GPU accelerated algorithms for computing matrix function vector products with applications to exponential integrators and fractional diffusion

Farquhar, Megan E., Moroney, Timothy J., Yang, Qianqian, & Turner, Ian W. (2016) GPU accelerated algorithms for computing matrix function vector products with applications to exponential integrators and fractional diffusion. SIAM Journal on Scientific Computing, 38(3), C127-C149.

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Abstract

The efficient computation of matrix function vector products has become an important area of research in recent times, driven in particular by two important applications: the numerical solution of fractional partial differential equations and the integration of large systems of ordinary differential equations. In this work we consider a problem that combines these two applications, in the form of a numerical solution algorithm for fractional reaction diffusion equations that after spatial discretisation, is advanced in time using the exponential Euler method. We focus on the efficient implementation of the algorithm on Graphics Processing Units (GPU), as we wish to make use of the increased computational power available with this hardware. We compute the matrix function vector products using the contour integration method in [N. Hale, N. Higham, and L. Trefethen. Computing Aα, log(A), and related matrix functions by contour integrals. SIAM J. Numer. Anal., 46(5):2505–2523, 2008]. Multiple levels of preconditioning are applied to reduce the GPU memory footprint and to further accelerate convergence. We also derive an error bound for the convergence of the contour integral method that allows us to pre-determine the appropriate number of quadrature points. Results are presented that demonstrate the effectiveness of the method for large two-dimensional problems, showing a speedup of more than an order of magnitude compared to a CPU-only implementation.

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ID Code: 84722
Item Type: Journal Article
Refereed: Yes
Additional URLs:
Keywords: Matrix functions, Contour integral method, fractional reaction-diffusion equation, fractional Laplacian, Exponential Euler Method, Krylov subspace methods
DOI: 10.1137/15M1021672
ISSN: 1095-7197
Divisions: Current > Research Centres > ARC Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS)
Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Funding:
Copyright Owner: Copyright 2015 Society for Industrial and Applied Mathematics
Deposited On: 14 Jun 2015 23:02
Last Modified: 03 Jun 2016 09:07

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