Information on the thesis

The thesis is devoted to the numerical solution of the problem of reconstructing the Cauchy data of a harmonic function in a three-dimensional double connected domain. In the paper, it is shown that the initial problem belongs to a class of ill-posed problems, in the sense of lack of stability on the input data. Therefore, in order to obtain a stable numerical solution, a class of regularization methods is considered. A brief overview of existing regularization numerical methods is made for solving incorrect problems, in particular, a two-dimensional or three-dimensional Cauchy problem for the Laplace equation with a brief analysis of advantages or disadvantages. Also, a brief description of the functional spaces required for theoretical studies is given.

For a numerical solution of the initial problem, the direct Tikhonov regularization method and two iterative methods (the alternating method and the generalized Landveber method) are considered.

In the case of Tikhonov’s method, the Cauchy problem is reduced to an ill-posed system of two-dimensional integral equations by means of the theory of potential and using the Green’s formula. In both variants, the integral representation of the solution, the corresponding ill-posed system of integral equations, as well as the appearance of the desired Cauchy data on the internal boundary is given. For two systems of integral equations, the possibility of applying Tikhonov’s regularization method has been investigated. The parametrization of the received systems of integral equations and the weak singularity in the kernel are investigated. By means of the discrete Galerkin projection method, the parametrized systems of integral equations are completely discretized, moreover unknown densities are approximated by a linear combination of spherical harmonics, and the integrals are approximated by means of corresponding cubature rules with superalgebraic order of convergence for analytic integral functions. As a result, the solutions of the systems of the integral equations are reduced to the search for unknown coefficients in the approximations of unknown densities, and the sought-after coefficients are the solution of the systems of linear algebraic equations obtained after the application of the Galerkin method. It is shown how to optimize the process of forming coefficients of systems of linear equations to reduce the number of calculations and how to use the obtained results in the formation of discrete Cauchy data approximation.

To the obtained systems of linear equations, the Tikhonov regularization method was used, and the regularization parameter was chosen using the L-curve method. The formulas for finding an approximate solution in the domain and for finding approximate values of the Cauchy data on the inner boundary of the domain for both approaches of the method of integral equations are given, and a brief comparative analysis of both approaches is presented.

Also, iterative regularizing methods for numerical solving of an ill-posed three-dimensional Cauchy problem for the Laplace equation are used in the thesis. First, the algorithm of the alternating method is given. This iteration method was proposed by V. Kozlov and V. Maz’ya for ill-posed problems with a self-adjoint operator and was used in some papers for numerical solving of a two-dimensional Cauchy problem. In this thesis, the idea of the method is extended to the case of three-dimensional domains. An algorithm for an iterative method is presented, which consists of sequential solving of two well-posed mixed Neumann-Dirichlet and Dirichlet-Neumann problems for the Laplace equation. The convergence and stability of this method are investigated.

The classic iterative regularizing method is the Landveber method, and as is known for the iteration procedure, the form of the adjoint operator is necessary. Two modifications of the generalized Landveber method for which no adjoint operator is required are constructed in the work. The algorithms of data of iterative procedures are presented, at each step of which it is necessary to solve the Dirichlet and Robin problem or the mixed Dirichlet-Neumann and Neumann-Dirichlet problems for the Laplace equation with the corresponding input data. The convergence and stability of the obtained iterative methods are investigated.

Since at every step of all iterative processes one needs to solve one of the well-posed three-dimensional Neumann-Dirichlet, Dirichlet-Neumann, Dirichlet or Robin problems for the Laplace equation, then the algorithms for their numerical solving by an indirect method of integral equations are considered. All well-posed problems are reduced to a system of two-dimensional integral equations, the well-posedness of the received systems in the corresponding spaces is studied, a weak singularity in the kernel are singled out. A discrete Galerkin projection method is considered for the discretization of the obtained systems of integral equations, the convergence of this method is investigated and the estimations of errors of the method are established. The specificity of three-dimensional areas is taken into account, as well as optimization of calculations.

For all regularization methods for solving the Cauchy problem for the Laplace equation, several numerical experiments are presented for various domain configurations and different input data, as well as for noise data, which prove the stability and convergence of the methods under consideration: the tables of changes of the errors of the desired Cauchy data and the corresponding graphs are given. For the considered correct boundary problems, several numerical experiments are shown which show the high efficiency of the method of integral equations in the case of well-posed problems.

Key words: Cauchy problem, Laplace equation, three-dimensional domain, Tikhonov method, alternating method, generalized Landveber method, L-curve method, integral equations method, Wienert method.