arXiv subjects

Hjørdis Schlüter

Publications and source records attributed to Hjørdis Schlüter.

9 recordsLinked to original sources

arXiv2026

Travel time tomography for elliptically anisotropic elastic planar media

We study the gauge of the travel time tomography problem for elliptically anisotropic planar media. Under this assumption the respective slowness surfaces consist of an inner ellipse and an outer circle. Hence, the singularities of seismic waves propagate on geodesics given by two Riemannian metrics of which the one related to shear waves is conformally Euclidean. We show that if both of these metrics are lens rigid then, up to boundary fixing diffeomorphisms, each elliptically anisotropic stiffness tensor is uniquely determined by the lens data whenever it has at least one isotropic point. If none of those points exists, then there are exactly two elliptically anisotropic stiffness tensors with the same lens data. Clearly, not all boundary fixing diffeomorphisms preserve the imposed structure of the slowness surface. We show that the set of well-behaved diffeomorphisms is never a subgroup of all boundary fixing diffeomorphisms. Our proofs are based on an interplay of algebra and geometry, and the key of the proofs is a characterization of the elliptic anisotropy via several polynomial conditions.

math.AP↗

arXiv2026

QVaR: a Quantum Variational Regularization method for Linear Inverse Problems

We present a tailored framework for solving regularized linear inverse problems using quantum optimization methods. By discretizing the solution space and encoding data fidelity and regularization terms into quadratic unconstrained binary optimization (QUBO) models, we formulate both Tikhonov- and sparsity-promoting regularized inverse problems within a unified quantum optimization framework. We further introduce a notion of quantum sensitivity that characterizes the effect of perturbations arising from approximate quantum evolution and discretization. We derive bounds relating these perturbations to stability properties of the underlying inverse problem, thereby establishing a theoretical connection between quantum solution variability and classical notions of ill-posedness. The framework is extended to variational inverse problems through wavelet-based representations and complemented by reduced-order modeling strategies in both parameter and Hamiltonian spaces to mitigate current hardware limitations. Numerical experiments on simulated and physical quantum hardware indicate that the resulting low-energy solution distributions retain information about the underlying inverse problem, while also revealing the limitations imposed by finite-time evolution, discretization, and hardware noise.

math.NA↗

arXiv2026

Bayesian Formulation of Acousto-Electric Tomography and Quantified Uncertainty in Limited View

Acousto-electric tomography (AET) is a hybrid imaging modality that combines electrical impedance tomography with focused ultrasound perturbations to obtain interior power density measurements, which provide additional information that can enhance the stability of conductivity reconstruction. In this work, we study the AET inverse problem within a Bayesian framework and compare statistical reconstruction with analytical approaches. The unknown conductivity is modeled as a random field, and inference is based on the posterior distribution conditioned on the measurements. We consider likelihood constructions based on both L1- and L2-type data misfit norms and establish Bayesian well-posedness for both formulations within the framework of Stuart (2010). Numerical experiments investigate the performance of the Bayesian method from noisy power density measurements using the L1 and L2 likelihood functions and a smooth prior and a piecewise-constant prior for different limited view configurations, including severely limited boundary access. In particular, we demonstrate that small inclusions near the accessible boundary can be reconstructed from AET data corresponding to a single EIT measurement, and we quantify reconstruction uncertainty through posterior statistics.

math.AP↗

arXiv2025

Reconstruction of the Dipole Amplitude in the Dipole Picture as a mathematical Inverse Problem

We show that the inference problem of constraining the dipole amplitude with inclusive deep inelastic scattering data can be written into a discrete linear inverse problem, in an analogous manner as can be done for computed tomography. To this formulation of the problem, we apply standard inverse problems methods and algorithms to reconstruct known dipole amplitudes from simulated reduced cross section data with realistic precision. The main difference of this approach to previous works is that this implementation does not require any fit parametrization of the dipole amplitude. The freedom from parametrization also enables us for the first time to quantify the uncertainties of the inferred dipole amplitude in a novel more general framework. This mathematical approach to small-$x$ phenomenology opens a path to parametrization bias free inference of the dipole amplitude from HERA and Electron--Ion Collider data.

hep-ph↗

arXiv2024

Feasibility of hybrid inverse problems in limited view

Hybrid inverse problems such as Acousto-Electric Tomography, Current Density Imaging or Magnetic Resonance Electric Impedance Tomography are concerned with reconstructing the electrical conductivity from interior measurements. For a two-dimensional object the measurements correspond to two different functions imposed as the Neumann boundary condition to an elliptic PDE. In limited view these functions are only non-zero on the part of the boundary that one can control. In this paper we address how to choose such boundary functions in limited view such that the reconstruction procedure is feasible. This is related to the corresponding Jacobian being non-zero. We supplement the theoretical results by numerical reconstructions following the approach of Acousto-Electric Tomography.

math.AP↗

arXiv2024

Gauge freedoms in the anisotropic elastic Dirichlet-to-Neumann map

We address the inverse problem of recovering the stiffness tensor and density of mass from the Dirichlet-to-Neumann map. We study the invariance of the Euclidean and Riemannian elastic wave equation under coordinate transformations. Furthermore, we present gauge freedoms between the parameters that leave the elastic wave equations invariant. We use these results to present gauge freedoms in the Dirichlet-to-Neumann map associated to the Riemannian elastic wave equation.

math.AP↗

arXiv2023

Jacobian of solutions to the conductivity equation in limited view

The aim of hybrid inverse problems such as Acousto-Electric Tomography or Current Density Imaging is the reconstruction of the electrical conductivity in a domain that can only be accessed from its exterior. In the inversion procedure, the solutions to the conductivity equation play a central role. In particular, it is important that the Jacobian of the solutions is non-vanishing. In the present paper we address a two-dimensional limited view setting, where only a part of the boundary of the domain can be controlled by a non-zero Dirichlet condition, while on the remaining boundary there is a zero Dirichlet condition. For this setting, we propose sufficient conditions on the boundary functions so that the Jacobian of the corresponding solutions is non-vanishing. In that regard we allow for discontinuous boundary functions, which requires the use of solutions in weighted Sobolev spaces. We implement the procedure of reconstructing a conductivity from power density data numerically and investigate how this limited view setting affects the Jacobian and the quality of the reconstructions.

math.FA↗

arXiv2023

Reconstructing anisotropic conductivities on two-dimensional Riemannian manifolds from power densities

We consider an electrically conductive compact two-dimensional Riemannian manifold with smooth boundary. This setting defines a natural conductive Laplacian on the manifold and hence also voltage potentials, current fields and corresponding power densities arising from suitable boundary conditions. Motivated by Acousto-Electric Tomography we show that if the manifold has genus zero and the metric is known, then the anisotropic conductivity can be recovered uniquely and constructively from knowledge of a few power densities. We illustrate the procedure numerically by reconstructing an anisotropic conductivity on the catenoid, i.e. the classical genus zero minimal surface in three-space.

math.AP↗

arXiv2022

Conductivity reconstruction from power density data in limited view

In Acousto-Electric tomography, the objective is to extract information about the interior electrical conductivity in a physical body from knowledge of the interior power density data generated from prescribed boundary conditions for the governing elliptic partial differential equation. In this note, we consider the problem when the controlled boundary conditions are applied only on a small subset of the full boundary. We demonstrate using the unique continuation principle that the Runge approximation property is valid also for this special case of limited view data. As a consequence, we guarantee the existence of finitely many boundary conditions such that the corresponding solutions locally satisfy a non-vanishing gradient condition. This condition is essential for conductivity reconstruction from power density data. In addition, we adapt an existing reconstruction method intended for the full data situation to our setting. We implement the method numerically and investigate the opportunities and shortcomings when reconstructing from two fixed boundary conditions.

math.AP↗