The dynamical inverse problem for a non-self-adjoint sturm-liouville operator

An approach to the inverse problem (the so-called BC-method) based on boundary-control theory is developed. A procedure of reconstructing a nonsymmetric matrix-function (a potential) given on a semiaxis by a dynamical response operator is described. The results of numerical tests are presented. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 7–21. Translated by T. N. Surkova.     

An anisotropic inverse boundary value problem

We consider the impedance tomography problem for anisotropic conductivities. Given a bounded region Ω in space, a diffeomorphism Ψ from Ω to itself which restricts to the identity on ? Ω, and a conductivity γ on Ω, it is easy to construct a new conductivity Ψ_*γ which will produce the same voltage and current measurements on ? Ω. We prove the converse in two dimensions (i.e., if γ₁ and γ₂ produce the same boundary measurements, then γ₁, = Ψ_*γ₂ for an appropriate Ψ) for conductivities which are near a constant.     

An inverse boundary value problem for the Oseen equation

Based on the two‐dimensional stationary Oseen equation we consider the problem to determine the shape of a cylindrical obstacle immersed in a fluid flow from a knowledge of the fluid velocity on some arc outside the obstacle. First, we obtain a uniqueness result for this ill‐posed and non‐linear inverse problem. Then, for the approximate solution we propose a regularized Newton iteration scheme based on a boundary integral equation of the first kind. For a foundation of Newton‐type methods we establish the Fréchet differentiability of the solution to the Dirichlet problem for the Oseen equation with respect to the boundary and investigate the injectivity of the linearized mapping. Some numerical examples for the feasibility of the method are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

An inverse diffusion problem with nonlocal boundary conditions

This article considers the inverse problem of identification of a time‐dependent thermal diffusivity together with the temperature in an one‐dimensional heat equation with nonlocal boundary and integral overdetermination conditions when a heat exchange takes place across boundary of the material. The well‐posedness of the problem is studied under some regularity, and consistency conditions on the data of the problem together with the nonnegativity condition on the Fourier coefficients of the initial data and source term. The inverse problem is also studied numerically by using the Crank–Nicolson finite difference scheme combined with predictor‐corrector technique. The numerical examples are presented and discussed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 564–590, 2016     

Global uniqueness for an inverse boundary problem arising in elasticity

Summary We prove that we can determine the Lamé parameters of an elastic, isotropic, inhomogeneous medium in dimensionsn3, by making measurements of the displacements and corresponding stresses at the boundary of the medium.Oblatum 14-VI-1993 & 22-XII-1993Partially supported by NSF Grant DMS-9100178 and ONR grant N 0014-93-1-0295     

On the unsolvability of the boundary inverse problem of heat conduction

It is shown that the boundary inverse nonstationary problem of heat conduction with given heat flux and temperature on one of the boundary surfaces is essentially ill-posed, since it has no solution in the class of continuous functions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 114–118.     

Reconstruction of inclusions for the inverse boundary value problem with mixed type boundary condition

We consider an inverse boundary value problem for identifying the inclusion inside a known anisotropic conductive medium. We give a reconstruction procedure for identifying the inclusion from the Dirichlet–Neumann map or the Neumann–Dirichlet map associated with the mixed type boundary condition.     

Two-dimensional inverse quasilinear parabolic problem with periodic boundary condition

In this study, we consider a coefficient problem of a quasi-linear two-dimensional parabolic inverse problem with periodic boundary and integral over determination conditions. We prove the existence, uniqueness and continuously dependence upon the data of the solution by iteration method. Also, we consider numerical solution for this inverse problem by using linearization and the implicit finite-difference scheme.     

Conditional stability estimation for an inverse boundary problem with non-smooth boundary in

In this paper, we investigate an inverse problem of determining a shape of a part of the boundary of a bounded domain in by a solution to a Cauchy problem of the Laplace equation. Assuming that the unknown part is a Lipschitz continuous surface, we give a logarithmic conditional stability estimate in determining the part of boundary under reasonably a priori information of an unknown part. The keys are the complex extension and estimates for a harmonic measure.