On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction–Diffusion–Advection-Type with Data on the Position of a Reaction Front

The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction–diffusion–advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.

Реализация и производительность алгоритмов волновой томографии на вычислительных платформах SIMD CPU и GPU

This paper is concerned with implementation of wave tomography algorithms on modern SIMD CPU and GPU computing platforms. The field of wave tomography, which is currently under development, requires powerful computing resources. Main applications of wave tomography are medical imaging, nondestructive testing, seismic studies. Practical applications depend on computing hardware. Tomographic image reconstruction via wave tomography technique involves solving coefficient inverse problems for the wave equation. Such problems can be solved using iterative gradient-based methods, which rely on repeated numerical simulation of wave propagation process. In this study, finite-difference time-domain (FDTD) method is employed for wave simulation. This paper discusses software implementation of the algorithms and compares the performance of various computing devices: multi-core Intel and ARM-based CPUs, NVidia graphics processors. В данной статье рассматривается реализация алгоритмов волновой томографии на современных вычислительных платформах SIMD CPU и GPU. Область волновой томографии, которая в настоящее время находится в стадии разработки, требует мощных вычислительных ресурсов. Основные области применения волновой томографии - это медицинская визуализация, неразрушающий контроль, сейсмические исследования. Практические приложения зависят от вычислительного оборудования. Восстановление томографического изображения методом волновой томографии включает решение коэффициентов обратной задачи для волнового уравнения. Такие проблемы могут быть решены с помощью итерационных градиентных методов, основанных на многократном численном моделировании процесса распространения волн. В этом исследовании для моделирования волн используется метод конечных разностей во временной области (FDTD). В статье обсуждается программная реализация алгоритмов и сравнивается производительность различных вычислительных устройств: многоядерных процессоров Intel и ARM, графических процессоров NVidia.

A triply cubic polynomials approach for globally convergent algorithm in coefficient inverse problems

In this paper, a triply cubic polynomials approach is proposed firstly for solving the globally convergent algorithm of coefficient inverse problems in three dimension. Using Taylor type expansion for three arguments to construct tripled cubic polynomials for the approximate solution as identifying coefficient function of parabolic initial-boundary problems, in which unknown coefficients appeared at nonlinear term. The presented computational approach is effective to execute in spatial 3D issues arising in real world. Further, the complete algorithm can be straightforwardly performed to lower or higher dimensions case

A triply cubic polynomials approach for globally convergent algorithm in coefficient inverse problems

In this paper, a triply cubic polynomials approach is proposed firstly for solving the globally convergent algorithm of coefficient inverse problems in three dimension. Using Taylor type expansion for three arguments to construct tripled cubic polynomials for the approximate solution as identifying coefficient function of parabolic initial-boundary problems, in which unknown coefficients appeared at nonlinear term. The presented computational approach is effective to execute in spatial 3D issues arising in real world. Further, the complete algorithm can be straightforwardly performed to lower or higher dimensions case

COEFFICIENT INVERSE PROBLEMS FOR THE PARABOLIC EQUATION WITH GENERAL WEAK DEGENERATION

It is investigated the inverse problems for the degenerate parabolic equation. The mi- nor coeffcient of this equation is a linear polynomial with respect to space variable with two unknown time-dependent functions. The degeneration of the equation is caused by the monotone increasing function at the time derivative. It is established conditions of existence and uniqueness of the classical solutions to the named problems in the case of weak degeneration.