Related papers: The inverse problem within free Electrodynamics an…
The enclosure method was originally introduced for inverse problems of concerning non-destructive evaluation governed by elliptic equations. It was developed as one of useful approaches in inverse problems and applied for various equations.…
An Electrodynamics solver for moving sources is introduced. The main challenges and formulation are highlighted. The solver enables the simulation of fields for sources undergoing arbitrary motion. Two examples of uniformly moving current…
The method of variational completion allows one to transform an (in principle, arbitrary) system of partial differential equations -- based on an intuitive ``educated guess'' -- into the Euler-Lagrange one attached to a Lagrangian, by…
The equations of electromagnetic fields in a medium is usually written in the rest frame of the medium. We outline a method of generalizing the discussion to arbitrary inertial frames. In the discussion, we also include the possibility that…
In the present work, firstly, we use a minimax equality to prove the existence of a solution of certain system of varitional equations and we provide a numerical approximation of such a solution. Then, we propose a numerical method to solve…
We give an overview of recent advances in analysis of equations of electrodynamics with the aid of biquaternionic technique. We discuss both models with constant and variable coefficients, integral representations of solutions, a numerical…
Anisotropy of a space naturally leads to direction dependent electromagnetic tensors and electromagnetic potentials. Starting from this idea and using variational approaches and exterior derivative formalism, we extend some of the classical…
We give examples on the use of the Stone-Weierstrass theorem in inverse problems. We show uniqueness in the linearized Calder\'on problem on holomorphically separable K\"ahler manifolds, and in the Calder\'on problem for nonlinear equations…
A simple mathematical procedure is introduced which allows redefining in an exact way divergent integrals and limits that appear in the basic equations of classical electrodynamics with point charges. In this way all divergences are at once…
This paper investigates the formulation and implementation of Bayesian inverse problems to learn input parameters of partial differential equations (PDEs) defined on manifolds. Specifically, we study the inverse problem of determining the…
We introduce a new approach to the anisotropic Calder\'on problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large…
We are concerned with the well-posedness of linear elliptic systems posed on $\mathbb{R}^d$. The concrete problem of interest, for which we require this theory, arises from the linearization of the equations of anisotropic finite…
We consider the inverse conductivity problem of identifying embedded objects in unbounded domains. The main tool is a set of special solutions to the Schroedinger equation, the complex spherical waves, which are constructed by a Carleman…
Describing the solutions of inverse problems arising in signal or image processing is an important issue both for theoretical and numerical purposes. We propose a principle which describes the solutions to convex variational problems…
The Lorentz transformations are represented by Einstein velocity addition on the ball of relativistically admissible velocities. This representation is by projective maps. The Lie algebra of this representation defines the relativistic…
A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as…
The bootstrap method has proven useful for a wide range of matrix models. Here, we show that the computed momenta can be used to reconstruct the underlying eigenvalue probability distribution, which in turn allows us to compute the free…
A complete solution for an inverse problem needs five main steps: choice of basis functions for discretization, determination of the order of the model, estimation of the hyperparameters, estimation of the solution, and finally,…
We are concerned with the reconstruction of inclusions in elastic bodies based on measurements from a laboratory experiment. In doing so, we solve the inverse problem of the time-harmonic elastic wave equation, in contrast to the stationary…
We propose a reformulation of electrodynamics in terms of a {\it physical} vector potential entirely free of gauge ambiguities. Quantizing the theory leads to a propagator that is gauge invariant by construction in this reformulation, in…