An efficient algorithm for the parallel solution of high-dimensional differential equations

An efficient algorithm for the parallel solution of high-dimensional differential equations Stefan Klusa, Tuhin Sahaib,∗, Cong Liub, Michael Dellnitza aInstitute for Industrial Mathematics, University of Paderborn, 33095 Paderborn, Germany bUnited Technologies Research Center, East Hartford, CT 06108, USA Abstract The study of high-dimensional differential equations is challenging and difficult due to the analyt- ical and computational intractability. Here, we improve the speed of waveform relaxation (WR), a method to simulate high-dimensional differential-algebraic equations. This new method termed adaptive waveform relaxation (AWR) is tested on a communication network example. Further we propose different heuristics for computing graph partitions tailored to adaptive waveform re- laxation. We find that AWR coupled with appropriate graph partitioning methods provides a speedup by a factor between 3 and 16. Keywords: waveform relaxation, adaptive windowing, graph partitioning, Petri nets, parallel algorithms 1. Introduction Over the past few years, several attempts have been made to study differential equations of high dimensionality. These equations naturally occur in models for systems as diverse as metabolic networks [1], communication networks [2], fluid turbulence [3], heart dynamics [4], chemical systems [5] and electrical circuits [6] to name but a few. Traditional approaches ap- proximate the full system by dynamical systems of lower dimension. These model reduction techniques [7] include proper orthogonal decomposition (POD) along with Galerkin projections [3], Krylov subspace methods [8], and balanced truncation or balanced POD (see e.g. [9]). In this work, we accelerate a parallel algorithm, for the simulation of differential-algebraic equations, called waveform relaxation [6, 10, 11]. In waveform relaxation, instead of approxi- mating the original system by a lower-dimensional model, the methodology is to distribute the computations for the entire system on multiple processors. Each processor solves only a part of the problem. The solutions corresponding to subsystems on other processors are regarded as inputs whose waveforms are given by the solution of the previous iteration. This step is one iteration of the procedure. At the end of each iteration the solutions are distributed among the processors. The procedure is repeated until convergence is achieved. The initial waveforms are typically chosen to be constant. This paper is organized as follows: Based on previously derived error bounds for waveform relaxation (cf. [13, 14]), we propose and demonstrate a new algorithm to break the time interval ∗Corresponding author. Email: SahaiT@utrc.utc.com Preprint submitted to Journal of Computational and Applied Mathematics October 28, 2018 arXiv:1003.5238v3 [cs.DC] 26 Oct 2010 for simulation [0, T] into smaller subintervals. We call this method adaptive waveform relax- ation. It is important to note that this method is different from windowing methods discussed in [10]. Subsequently, we analyze and present time and memory complexity of waveform relax- ation techniques and the dependence of the convergence behavior on the decomposition of the system. Furthermore, we introduce different graph partitioning heuristics in order to efficiently generate an appropriate splitting. We demonstrate that the combination of graph partitioning along with adaptive waveform relaxation results in an improved performance over traditional waveform relaxation and standard windowing techniques. 2. Error bounds For an ordinary differential equation of the form ˙x = f(x), f : Rn 7→Rn, the iteration method described in the introduction can be written as ˙xk+1 = φ(xk+1, xk), (1) with φ : Rn × Rn 7→Rn and φ(x, x) = f(x). The standard Picard–Lindel¨of iteration, for example, is given by φ(x, y) = f(y). Convergence is, by definition, achieved if ∥xk+1 −xk∥< ε for a predefined threshold ε. This procedure can be used to solve differential-algebraic equations as well. For a more detailed overview on waveform relaxation we refer to [6, 10]. We assume that the splitting φ is Lipschitz continuous, i.e. there exist constants µ ≥0 and η ≥0 such that ∥φ(x, y) −φ(˜x, ˜y)∥≤µ∥x −˜x∥+ η∥y −˜y∥. (2) Let ¯x be the exact solution of the differential equation and define Ek to be the error of the k-th iterate, that is Ek = xk −¯x. (3) It is well known that the iteration given by Eqn. 1 converges superlinearly (evident in Proposition 2.1) to the exact solution and that the error is bounded. Convergence results and error bounds for waveform relaxation have previously been derived in [10, 11, 13, 14]. For the purpose of this paper the following version of the convergence result will be useful. Proposition 2.1. Assuming that the splitting φ satisfies the Lipschitz condition, the norm of the error ∥Ek∥on the interval [0, T] is bounded as follows ∥Ek∥≤CkηkT k k! ∥E0∥, (4) with C = eµT. Remark 2.2. In Eqn. 4 it is important to note that k! will eventually dominate the numera

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