In many applications, such as the heat conduction and hydrology, there is a need to recover the (possibly discontinuous) diffusion coefficient a from boundary measurements of solutions of a parabolic equation. The complete inverse problem is ill posed and nonlinear, so numerical solution is quite difficult, and we linearize the problem around constant a. We study and solve numerically the linea...

The inverse problem of identifying an unknown source control parameter in a semilinear parabolic equation under an integral overdetermination condition is considered. The series pattern solution of the proposed problem is obtained by using the weighted homotopy analysis method (WHAM). A description of the method for solving the problem and finding the unknown parameter is derived. Finally, two ...

We consider the inverse problem of reconstructing the diiusion coeecient in a quasilinear parabolic diierential equation in divergence form from measurements of the solution at a nite number of points in the interior of the domain. An equation error method is developed which transforms the inverse problem into a system of linear operator equations for the diiusion coeecient, and which can be so...

This paper develops truncated Newton methods as an appropriate tool for nonlinear inverse problems which are ill-posed in the sense of Hadamard. In each Newton step an approximate solution for the linearized problem is computed with the conjugate gradient method as an inner iteration. The conjugate gradient iteration is terminated when the residual has been reduced to a prescribed percentage. U...

The focus of this paper is on conditional stability estimates for illposed inverse problems in partial differential equations. Conditional stability estimates have been obtained in the literature by a couple different methods. In this paper we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We are going to work out the theoretical backgr...

Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems, tasks of recovering the dependence of the right-hand side on time and spatial variables can be treated as independent. These tasks relate to a class of linear ...

The main purpose of this work is to propose a new network architecture model for deep learning applied solve an inverse source problem two-dimensional degenerate parabolic equation from final observations with degeneracy occurring anywhere in the spatial domain.

In the theory and practice of inverse problems for partial differential equations (PDEs) much attention is paid to the problem of the identification of coefficients from some additional information. This work deals with the problem of determining in a multidimensional parabolic equation the lower coefficient that depends on time only. To solve numerically a nonlinear inverse problem, linearized...

Stability of the solutions to 3D inverse scattering problems with fixed-energy data. Abstract A review of the author's results is given. Inversion formulas and stability results for the solutions to 3D inverse scattering problems with fixed energy data are obtained. Inversion of exact and noisy data is considered. The inverse potential scattering problem with fixed-energy scattering data is dis...

In this paper we investigate stability and convergence rates of the Tikhonov regularization method for the identiication of the diiusion parameter in a multi{dimensional parabolic equation. By choosing a problem{adapted approach, we obtain much better results than by applying the general theory of Tikhonov regularization of nonlinear inverse problems.