arXiv subjects

Andreas Tataris

Publications and source records attributed to Andreas Tataris.

4 recordsLinked to original sources

arXiv2026

Random tree Besov priors: Data-driven regularisation parameter selection

We develop a data-driven algorithm for automatically selecting the regularisation parameter in Bayesian inversion under random tree Besov priors. One of the key challenges in Bayesian inversion is the construction of priors that are both expressive and computationally feasible. Random tree Besov priors, introduced in Kekkonen et al. (2023), provide a flexible framework for capturing local regularity properties and sparsity patterns in a wavelet basis. In this paper, we extend this approach by introducing a hierarchical model that enables data-driven selection of the wavelet density parameter, allowing the regularisation strength to adapt across scales while retaining computational efficiency. We focus on nonparametric regression and also present preliminary plug-and-play results for a deconvolution problem.

math.ST↗

arXiv2025

A variational Lippmann-Schwinger-type approach for the Helmholtz impedance problem on bounded domains

Recently, reduced order modeling methods have been applied to solving inverse boundary value problems arising in frequency domain scattering theory. A key step in projection-based reduced order model methods is the use of a sesquilinear form associated with the forward boundary value problem. However, in contrast to scattering problems posed in $\mathbb{R}^d$, boundary value formulations lose certain structural properties, most notably the classical Lippmann-Schwinger integral equation is no longer available. In this paper we derive a Lippmann-Schwinger type equation aimed at studying the solution of a Helmholtz boundary value problem with a variable refractive index and impedance boundary conditions. In particular, we start from the variational formulation of the boundary value problem and we obtain an equivalent operator equation which can be viewed as a bounded domain analogue of the classical Lippmann-Schwinger equation. We first establish analytical properties of our variational Lippmann-Schwinger type operator. Based on these results, we then show that the parameter-to-state map, which maps a refractive index to the corresponding wavefield, maps weakly convergent sequences to strongly convergent ones when restricted to refractive indices in Lebesgue spaces with exponent greater than 2. Finally, we use the derived weak to strong sequential continuity to show existence of minimizers for a reduced order model based optimization methods aimed at solving the inverse boundary value problem as well as for a conventional data misfit based waveform inversion method.

math.AP↗

arXiv2025

Inverse scattering for Schr\"{o}dinger equation in the frequency domain via data-driven reduced order modeling

In this paper we develop a numerical method for solving an inverse scattering problem of estimating the scattering potential in a Schr\"{o}dinger equation from frequency domain measurements based on reduced order models (ROM). The ROM is a projection of Schr\"{o}dinger operator onto a subspace spanned by its solution snapshots at certain wavenumbers. Provided the measurements are performed at these wavenumbers, the ROM can be constructed in a data-driven manner from the measurements on a surface surrounding the scatterers. Once the ROM is computed, the scattering potential can be estimated using non-linear optimization that minimizes the ROM misfit. Such an approach typically outperforms the conventional methods based on data misfit minimization. We develop two variants of ROM-based algorithms for inverse scattering and test them on a synthetic example in two spatial dimensions.

math.NA↗

arXiv2023

A data-driven approach to solving a 1D inverse scattering problem

In this paper, we extend the ROM-based approach for inverse scattering with Neumann boundary conditions, introduced by Druskin at. al. (Inverse Problems 37, 2021), to the 1D Schr{ö}dinger equation with impedance (Robin) boundary conditions. We also propose a novel data-assimilation (DA) inversion method based on the ROM approach, thereby avoiding the need for a Lanczos-orthogonalization (LO) step. Furthermore, we present a detailed numerical study and comparison of the accuracy and stability of the DA and LO methods.

math.NA↗