A. M. Denisov

Elements of the Theory of Inverse Problems

The basic concepts of inverse and ill-posed problems inverse problems for ordinary differential equations linear inverse problems for partial differential equations inverse coefficient problems for partial differential equations problems of determining the function from the values of integrals methods of solution of inverse problems

Numerical methods for some inverse problems of heart electrophysiology

We consider numerical methods for solving inverse problems that arise in heart electrophysiology. The first inverse problem is the Cauchy problem for the Laplace equation. Its solution algorithm is based on the Tikhonov regularization method and the method of boundary integral equations. The second inverse problem is the problem of finding the discontinuity surface of the coefficient of conductivity of a medium on the basis of the potential and its normal derivative given on the exterior surface. For its numerical solution, we suggest a method based on the method of boundary integral equations and the assumption on a special representation of the unknown surface.

A numerical method for calculating isotherms of ion exchange on impregnated sulfonate ion-exchangers based on the data of dynamic experiments

Abstract The inverse problem of searching ion-exchange isotherms from effluent concentration-time histories obtained by column experiments in displacing one ion by the other based on the mathematical model of nonequilibrium sorption dynamics has been solved. Elution curves for binary Na+-H+ and H+-Na+ exchange on impregnated sulfonate ion-exchangers with different capacities have been obtained for 0.001 M solutions of chlorides. Frontal analysis (direct and reverse) of 0.001 M HCl-NaCl mixtures of different compositions on the same resin samples have been carried out. The results of the first series of experiments have been used for calculating ion-exchange isotherms and for comparison with equilibrium data obtained in the second series. Computed isotherms demonstrate satisfactory fit with equilibrium data obtained for the same couple of ions (Na+ and H+) under the same conditions in independent experiments.

Numerical Solution of an Inverse Electrocardiography Problem for a Medium with Piecewise Constant Electrical Conductivity

A numerical method is proposed for solving an inverse electrocardiography problem for a medium with a piecewise constant electrical conductivity. The method is based on the method of boundary integral equations and Tikhonov regularization.

Inverse problem for a quasilinear system of partial differential equations with a nonlocal boundary condition

An initial boundary-value problem for a quasilinear system of partial differential equations with a nonlocal boundary condition involving a delayed argument is considered. The existence of a unique solution to this problem is proved by reducing it to a system of nonlinear integral-functional equations. The inverse problem of finding a solution-dependent coefficient of the system from additional information on a solution component specified at a fixed point of space as a function of time is formulated. The uniqueness of the solution of the inverse problem is proved. The proof is based on the derivation and analysis of an integral-functional equation for the difference between two solutions of the inverse problem.

Inverse problem for the diffusion equation in the case of spherical symmetry

The initial boundary value problem for the diffusion equation is considered in the case of spherical symmetry and an unknown initial condition. Additional information used for determining the unknown initial condition is an external volume potential whose density is the Laplace operator applied to the solution of the initial boundary value problem. The uniqueness of the solution of the inverse problem is studied depending on the parameters entering into the boundary conditions. It is shown that the solution of the inverse problem is either unique or not unique up to a one-dimensional linear subspace.

The inverse problem for mathematical models of heart excitation

The inverse problem for mathematical models of heart excitation is stated; this problem is to determine the initial condition in the initial-boundary value problem for an evolutionary system of partial differential equations given the volume potential whose density is determined by the solution to the evolutionary system. It is proved that the solution of the inverse problem in the generic statement is not unique.

Numerical method for determining the inhomogeneity boundary in the Dirichlet problem for Laplace’s equation in a piecewise homogeneous medium

The Dirichlet problem for Laplace’s equation in a two-dimensional domain filled with a piecewise homogeneous medium is considered. The boundary of the inhomogeneity is assumed to be unknown. The inverse problem of determining the inhomogeneity boundary from additional information on the solution of the Dirichlet problem is considered. A numerical method based on the linearization of the nonlinear operator equation for the unknown boundary is proposed for solving the inverse problem. The results of numerical experiments are presented.

Numerical solution of the inverse electrocardiography problem with the use of the Tikhonov regularization method

The inverse electrocardiography problem related to medical diagnostics is considered in terms of potentials. Within the framework of the quasi-stationary model of the electric field of the heart, the solution of the problem is reduced to the solution of the Cauchy problem for the Laplace equation in R3. A numerical algorithm based on the Tikhonov regularization method is proposed for the solution of this problem. The Cauchy problem for the Laplace equation is reduced to an operator equation of the first kind, which is solved via minimization of the Tikhonov functional with the regularization parameter chosen according to the discrepancy principle. In addition, an algorithm based on numerical solution of the corresponding Euler equation is proposed for minimization of the Tikhonov functional. The Euler equation is solved using an iteration method that involves solution of mixed boundary value problems for the Laplace equation. An individual mixed problem is solved by means of the method of boundary integral equations of the potential theory. In the study, the inverse electrocardiography problem is solved in region Ω close to the real geometry of the torso and heart.

Inverse Problem for a Quasilinear Wave Equation

The present paper deals with the proof of the existence and uniqueness of a solution of an inverse problem for a quasilinear wave equation with an unknown coefficient q(x) multiplying a lower-order derivative. The values of the solution of the Cauchy problem for this equation on some curve serve as additional information for the solution of the inverse problem. The proof of the existence and uniqueness of the solution of the inverse problem is based on the reduction of the problem to an integro-functional equation for the unknown function q(x). A similar approach was used in [1, 2] for the analysis of the inverse problem of finding an unknown function occurring in the nonlinear term specifying the source in the wave equation. Inverse problems for the wave equation were considered in a number of papers (e.g., see [3–7]). 1. STATEMENT OF THE PROBLEM AND THE MAIN RESULT Consider the Cauchy problem for the quasilinear wave equation utt(x, t )= a 2 uxx(x, t )+ f (x, ux(x, t)) q(x), (x, t) ∈ ∆d, (1.1) u(x, 0) = ϕ(x) ,u t(x, 0) = ψ(x), 0 ≤ x ≤ d,