Abstract
It is the purpose of this paper to provide an acceleration of waveform relaxation (WR) methods for the numerical solution of large systems of ordinary differential equations. The introduced technique is based on the employ of graphics processing units (GPUs) in order to speed-up the numerical integration process. A CUDA solver based on WR-Picard, WR-Jacobi and red-black WR-Gauss-Seidel iterations is presented and some numerical experiments realized on a multi-GPU machine are provided.
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References
Burrage, K.: Parallel and sequential methods for ordinary differential equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (1995)
Burrage, K., Dyke, C., Pohl, B.: On the performance of parallel waveform relaxations for differential systems. Appl. Numer. Math. 20(1-2), 39–55 (1996)
Burrage, K., Jackiewicz, Z., Norsett, S.P., Renaut, R.A.: Preconditioning waveform relaxation iterations for differential systems. BIT 36(1), 54–76 (1996)
Burrage, K., Sand, J.: A Jacobi Waveform Relaxation Method for ODEs. SIAM J. Sci. Comput. 20(2), 534–552 (1998)
Capobianco, G., Cardone, A.: A parallel algorithm for large systems of Volterra integral equations of Abel type. J. Comput. Appl. Math. 220, 749–758 (2008). doi:10.1016/j.cam.2008.05.026
Capobianco, G., Conte, D.: An efficient and fast parallel method for Volterra integral equations of Abel type. J. Comput. Appl. Math. 189/1–2, 481–493 (2006). doi:10.1016/j.cam.2005.03.056
Capobianco, G., Conte, D., Del Prete, I.: High performance numerical methods for Volterra equations with weakly singular kernels. J. Comput. Appl. Math. 228, 571–579 (2009). doi:10.1016/j.cam.2008.03.027
Cardone, A., Messina, E., Russo, E.: A fast iterative method for discretized Volterra-Fredholm integral equations. J. Comput. Appl. Math 189, 568–579 (2006)
Cardone, A., Messina, E., Vecchio, A.: An adaptive method for Volterra-Fredholm integral equations on the half line. J. Comput. Appl. Math. 228 (2), 538–547 (2009)
Conte, D., D’Ambrosio, R., Jackiewicz, Z.: Two-step Runge-Kutta methods with quadratic stability functions. J. Sci. Comput. 44(2), 191–218 (2010)
Conte, D., D’Ambrosio, R., Jackiewicz, Z., Paternoster, B.: A pratical approach for the derivation of algebraically stable two-step Runge-Kutta methods. Math. Model. Anal. 17(1), 65–77 (2012)
Conte, D., D’Ambrosio, R., Jackiewicz, Z., Paternoster, B.: Numerical search for algebrically stable two-step continuous Runge-Kutta methods. J. Comput. Appl. Math. 239, 304–321 (2013)
Conte, D., D’Ambrosio, R., Paternoster, B.: Two-step diagonally-implicit collocation based methods for Volterra Integral Equations. Appl. Numer. Math. 62 (10), 1312–1324 (2012). doi:10.1016/j.apnum.2012.06.007
Conte, D., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for Volterra integral equations. Appl. Math. Comp. 204, 839–853 (2008)
Conte, D., Paternoster, B.: Multistep collocation methods for Volterra Integral Equations. Appl. Numer. Math. 59, 1721–1736 (2009)
Courvoisier, Y., Gander, M.J.: Optimization of Schwarz waveform relaxation over short time windows. Numerical Algorithms 64(2), 221–243 (2013)
Crisci, M.R., Paternoster, B., Russo, E.: Fully parallel Runge-Kutta-Nyström methods for ODEs with oscillating solutions. Appl. Numer. Math. 11(1–3), 143–158 (1993)
D’Ambrosio, R., Ferro, M., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for ordinary differential equations. Numer. Algorithms 53(2-3), 195–217 (2010)
D’Ambrosio, R., Jackiewicz, Z.: Continuous Two-Step Runge-Kutta Methods for Ordinary Differential Equations. Numer. Algorithms 54(2), 169–193 (2010)
D’Ambrosio, R., Jackiewicz, Z.: Construction and implementation of highly stable two-step continuous methods for stiff differential systems. Math. Comput. Simul. 81(9), 1707–1728 (2011)
D’Ambrosio, R., Paternoster, B.: Two-step modified collocation methods with structured coefficient matrices for ordinary differential equations. Appl. Numer. Math. 62(10), 1325–1334 (2012)
Davison, E.J.: An Algorithm for the Computer Simulation of Very Large Dynamic Systems. Automatica 9, 665–675 (1973)
Enright, W.: On the efficient and reliable numerical solution of large linear systems of O.D.E.’s. IEEE Trans. Autom. Control AC–24(6), 905–908 (1979)
Everstine: Numerical Solution of PDE, gwu.geverstine.com/pdenum.pdf (2010)
Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations. I. Nonstiff problems. Second edition. Springer Series in Computational Mathematics, 8. Springer-Verlag, Berlin (1993)
Hairer, E., Wanner, G.: Solving ordinary differential equations. II. Stiff and differential-algebraic problems. Second revised edition, paperback. Springer Series in Computational Mathematics, 14. Springer-Verlag, Berlin (2010)
Jackson, K.R., Norsett, S.P.: The Potential for Parallelism in Runge-Kutta Methods. Part 1: RK Formulas in Standard Form. SIAM J. Numer. Anal. 32, 49–82 (1990)
Janssen, J.: Acceleration of waveform relaxation methods for linear ordinary and partial differential equations, PhD-thesis. Katholieke Universiteit Leuven, Belgium (1997)
Lelarasmee, E.: The Waveform Relaxation Method for the Time-Domain Analysis of Large-Scale Nonlinear Systems, Ph.D. thesis Department of Electrical Engineering and Computer Science. University of California, Berkeley (1982)
Leimkuhler, B.: Estimating Waveform Relaxation Convergence. SIAM J. Sci. Comput. 14, 872–889 (1993)
Leimkuhler, B., Ruehli, A.: Rapid convergence of waveform relaxation. Appl. Numer. Math. 11(1–3), 211–224 (1993)
Lopez, L.: Methods based on boundary value techniques for solving parabolic equations on parallel computers. Parallel Comput. 19(9), 979–991 (1993)
Lopez, L., Politi, T.: Parallel methods in the numerical treatment of population dynamic models. Parallel Comput. 18(7), 767–777 (1992)
Lopez, L., Politi, T.: Tridiagonal splittings in the conditioning and parallel solution of banded linear systems. Linear Algebra Appl. 251, 249–265 (1997)
Lopez, L., Trigiante, D.: A finite difference scheme for a stiff problem arising in the numerical solution of a population dynamic model with spatial diffusion. Nonlinear Anal. 9(1), 1–12 (1985)
Miekkala, U., Nevanlinna, O.: Convergence of dynamic iterations for initial value problems. SIAM J. Sci. Stat. Comp. 8, 459–482 (1987)
Miekkala, U., Nevanlinna, O.: Iterative Solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996)
NVIDIA Corporation: NVIDIA CUDA compute unified device architecture programming guide, http://developer.download.nvidia.com/compute/DevZone/docs/html/C/doc/CUDA_C_Programming_Guide.pdf (2012)
Ortega, J.: Introduction to Parallel and Vector Solution of Linear Systems Springer (1988)
Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations. Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin (1994)
Reichelt, M., White, J., Allen, J.: Waveform relaxation for transient simulation of two-dimensional MOS devices. In: Proceedings of the IEEE International Conference on Comparative-Aided Design (1989)
Sanders, J., Kandrot, E.: CUDA by example, An Introduction to General-Purpose GPU Programming, NVIDIA Corporation. Addison-Wesley (2011)
Simek, V., Dvorak, R., Zboril, F., Kunovsky, J.: Towards Accelerated Computation of Atmospheric Equations Using CUDA, UKSIM ’09 11th International Conference on Computer Modelling and Simulation, pp. 449–454 (2009)
Spiteri, R.J., Dean, R.C.: On the Performance of an Implicit-Explicit Runge-Kutta Method in Models of Cardiac Electrical Activity. IEEE Trans. Biomed. Eng. 55(5), 1488–1495 (2008)
van der Houwen, P.J., Sommeijer, B.P.: CWI contributions to the development of parallel Runge-Kutta methods. Appl. Numer. Math. 22(1-3), 327–344 (1996)
Van Lent, J., Vandewalle, S.: Multigrid waveform relaxation for anisotropic partial differential equations. Numer. Algorithms 31(1-4), 361–380 (2002)
Zhang, H., Jiang, Y.-L.: A note on the H1-convergence of the overlapping Schwarz waveform relaxation method for the heat equation. Numer. Algorithms 66 (2), 299–307 (2014)
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Conte, D., D’Ambrosio, R. & Paternoster, B. GPU-acceleration of waveform relaxation methods for large differential systems. Numer Algor 71, 293–310 (2016). https://doi.org/10.1007/s11075-015-9993-6
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DOI: https://doi.org/10.1007/s11075-015-9993-6