arXiv subjects

Vladimir G. Romanov

Publications and source records attributed to Vladimir G. Romanov.

12 recordsLinked to original sources

arXiv2026

The continuity properties of the solutions of the wave equation at different boundary conditions across the characteristic line

It is well-known the continuity properties of solutions to the wave equation depend strongly on both the initial data and the boundary conditions, particularly across the characteristic lines, where information propagates. Moreover, the continuity properties of solutions to the wave equation depend strongly on the type of boundary condition imposed and on the interaction of the characteristics with the boundary. Across the characteristic lines, singularities and discontinuities propagate according to the geometry of the problem, while the boundary conditions determine how these singularities are re?ected or transmitted. In this study, we will examine the behavior of solutions of the wave equation, corresponding to different boundary conditions, on its characteristic line, using the formulas obtained as a result of d'Alembert's formula, for all the three, Dirichlet, Neumann and Robin, types of boundary conditions.

math.AP↗

arXiv2026

Global Convergence and Uniqueness for an Inverse Problem Posed by Gelfand

The first globally convergent numerical method is developed for a coefficient inverse problem (CIP) for the $n-$d, $n\geq 2$ wave equation with the unknown potential in the most challenging case when the $δ-$ function is present in the initial condition with a single location of the point source. In fact, an approximate mathematical model for that CIP is derived. That globally convergent numerical method is developed for this model. This is a new version of the so-called convexification numerical method. Uniqueness theorem is proven as well within the framework of that approximate mathematical model. The question about uniqueness of this CIP was first posed by a famous mathematician I. M. Gelfand in 1954 as an $n-$d ($n=2,3$) extension of the fundamental theorem of V.A. Marchenko in the 1-d case (1950). Based on a Carleman estimate, convergence analysis is carried out. This analysis ensures the global convergence of the proposed numerical method, i.e. it is not necessary to have a good first guess for the solution. Exhaustive computational experiments with noisy data demonstrate a high reconstruction accuracy of complicated structures. In particular, this accuracy points towards a high adequacy of that approximate mathematical model.

math.NA↗

arXiv2025

Convexification Numerical Method for Imaging of Moving Targets

The problem of imaging of a moving target is formulated as a Coefficient Inverse Problem for a hyperbolic equation with its coefficient depending on all three spatial variables and time. As the initial condition, the point source running along a straight line is used. Lateral Cauchy data are known for each position of the point source. A truncated Fourier series with respect to a special orthonormal basis is used. First, Lipschitz stability estimate is obtained. Next, a globally convergent numerical method, the so-called convexification method, is developed and its convergence analysis is carried out. The convexification method is based on a Carleman estimate. Results of numerical experiments are presented.

math.NA↗

arXiv2024

Convexification for the 3D Problem of Travel Time Tomography

The travel time tomography problem is a coefficient inverse problem for the eikonal equation. This problem has well known applications in seismic. The eikonal equation is considered here in the circular cylinder, where point sources run along its axis and measurements of travel times are conductes on the whole surface of this cylinder. A new version of the globally convergent convexification numerical method for this problem is developed. Results of numerical studies are presented.

math.NA↗

arXiv2023

Convexification Numerical Method for a Coefficient Inverse Problem for the Riemannian Radiative Transfer Equation

The first globally convergent numerical method for a Coefficient Inverse Problem (CIP) for the Riemannian Radiative Transfer Equation (RRTE) is constructed. This is a version of the so-called \textquotedblleft convexification" method, which has been pursued by this research group for a number of years for some other CIPs for PDEs. Those PDEs are significantly different from the RRTE. The presence of the Carleman Weight Function (CWF) in the numerical scheme is the key element which insures the global convergence. Convergence analysis is presented along with the results of numerical experiments, which confirm the theory. RRTE governs the propagation of photons in the diffuse medium in the case when they propagate along geodesic lines between their collisions. Geodesic lines are generated by the spatially variable dielectric constant of the medium.

math.NA↗

arXiv2022

A Hölder Stability Estimate for a 3D Coefficient Inverse Problem for a Hyperbolic Equation With a Plane Wave

A 3D coefficient inverse problem for a hyperbolic equation with non-overdetermined data is considered. The forward problem is the Cauchy problems with the initial condition the delta function concentrated at a single plane (i.e. the plane wave). A certain associated operator is written in finite differences with respect to two out of three spatial variables, i.e. "partial finite differences". The grid step size is bounded from the below by a fixed number. A Carleman estimate is applied to obtain, for the first time, a Hölder stability estimates for this problem. Another new result is an estimate from the below of the amplitude of the first term of the expansion of the solution of the forward problem near the characteristic wedge.

math.AP↗

arXiv2021

Can a single PDE govern well the propagation of the electric wave field in a heterogeneous medium in 3D?

It is demonstrated in this paper that the propagation of the electric wave field in a heterogeneous medium in 3D can sometimes be governed well by a single PDE, which is derived from the Maxwell's equations. The corresponding component of the electric field dominates two other components. This justifies some past results of the second author with coauthors about numerical solutions of coefficient inverse problems with experimental electromagnetic data. In addition, since it is simpler to work in applications with a single PDE rather than with the complete Maxwell's system, then the result of this paper might be useful to researchers working on applied issues of the propagation of electromagnetic waves in inhomogeneous media.

physics.class-ph↗

arXiv2017

Uniqueness of a 3-D coefficient inverse scattering problem without the phase information

We use a new method to prove uniqueness theorem for a coefficient inverse scattering problem without the phase information for the 3-D Helmholtz equation. We consider the case when only the modulus of the scattered wave field is measured and the phase is not measured. The spatially distributed refractive index is the subject of the interest in this problem. Applications of this problem are in imaging of nanostructures and biological cells.

math-ph↗

arXiv2016

Inverse source problem for wave equation and GPR data interpretation problem

The inverse problem of identifying the unknown spacewise dependent source F(x) in 1D wave equation is considered. Measured data are taken in the form g(t) := u(0; t). The relationship between that problem and Ground Penetrating Radar (GRR) data interpretation problem is shown. The non-iterative algorithm for reconstructing the unknown source F(x) is developed. The algorithm is based on the Fourier expansion of the source F(x) and the explicit representation of the direct problem solution via the function F(x). Then the minimization problem for discrete form of the Tikhonov functional is reduced to the linear algebraic system and solved numerically. Calculations show that the proposed algorithm allows to reconstruct the spacewise dependent source F(x) with enough accuracy for noise free and noisy data.

math.NA↗

arXiv2015

Two reconstruction procedures for a 3-d phaseless inverse scattering problem for the generalized Helmholtz equation

The 3-d inverse scattering problem of the reconstruction of the unknown dielectric permittivity in the generalized Helmholtz equation is considered. The main difference with the conventional inverse scattering problems is that only the modulus of the scattering wave field is measured. The phase is not measured. The initializing wave field is the incident plane wave. On the other hand, in the previous recent works of the authors about the "phaseless topic" the case of the point source was considered [20,21,22]. Two reconstruction procedures are developed for a linearized case. However, the linearization is not the Born approximation. This means that, unlike the Born approximation, our linearization does not break down when the frequency tends to the infinity. Applications are in imaging of nanostructures and biological cells.

math-ph↗

arXiv2015

Reconstruction procedures for two inverse scattering problems without the phase information

This is a continuation of two recent publications of the authors about reconstruction procedures for 3-d phaseless inverse scattering problems. The main novelty of this paper is that the Born approximation for the case of the wave-like equation is not considered. It is shown here that the phaseless inverse scattering problem for the 3-d wave-like equation in the frequency domain leads to the well known Inverse Kinematic Problem. Uniqueness theorem follows. Still, since the Inverse Kinematic Problem is very hard to solve, a linearization is applied. More precisely, geodesic lines are replaced with straight lines. As a result, an approximate explicit reconstruction formula is obtained via the inverse Radon transform. The second reconstruction method is via solving a problem of the integral geometry using integral equations of the Abel type.

math-ph↗

arXiv2014

Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrodinger equation

The inverse scattering problem of the reconstruction of the unknown potential with compact support in the 3-d Schrödinger equation is considered. Only the modulus of the scattering complex valued wave field is known, whereas the phase is unknown. It is shown that the unknown potential can be reconstructed via the inverse Radon transform. Therefore, a long standing problem posed in 1977 by K. Chadan and P.C. Sabatier in their book "Inverse Problems in Quantum Scattering Theory" is solved.

math-ph↗