Roman Chapko
@lnu.edu.ua
Faculty of Applied Mathematics and Informatics
Ivan Franko National University of Lviv
EDUCATION
Lviv National University. Studies in Faculty of Applied Mathematics and Mechanics 1980-1985. Dipl. (equivalent to M.S.) in Applied Mathematics
RESEARCH INTERESTS
Numerical solution of linear and non-linear evolution inverse problems, integral equation approach
Scopus Publications
Scopus Publications
Ihor Borachok, Roman Chapko, and B. Tomas Johansson
Springer Science and Business Media LLC
R. Chapko and B. T. Johansson
Springer Science and Business Media LLC
Ihor Borachok, Roman Chapko, and B. Tomas Johansson
Springer Science and Business Media LLC
Ihor Borachok, Roman Chapko, and B. Tomas Johansson
Springer Science and Business Media LLC
Roman Chapko and Leonidas Mindrinos
MDPI AG
We consider the inverse problem of reconstructing the boundary curve of a cavity embedded in a bounded domain. The problem is formulated in two dimensions for the wave equation. We combine the Laguerre transform with the integral equation method and we reduce the inverse problem to a system of boundary integral equations. We propose an iterative scheme that linearizes the equation using the Fréchet derivative of the forward operator. The application of special quadrature rules results to an ill-conditioned linear system which we solve using Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.
Roman Chapko, B. Tomas Johansson, and Mariia Vlasiuk
Springer Science and Business Media LLC
Ihor Borachok, Roman Chapko, and B. Tomas Johansson
Springer Science and Business Media LLC
Roman Chapko, B. Tomas Johansson, and Leonidas Mindrinos
Elsevier BV
Roman Chapko and Leonidas Mindrinos
Springer Science and Business Media LLC
Andriy Beshley, Roman Chapko, and B. Tomas Johansson
Elsevier BV
Roman Chapko, B. Tomas Johansson, Yuriy Muzychuk, and Andriy Hlova
Elsevier BV
Roman Chapko, , B. Tomas Johansson, and
American Institute of Mathematical Sciences (AIMS)
A boundary integral based method for the stable reconstruction of missing boundary data is presented for the biharmonic equation. The solution (displacement) is known throughout the boundary of an annular domain whilst the normal derivative and bending moment are specified only on the outer boundary curve. A recent iterative method is applied for the data completion solving mixed problems throughout the iterations. The solution to each mixed problem is represented as a biharmonic single-layer potential. Matching against the given boundary data, a system of boundary integrals is obtained to be solved for densities over the boundary. This system is discretised using the Nystrom method. A direct approach is also given representing the solution of the ill-posed problem as a biharmonic single-layer potential and applying the similar techniques as for the mixed problems. Tikhonov regularization is employed for the solution of the corresponding discretised system. Numerical results are presented for several annular domains showing the efficiency of both data completion approaches.
Andriy Beshley, Roman Chapko, and B. Tomas Johansson
Springer International Publishing
Roman Chapko and B. Tomas Johansson
Wiley
An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown.
Andriy Beshley, Roman Chapko, and B. Tomas Johansson
Springer Science and Business Media LLC
George Baravdish, Ihor Borachok, Roman Chapko, B. Tomas Johansson, and Marián Slodička
Elsevier BV
Roman Chapko and B. Tomas Johansson
Elsevier BV
Roman Chapko, Drossos Gintides, and Leonidas Mindrinos
Springer Science and Business Media LLC
Roman Chapko and Leonidas Mindrinos
Rocky Mountain Mathematics Consortium
A numerical method for the Dirichlet initial boundary value problem for the elastic equation in the exterior and unbounded region of a smooth closed simply connected 2-dimensional domain, is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and a boundary integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the time-depended problem to a sequence of stationary boundary value problems, which are solved by a boundary layer approach resulting to a sequence of boundary integral equations of the first kind. The numerical discretization and solution are obtained by a trigonometrical quadrature method. Numerical results are included.
R. Chapko and B. T. Johansson
Springer Science and Business Media LLC
Roman Chapko and B. Tomas Johansson
Springer Science and Business Media LLC
Ihor Borachok, Roman Chapko, and B. Tomas Johansson
Walter de Gruyter GmbH
AbstractWe consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert’s method [
Ihor Borachok, Roman Chapko, and B. Tomas Johansson
Informa UK Limited
A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.
Roman Chapko
IEEE
We consider two types of inverse boundary problems related to the Laplace equation in planar double connected domains. The first one consists in the determining the Cauchy data on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the function (or the normal derivative) is known and, additionally, on a portion of this exterior boundary the normal derivative (or the function) is also given. For this linear inverse problem we propose a direct boundary integral approach in combination with Tikhonov regularization for stable determination of the Cauchy data on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green's functions. We outline an effective way of discretizing these boundary integral equations based on the Nyström method and trigonomtric approximations. The second set of inverse problems is the reconstruction of the interior curve from the given Cauchy data of a harmonic function on the exterior boundary. With the help of Green's function and potential theory this non-linear boundary problem is reduced to the system of non-linear boundary integral equations. We develop three iterative algorithms for its numerical solution. Full discretization of the linearized systems is realized by a trigonometric quadrature method. Due to the inherited ill-possedness in the obtained system of linear equations we apply the Tikhonov regularization. Numerical results show that the proposed approach gives good accuracy of reconstructions with economical computational cost.
Roman Chapko, Christina Babenko, Volodymyr Khlobystov, and Volodymyr Makarov
Informa UK Limited
We consider the problem of interpolating a multivariable function defined on a bounded domain using its traces on parametric hypersurfaces. Our approach is based on the theory of operator polynomial interpolation. We construct the corresponding operator interpolation polynomial for a given function and analyse in detail particular two- and three-dimensional cases. Numerical examples presented in the paper show the flexibility of the proposed approach.