ABSTRACT: A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness
of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method
of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity.
A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.
Computational Mathematics and Mathematical Physics 04/2012; 49(7):1141-1150. · 0.30 Impact Factor
ABSTRACT: 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.
Differential Equations 04/2012; 45(7):1034-1043. · 0.42 Impact Factor
ABSTRACT: 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 R
3. 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.
Moscow University Computational Mathematics and Cybernetics 04/2012; 32(2):61-68.