Adaptation and Evaluation of the Multisplitting-Newton and Waveform Relaxation Methods Over Distributed Volatile Environments

International audienceThis paper presents new adaptations of two methods that solve large differential equations systems, to the grid context. The first method isbased on the Multisplitting concept and the second on the Waveform Relaxation concept. Their adaptations are implemented according to the asynchronous iteration model which is well suited to volatile architectures that suffer from high latency networks. Many experiments were conducted to evaluate and compare the accuracy and performance of both methods while solving the advection-diffusion problem over heterogeneous, distributed and volatile architectures. The JACEP2P-V2 middleware provided the fault tolerant asynchronous environment, required for these experiments

Parallelization of direct algorithms using multisplitting methods in grid environments

The goal of this paper is to introduce a new approach to the building of efficient distributed linear system solvers. The starting point of the results of this paper lies in the fact that the parallelization of direct algorithms requires frequent synchronizations in order to obtain the solution for a linear problem. In a grid computing environment, communication times are significant and the bandwidth is variable, therefore frequent synchronizations slow down performances. Thus it is desirable to reduce the number of synchronizations in a parallel direct algorithm. Inspired from multisplitting techniques, the method we present consists in solving several linear problems obtained by splitting the original one. Each linear system is solved independently on a cluster by using the direct method. This paper uses the theoretical results of \cite{BMR97} in order to build coarse grained algorithms designed for solving linear systems in the grid computing context

Asynchronous iterations with flexible communication: contracting operators

AbstractThe concept of flexible communication permits one to model efficient asynchronous iterations on parallel computers. This concept is particularly useful in two practical situations. Firstly, when communications are requested while a processor has completed the current update only partly, and secondly, in the context of inner/outer iterations, when processors are also allowed to make use of intermediate results obtained during the inner iteration in other processors.In the general case of nonlinear or linear fixed point problems, we give a global convergence results for asynchronous iterations with flexible communication whereby the iteration operators satisfy certain contraction hypotheses. In this manner we extend to a contraction context previous results obtained for monotone operators with respect to a partial ordering

Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations

[EN] The benefits and properties of iterative splitting methods, which are based on serial versions, have been studied in recent years, this work, we extend the iterative splitting methods to novel classes of parallel versions to solve nonlinear fractional convection-diffusion equations. For such interesting partial differential examples with higher dimensional, fractional, and nonlinear terms, we could apply the parallel iterative splitting methods, which allow for accelerating the solver methods and reduce the computational time. Here, we could apply the benefits of the higher accuracy of the iterative splitting methods. We present a novel parallel iterative splitting method, which is based on the multi-splitting methods, The flexibilisation with multisplitting methods allows for decomposing large scale operator equations. In combination with iterative splitting methods, which use characteristics of waveform-relaxation (WR) methods, we could embed the relaxation behavior and deal better with the nonlinearities of the operators. We consider the convergence results of the parallel iterative splitting methods, reformulating the underlying methods with a summation of the individual convergence results of the WR methods. We discuss the numerical convergence of the serial and parallel iterative splitting methods with respect to the synchronous and asynchronous treatments. Furthermore, we present different numerical applications of fluid and phase field problems in order to validate the benefit of the parallel versions.This research was partially supported by Ministerio de Economia y Competitividad, Spain, under grant PGC2018-095896-B-C21-C22 and German Academic Exchange Service grant number 91588469.Geiser, J.; Martínez Molada, E.; Hueso, JL. (2020). Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations. Mathematics. 8(11):1-42. https://doi.org/10.3390/math8111950S142811Farago, I., & Geiser, J. (2007). Iterative operator-splitting methods for linear problems. International Journal of Computational Science and Engineering, 3(4), 255. doi:10.1504/ijcse.2007.018264Frommer, A., & Szyld, D. B. (2000). On asynchronous iterations. Journal of Computational and Applied Mathematics, 123(1-2), 201-216. doi:10.1016/s0377-0427(00)00409-xO’Leary, D. P., & White, R. E. (1985). Multi-Splittings of Matrices and Parallel Solution of Linear Systems. SIAM Journal on Algebraic Discrete Methods, 6(4), 630-640. doi:10.1137/0606062White, R. E. (1986). Parallel Algorithms for Nonlinear Problems. SIAM Journal on Algebraic Discrete Methods, 7(1), 137-149. doi:10.1137/0607017Geiser, J. (2016). Picard’s iterative method for nonlinear multicomponent transport equations. Cogent Mathematics, 3(1), 1158510. doi:10.1080/23311835.2016.1158510Miekkala, U., & Nevanlinna, O. (1987). Convergence of Dynamic Iteration Methods for Initial Value Problems. SIAM Journal on Scientific and Statistical Computing, 8(4), 459-482. doi:10.1137/0908046Miekkala, U., & Nevanlinna, O. (1996). Iterative solution of systems of linear differential equations. Acta Numerica, 5, 259-307. doi:10.1017/s096249290000266xGeiser, J. (2010). Iterative operator-splitting methods for nonlinear differential equations and applications. Numerical Methods for Partial Differential Equations, 27(5), 1026-1054. doi:10.1002/num.20568He, D., Pan, K., & Hu, H. (2020). A spatial fourth-order maximum principle preserving operator splitting scheme for the multi-dimensional fractional Allen-Cahn equation. Applied Numerical Mathematics, 151, 44-63. doi:10.1016/j.apnum.2019.12.018Giona, M., Cerbelli, S., & Roman, H. E. (1992). Fractional diffusion equation and relaxation in complex viscoelastic materials. Physica A: Statistical Mechanics and its Applications, 191(1-4), 449-453. doi:10.1016/0378-4371(92)90566-9Nigmatullin, R. R. (1986). The realization of the generalized transfer equation in a medium with fractal geometry. physica status solidi (b), 133(1), 425-430. doi:10.1002/pssb.2221330150Allen, S. M., & Cahn, J. W. (1979). A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica, 27(6), 1085-1095. doi:10.1016/0001-6160(79)90196-2Yue, P., Feng, J. J., Liu, C., & Shen, J. (2005). Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids. Journal of Non-Newtonian Fluid Mechanics, 129(3), 163-176. doi:10.1016/j.jnnfm.2005.07.002Sommacal, L., Melchior, P., Oustaloup, A., Cabelguen, J.-M., & Ijspeert, A. J. (2008). Fractional Multi-models of the Frog Gastrocnemius Muscle. Journal of Vibration and Control, 14(9-10), 1415-1430. doi:10.1177/1077546307087440Moshrefi-Torbati, M., & Hammond, J. K. (1998). Physical and geometrical interpretation of fractional operators. Journal of the Franklin Institute, 335(6), 1077-1086. doi:10.1016/s0016-0032(97)00048-3El-Nabulsi, R. A. (2009). Fractional Dirac operators and deformed field theory on Clifford algebra. Chaos, Solitons & Fractals, 42(5), 2614-2622. doi:10.1016/j.chaos.2009.04.002Kanney, J. F., Miller, C. T., & Kelley, C. T. (2003). Convergence of iterative split-operator approaches for approximating nonlinear reactive transport problems. Advances in Water Resources, 26(3), 247-261. doi:10.1016/s0309-1708(02)00162-8Geiser, J., Hueso, J. L., & Martínez, E. (2020). Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations. Mathematics, 8(3), 302. doi:10.3390/math8030302Meerschaert, M. M., Scheffler, H.-P., & Tadjeran, C. (2006). Finite difference methods for two-dimensional fractional dispersion equation. Journal of Computational Physics, 211(1), 249-261. doi:10.1016/j.jcp.2005.05.017Irreversibility, Least Action Principle and Causality. Preprint, HAL, 2008 https://hal.archives-ouvertes.fr/hal-00348123v1Cresson, J. (2007). Fractional embedding of differential operators and Lagrangian systems. Journal of Mathematical Physics, 48(3), 033504. doi:10.1063/1.2483292Meerschaert, M. M., & Tadjeran, C. (2004). Finite difference approximations for fractional advection–dispersion flow equations. Journal of Computational and Applied Mathematics, 172(1), 65-77. doi:10.1016/j.cam.2004.01.033Geiser, J. (2011). Computing Exponential for Iterative Splitting Methods: Algorithms and Applications. Journal of Applied Mathematics, 2011, 1-27. doi:10.1155/2011/193781Geiser, J. (2008). Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations. Journal of Computational and Applied Mathematics, 217(1), 227-242. doi:10.1016/j.cam.2007.06.028Ladics, T. (2015). Error analysis of waveform relaxation method for semi-linear partial differential equations. Journal of Computational and Applied Mathematics, 285, 15-31. doi:10.1016/j.cam.2015.02.003Yuan, D., & Burrage, K. (2003). Convergence of the parallel chaotic waveform relaxation method for stiff systems. Journal of Computational and Applied Mathematics, 151(1), 201-213. doi:10.1016/s0377-0427(02)00749-5Ladics, T., & Faragó, I. (2013). Generalizations and error analysis of the iterative operator splitting method. Open Mathematics, 11(8). doi:10.2478/s11533-013-0246-4Moler, C., & Van Loan, C. (2003). Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later. SIAM Review, 45(1), 3-49. doi:10.1137/s00361445024180Najfeld, I., & Havel, T. F. (1995). Derivatives of the Matrix Exponential and Their Computation. Advances in Applied Mathematics, 16(3), 321-375. doi:10.1006/aama.1995.1017Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. doi:10.1017/s0962492910000048Casas, F., & Iserles, A. (2006). Explicit Magnus expansions for nonlinear equations. Journal of Physics A: Mathematical and General, 39(19), 5445-5461. doi:10.1088/0305-4470/39/19/s07Magnus, W. (1954). On the exponential solution of differential equations for a linear operator. Communications on Pure and Applied Mathematics, 7(4), 649-673. doi:10.1002/cpa.3160070404Jeltsch, R., & Pohl, B. (1995). Waveform Relaxation with Overlapping Splittings. SIAM Journal on Scientific Computing, 16(1), 40-49. doi:10.1137/0916004Faragó, I. (2008). A modified iterated operator splitting method. Applied Mathematical Modelling, 32(8), 1542-1551. doi:10.1016/j.apm.2007.04.018Li, J., Jiang, Y., & Miao, Z. (2019). A parareal approach of semi‐linear parabolic equations based on general waveform relaxation. Numerical Methods for Partial Differential Equations, 35(6), 2017-2043. doi:10.1002/num.22390Trotter, H. F. (1959). On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10(4), 545-545. doi:10.1090/s0002-9939-1959-0108732-6Strang, G. (1968). On the Construction and Comparison of Difference Schemes. SIAM Journal on Numerical Analysis, 5(3), 506-517. doi:10.1137/0705041Geiser, J. (2009). Operator-splitting methods in respect of eigenvalue problems for nonlinear equations and applications for Burgers equations. Journal of Computational and Applied Mathematics, 231(2), 815-827. doi:10.1016/j.cam.2009.05.00