The Waveform Relaxation Method for Time-Domain Analysis of Large Scale Integrated Circuits

Lelarasmee, E.; Ruehli, A.E.; Sangiovanni‐Vincentelli, Alberto

doi:10.1109/tcad.1982.1270004

Cited by 715 publications

(371 citation statements)

References 9 publications

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“…3 (a)-(f) compare the errors in the solutions from ode45 function in Matlab (with parameters of absolute error = 10 −6 and relative error = 10 −6 ) and Runge-Kutta 4th order method with two different time step sizes for the six ODE examples. For ODEs (1, 3, 5, 6), the step size is determined as 10 −7 , and as 10 − 6 for ODEs (2,4). The same analysis and step sizes are also used in the multi-step Adams-BashforthMoulton method and the Simpson's rule for the quadrature evaluations in Picard iterations, as they are all the same order with respect to time step size.…”

Section: Choice Of Step Sizementioning

confidence: 99%

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Hybrid dynamic iterations for the solution of initial value problems

Yu¹,

Srinivasan²

2011

Journal of Parallel and Distributed Computing

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Many scientific problems are posed as Ordinary Differential Equations (ODEs). A large subset of these are initial value problems, which are typically solved numerically. The solution starts by using a known state-space of the ODE system to determine the state at a subsequent point in time. This process is repeated several times. When the computational demand is high due to large state space, parallel computers can be used efficiently to reduce the time to solution. Conventional parallelization strategies distribute the state space of the problem amongst cores and distribute the task of computing for a single time step amongst the cores. They are not effective when the computational problems have fine granularity, for example, when the state space is relatively small and the computational effort arises largely from the long time span of the initial value problem. We propose a hybrid dynamic iterations method 1 which combines conventional sequential ODE solvers with dynamic iterations to parallelize the time domain. Empirical results demonstrate a factor of two to four improvement in performance of the hybrid dynamic iterations method over a conventional ODE solver on an 8 core processor. Compared to Picard iterations (also parallelized in the time domain), the proposed method shows better convergence and speedup results when high accuracy is required. Keywords: time parallelization, hybrid dynamic iterations, ODE solverEmail address: {yu,asriniva}@cs.fsu.edu (Yanan Yu and Ashok Srinivasan) 1 This paper is an extended version of a conference paper [1]. In this paper, we have considered an additional underlying ODE solver for the hybrid method, empirically evaluated with more ODE systems, and also evaluated the relative performance with low and high accuracy requirements.

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Section: Choice Of Step Sizementioning

confidence: 99%

“…Dynamic iterations [2,3], were well studied in the 1980s and 1990s. These methods start out with a guess for the solution over the entire time span, and in each iteration, they refine the approximate solution from the previous iteration.…”

Section: Introductionmentioning

confidence: 99%

Hybrid dynamic iterations for the solution of initial value problems

Yu¹,

Srinivasan²

2011

Journal of Parallel and Distributed Computing

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“…WR was first discussed in [153], and a global convergence proof was given for circuits with a grounded capacitor at each node, provided that some Lipschitz conditions hold and the time windows used are sufficiently small.…”

Section: T)mentioning

confidence: 99%

Modelling and Discretization of Circuit Problems

Günther

Feldmann

Maten

2005

Handbook of Numerical Analysis

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“…The waveform relaxation (WR) method is proposed by Learsmee and so forth [19]. The main properties of the method are flexibility and good parallelism, which is used to solve differential equations [20,21], fractional differential equations [22,23], delay differential equations [24,25], and differential-algebraic equations [26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Discrete Waveform Relaxation Method for Linear Fractional Delay Differential-Algebraic Equations

Liu¹,

Fu²,

Li³

2017

Discrete Dynamics in Nature and Society

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Fractional order delay differential-algebraic equations have the characteristics of time lag and memory and constraint limit. These yield some difficulties in the theoretical analysis and numerical computation. In this paper, we are devoted to solving them by the waveform relaxation method. The corresponding convergence results are obtained, and some numerical examples show the efficiency of the method.

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The Waveform Relaxation Method for Time-Domain Analysis of Large Scale Integrated Circuits

Cited by 715 publications

References 9 publications

Hybrid dynamic iterations for the solution of initial value problems

Hybrid dynamic iterations for the solution of initial value problems

Modelling and Discretization of Circuit Problems

Discrete Waveform Relaxation Method for Linear Fractional Delay Differential-Algebraic Equations

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