Parallel Algorithm for Solving the Inverse Two-Dimensional Fractional Diffusion Problem of Identifying the Source Term

Akimova E.N. Sultanov M.A. Misilov V.E. Nurlanuly Y.
November 2023 Multidisciplinary Digital Publishing Institute (MDPI)

Fractal and Fractional
2023 #7 Issue 11

This paper is devoted to the development of a parallel algorithm for solving the inverse problem of identifying the space-dependent source term in the two-dimensional fractional diffusion equation. For solving the inverse problem, the regularized iterative conjugate gradient method is used. At each iteration of the method, we need to solve the auxilliary direct initial-boundary value problem. By using the finite difference scheme, this problem is reduced to solving a large system of a linear algebraic equation with a block-tridiagonal matrix at each time step. Solving the system takes almost the entire computation time. To solve this system, we construct and implement the direct parallel matrix sweep algorithm. We establish stability and correctness for this algorithm. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to study the performance of parallel implementations.

block-elimination method , Caputo fractional derivative , finite-difference scheme , inverse problems , parallel computing , parallel matrix sweep method , source term identification , time-fractional diffusion equation

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Ural Branch of RAS, Krasovskii Institute of Mathematics and Mechanics, S. Kovalevskaya Street 16, Ekaterinburg, 620108, Russian Federation
Department of Information Technologies and Control Systems, Institute of Radioelectronics and Information Technology, Ural Federal University, Mira Street 19, Ekaterinburg, 620002, Russian Federation
Department of Mathematics, Faculty of Natural Science, Khoja Akhmet Yassawi International Kazakh-Turkish University, Turkistan, 160200, Kazakhstan
Department of High Performance Computing Technologies, Institute of Natural Sciences and Mathematics, Ural Federal University, Mira Street 19, Ekaterinburg, 620002, Russian Federation

Ural Branch of RAS
Department of Information Technologies and Control Systems
Department of Mathematics
Department of High Performance Computing Technologies

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