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We consider an inverse shape problem arising in electrical impedance tomography (EIT) for nondestructive testing, in which interior defects are modeled through Robin transmission conditions. Unlike classical formulations, we impose Robin…
Neural network approaches have been demonstrated to work quite well to solve partial differential equations in practice. In this context approaches like physics-informed neural networks and the Deep Ritz method have become popular. In this…
We give identities for the voltage and resistance functions on a metrized graph to show how these functions behave under any edge deletion/contraction and the identification of any two vertices. This leads to explicit versions of Rayleigh's…
This work considers using reduced basis techniques in connection to (smoothened) total variation regularization in electrical impedance tomography, but analogous ideas can also be used for other inverse elliptic boundary value problems. It…
In graph signal processing, data samples are associated to vertices on a graph, while edge weights represent similarities between those samples. We propose a convex optimization problem to learn sparse well connected graphs from data. We…
Tensor network contraction is central to problems ranging from many-body physics to computer science. We describe how to approximate tensor network contraction through bond compression on arbitrary graphs. In particular, we introduce a…
This paper deals with the solution of Maxwell's equations to model the electromagnetic fields in the case of a layered earth. The integrals involved in the solution are approximated by means of a novel approach based on the splitting of the…
Graph Neural Networks struggle to capture long-range dependencies due to over-squashing, where information from exponentially growing neighborhoods must pass through a small number of structural bottlenecks. While recent rewiring methods…
In this work, we consider the inverse electromagnetic scattering problem for a magneto-dielectric cylinder covering an impedance cylinder of arbitrary shape. We solve it by introducing a divide-and-conquer framework using specially designed…
In recent years, with the increase in renewable energy and storage penetration, power flow studies in low-voltage networks have become of interest in both industry and academia. Many studies use impedance represented by sequence components…
This work implements and numerically tests the direct reconstruction algorithm introduced in [Garde & Hyv\"onen, SIAM J. Math. Anal., 2024] for two-dimensional linearized electrical impedance tomography. Although the algorithm was…
Hybrid inverse problems are mathematical descriptions of coupled-physics (also called multi-waves) imaging modalities that aim to combine high resolution with high contrast. The solution of a high-resolution inverse problem, a first step…
In this work, we propose and analyse a weak Galerkin method for the electrical impedance tomography based on a bounded variation regularization. We use the complete electrode model as the forward system that is approximated by a weak…
Inversion of electromagnetic data finds applications in many areas of geophysics. The inverse problem is commonly solved with either deterministic optimization methods (such as the nonlinear conjugate gradient or Gauss-Newton) which are…
We propose a non-stationary iterated network Tikhonov (iNETT) method for the solution of ill-posed inverse problems. The iNETT employs deep neural networks to build a data-driven regularizer, and it avoids the difficult task of estimating…
The collection of all the strongly connected components in a directed graph, among each cluster of which any node has a path to another node, is a typical example of the intertwining structure and dynamics in complex networks, as its…
We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial…
Electrical impedance tomography (EIT) is a non-invasive imaging technique, which has been widely used in the fields of industrial inspection, medical monitoring and tactile sensing. However, due to the inherent non-linearity and…
We study the inverse problem for a semilinear wave equation on metric tree graphs. From the Dirichlet-to-Neumann map defined at all but one of the boundary vertices, we recover unknown connectivity of the graph, lengths of the edges, the…
This paper is devoted to the analysis of a second order method for recovering the \emph{a priori} unknown shape of an inclusion $\omega$ inside a body $\Omega$ from boundary measurement. This inverse problem - known as electrical impedance…