Abstract
In this work we deal with the solution of a two-dimensional inverse time fractional diffusion equation, involving a Caputo fractional derivative in his expression. Since we deal with a huge practical problem with a large domain, by starting from an accurate meshless localized collocation method using RBFs, here we propose a fast algorithm, implemented in a multicore architecture, which exploits suitable parallel computational kernels. More in detail, we firstly developed, a C code based on the numerical library LAPACK to perform the basic linear algebra operations and to solve linear systems, then, due to the high computational complexity and the large size of the problem, we propose a parallel algorithm specifically designed for multicore architectures and based on the Pthreads library. Performance analysis will show accuracy and reliability of our parallel implementation.
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References
Yan, L., Yang, F.: The method of approximate particular solutions for the time-fractional diffusion equation with a non-local boundary condition. Comput. Math. Appl. 70, 254–264 (2015)
Abbasbandy, S., Ghehsareh, H.R., Alhuthali, M.S., Alsulami, H.H.: Comparison of meshless local weak and strong forms based on particular solutions for a non-classical 2-D diffusion model. Eng. Anal. Bound. Elem. 39, 121–128 (2014)
Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703–1727 (2015)
Kilbas, A., Srivastava, M.H., Trujillo, J.J.: Theory and application of fractional differential equations. North Holl. Math. Stud. vol. 204 (2006)
Liu, Q., Gu, Y.T., Zhuang, P., Liu, F., Nie, Y.F.: An implicit RBF meshless approach for time fractional diffusion equations. Comput Mech 48, 1–12 (2011)
Cuomo, S., De Michele, P., Galletti, A., Marcellino, L.: A GPU parallel implementation of the local principal component analysis overcomplete method for DW image denoising. In: Proceedings - IEEE Symposium on Computers and Communications (2016), Vol. 2016-August, pp. 26–31 (2016)
D’Amore, L., Arcucci, R., Marcellino, L., Murli, A.: A parallel three-dimensional variational data assimilation scheme. AIP Conf. Proc. 1389, 1829–1831 (2011)
Cuomo, S., Galletti, A., Giunta, G., Marcellino, L.: Reconstruction of implicit curves and surfaces via RBF interpolation. Appl. Numer. Math. 116, 157–171 (2017). https://doi.org/10.1016/j.apnum.2016.10.016
Ala, G., Ganci, S., Francomano, E.: Unconditionally stable meshless integration of time domain Maxwell’s curl equations. Appl. Math. Comput. 255, 157–164 (2015)
Cuomo, S., Galletti, A., Giunta, G., Starace, A.: Surface reconstruction from scattered point via RBF interpolation on GPU. In: 2013 Federated Conference on Computer Science and Information Systems, FedCSIS 2013, art. no. 6644037, pp. 433–440 (2013)
Francomano, E., Paliaga, M., Galletti, A., Marcellino, L.: First experiences on an accurate SPH method on GPUs. In: (2018) Proceedings - 13th International Conference on Signal-Image Technology and Internet-Based Systems, SITIS 2017, 2018-January, pp. 445–449. https://doi.org/10.1109/SITIS.2017.79
Ala, G., Di Paola, M., Francomano, E., Pinnola, F.: Electrical analogous in viscoelasticity. Commun. Nonlinear Sci. Numer. Simul. 19(7), 2513–2527 (2014)
Podlubny, I.: Fractional Differential Equation, Academic Press, Cambridge (1999)
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De Luca, P., Galletti, A., Giunta, G., Marcellino, L., Raei, M. (2020). Performance Analysis of a Multicore Implementation for Solving a Two-Dimensional Inverse Anomalous Diffusion Problem. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11973. Springer, Cham. https://doi.org/10.1007/978-3-030-39081-5_11
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DOI: https://doi.org/10.1007/978-3-030-39081-5_11
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