A minimal communication approach to parallel time integration

We explore an approach due to Nievergelt of decomposing a time-evolution equation along the time dimension and solving it in parallel with as little communication as possible between the processors. This method computes a map from initial conditions to final conditions locally on slices of the time-domain, and then patches these operators together into a global solution using a single communication step. A basic error analysis is given, and some comparisons are made with other parallel in time methods. Based on the assumption that parallel computation is cheap but communication is very expensive, it is shown that this method can be competitive for some problems. We present numerical simulations on graphic chips and on traditional parallel clusters using hundreds of processors for a variety of problems to show the practicality and scalability of the proposed method.

[1]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[2]  M. Hochbruck,et al.  Exponential Runge--Kutta methods for parabolic problems , 2005 .

[3]  A. Zients.  Andy , 2003 .

[4]  Charbel Farhat,et al.  Time‐parallel implicit integrators for the near‐real‐time prediction of linear structural dynamic responses , 2006 .

[5]  Jürg Nievergelt,et al.  Parallel methods for integrating ordinary differential equations , 1964, CACM.

[6]  Colin B. Macdonald,et al.  Parallel High-Order Integrators , 2010, SIAM J. Sci. Comput..

[7]  Yousef Saad,et al.  Efficient Solution of Parabolic Equations by Krylov Approximation Methods , 1992, SIAM J. Sci. Comput..

[8]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

[9]  Yvon Maday,et al.  A parareal in time iterative solver: a further direction to parallel implementation. Domain decomposition methods in science and engineering , 2005 .

[10]  Yvon Maday,et al.  The Parareal in Time Iterative Solver: a Further Direction to Parallel Implementation , 2005 .

[11]  Vu Thai Luan,et al.  Explicit exponential Runge-Kutta methods of high order for parabolic problems , 2013, J. Comput. Appl. Math..

[12]  Marc Massot,et al.  Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies , 2011 .

[13]  Martin J. Gander,et al.  Analysis of a Krylov subspace enhanced parareal algorithm for linear problems , 2008 .

[14]  J. Lions,et al.  Résolution d'EDP par un schéma en temps « pararéel » , 2001 .

[15]  P. Henrici.  Discrete Variable Methods in Ordinary Differential Equations , 1962 .

[16]  Charbel Farhat,et al.  Time‐decomposed parallel time‐integrators: theory and feasibility studies for fluid, structure, and fluid–structure applications , 2003 .

[17]  Martin J. Gander,et al.  Analysis of the Parareal Algorithm Applied to Hyperbolic Problems Using Characteristics , 2008 .

[18]  Ivan P. Gavrilyuk,et al.  Exponentially Convergent Algorithms for the Operator Exponential with Applications to Inhomogeneous Problems in Banach Spaces , 2005, SIAM J. Numer. Anal..

[19]  Martin J. Gander,et al.  PARAEXP: A Parallel Integrator for Linear Initial-Value Problems , 2013, SIAM J. Sci. Comput..

[20]  Michael L. Minion,et al.  A HYBRID PARAREAL SPECTRAL DEFERRED CORRECTIONS METHOD , 2010 .

[21]  Martin J. Gander,et al.  Analysis of the Parareal Time-Parallel Time-Integration Method , 2007, SIAM J. Sci. Comput..

[22]  Yvon Maday,et al.  A Parareal in Time Semi-implicit Approximation of the Navier-Stokes Equations , 2005 .

[23]  Jianwei Hu,et al.  Modified propagators of parareal in time algorithm and application to Princeton Ocean model , 2008 .

[24]  M. Gander,et al.  Analysis of a Modified Parareal Algorithm for Second-Order Ordinary Differential Equations , 2007 .

[25]  Dongwoo Sheen,et al.  A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature , 2003 .