Bulletin of Dnipropetrovsk University. Series: Communications in Mathematical Modeling and Differential Equations Theory

27-35

Authors: O. V. Museyko, G. Leugering, P. Hastreiter

The problem of non-rigid registration of images, where certain parts are missing, is considered in the context of pre- and intraoperative or damaged data. A variational formulation of the problem is proposed such that a counterpart of the missing data can be restored in a template image along with a corresponding registration transforma­tion between the images. The problem is a registrational variant of a free-discontinuity Mumford-Shah segmentation problem with the unknown discontinuity set representing the boundary of missing data; thus, no preliminary segmentation is needed to detect the missing information. An approximation of the free-discontinuity problem by an Ambrosio-Tortorelli-type approximation is used to compute the numerical solutions. Ex­periments with 2D examples demonstrate the efficiency of the proposed approach.

45-63

Authors: I. G. Balanenko, P. I. Kogut

In the paper the Dirichlet optimal control problem associated with a linear parabolic equation the coefficients of which we take as controls in L1(Ω) has been studied. Since equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, it is shown that the optimal control problem in the coefficients can be stated in different settings depending on the choice of the class of admissible solutions. Using the direct method in the Calculus of Variations, the solvability of the above optimal control problems in the so-called class of inadmissible solutions has been discussed.

73-89

Authors: V. A. Ostapenko

Nonlinear initial-boundary filtering problems for filters containing variable parame­ters are considered. Every such problem is transformed to two independent boundary problems of Goursat for the concentration of sorbate in a chemical solution and con­centration of sorbate, being absorbed by a sorbent.Each of the formulated problems is solved applying the Riemann method. Method of constructing the Riemannn functions for equations with variable coefficients is developed. It is showed that it is sufficient to obtain by Riemann method only one of such variable. Second one can be obtained than by solution of Cauchy problem.It is proved the obtained expansion is really asymptotic.

103-113

Authors: V. A. Ostapenko

The inhomogeneous Hill equation in which coefficients and right hand part are pe­riodical with the same period T is considered. It is shown, that only some particular solutions of this equation can be periodical with period T. At the same time it is succeed to discover at certain values of equation’s parameters the general solution of inhomo-geneous Hill equation becomes periodical with period multiple of T. Necessary and sufficient conditions for periodicity of general solution of inhomogeneous Hill equation is obtained. A numerical algorithm for constructing of periodical fundamental system solutions to corresponding homogeneous Hill equation is worked out. It is shown, as the periodical general solution of inhomogeneous Hill equation can be obtained from this fundamental system. Some applications of obtained results in engineering practice are specified.

124-136

Authors: T. A. Boganova

We studied traffic flow models in vector-valued optimization statement where the flow is controlled at the nodes of network. We considered the case when an objective mapping possesses a weakened property of upper semicontinuity and made no assumptions on the interior of the ordering cone. The sufficient conditions for the existence of efficient controls of the traffic problems are derived. We discussed the simplest scalarization approach to construction of efficient solutions of vector optimization problem.