2)u = 0 \quad \text{in }\mathbb{R}^n,\; n\geq 2,

where the aim is to recover the unknown potential q from the scattering amplitude. We obtain uniqueness for full data and partial data cases of backscattering, fixed angle scattering, and fixed energy scattering. We are able to explicitly reconstruct individual Fourier modes of the potential, and if the measured directions and energies cover an open subset, we recover q.
The simplicity of the approach leads to an efficient numerical method, and numerical experiments show accurate reconstructions, even in the presence of noise.

This is joint work with Teemu Tyni, Valter Pohjola, Tony Liimatainen, and Matti Lassas.

Speaker: Roland Griesmaier, Marvin Knöller, Eliane Ariadne Felicitas Kummer (Karlsruhe Institute of Technology)
Title: Domain derivative and shape reconstruction for a transient inverse backscattering problem
Abstract: We are interested in the shape reconstruction of a three-dimensional penetrable scattering obstacle from backscattering measurements of the associated time-dependent far field.

To this avail, we study the temporal domain derivative (TDD) for a time-dependent acoustic scattering problem, namely the transmission problem for the wave-equation. We can then employ this domain derivative in an iterative reconstruction algorithm similar to the TDD for sound-soft scattering objects and near field measurements for a single incident wave, as has been studied in Knöller and Nick, Numer. Math. 158(1), 2026. For this, we first consider the operator, which maps the boundary of the penetrable scattering object to the far field, measured at a backscattering point \(-d \in \mathbb{S^2}\) at time \(t \in \mathbb{R}\). The TDD is then given by the point-evaluation of the Fréchet derivative of this operator with respect to perturbations of the boundary. To obtain the existence of the domain derivative in the time domain, we first study the corresponding problem in the Laplace domain. By establishing frequency bounds for the domain derivative in the frequency domain and by utilizing Paley-Wiener's theorem we prove the existence of the TDD in the time domain. By applying Runge-Kutta convolution quadrature as well as a regularized Gauß-Newton algorithm, we employ the time-discrete temporal domain derivative to describe an algorithm to reconstruct the boundary of a three-dimensional penetrable scattering object from time domain back-scattering measurements of the far field.

We demonstrate the efficiency of our algorithm by presenting numerical simulations.

Speaker: Spyridon Filippas (University of Helsinki)
Title: Recovering a matrix-valued potential in stationary spacetimes
Abstract: In this talk we consider the problem of recovering a time dependent matrix-valued potential on a general globally hyperbolic manifold from the
knowledge of the source to solution map of a wave equation including a connection 1-form term. We show that this problem can be reduced to studying the injectivity of a non-Abelian light ray transform. Under the assumption that our manifold is stationary, we then prove that injectivity for an appropriately defined Riemannian transform allows to uniquely determine the potential. This is based on a joint work with Lauri Oksanen and Miika Sarkkinen.

Speaker: Hjørdis Schlüter (University of Helsinki)
Title: Boundary determination in anisotropic elasticity
Abstract: We address the inverse problem of recovering an anisotropic stiffness tensor c and the density of mass \( {\rho} \) from the associated Dirichlet-to-Neumann map for n-dimensional domains with \( n \geq 2 \). We show that for generic stiffness tensors c and density \( {\rho} \) the associated Dirichlet-to-Neumann map determines c and \( {\rho} \) uniquely at the boundary. This result builds on a construction of special solutions that concentrate near a boundary point and are highly oscillatory. The generic assumptions are needed in order to ensure that this construction can be applied and to arrive at uniqueness from an equation derived from inserting the special solutions in the Alessandrini identity.