Implementing Radau IIA Methods for Stiff Delay Differential Equations

Guglielmi, Nicola; Hairer, Ernst

doi:10.1007/s006070170013

Cited by 109 publications

(84 citation statements)

References 11 publications

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“…We include results for a new version [10] of RADAR5 (Guglielmi and Hairer [9]), and DDE SOLVER (Thompson and Shampine [20]). We have set all absolute tolerances and the relative tolerance to TOL with TOL=10 −6 ,10 −9 for all solvers.…”

Section: Resultsmentioning

confidence: 99%

“…Delay differential equations (DDEs) are a class of differential equations that have received considerable recent attention and been proven to model many real life problems, traditionally formulated as systems of ordinary differential equations (ODEs), more naturally and more accurately. Several DDE solvers have been implemented during the past twenty years, based on the extension or modification of traditional ODE techniques such as those based on Runge-Kutta or linear multi-step formulas ( [2], [4], [9], [11], [17], [20], [21]). The implementations of these solvers are usually based on adapting an existing initial value problem (IVP) solver.…”

Section: Introductionmentioning

confidence: 99%

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An efficient unified approach for the numerical solution of delay differential equations

ZivariPiran

Enright

2009

Numer Algor

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In this paper we propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a standard step-by-step approach, such as Runge-Kutta or linear multi-step methods, and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. Using this interpretation we develop an efficient technique to solve the resulting discontinuous IVP. We also give a more clear process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes.The new modular design for the resulting simulator we introduce, helps to accelerate the utilization of advances in the different components of an effective numerical method. Such components include the underlying discrete formula, the interpolant for dense output, the strategy for handling discontinuities and the iteration scheme for solving any implicit equations that arise.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An efficient unified approach for the numerical solution of delay differential equations

ZivariPiran

Enright

2009

Numer Algor

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“…1. The domain decomposition results in 32 ODEs solved implicitly using a solver that has been proven to be highly effective for ODE systems (Guglielmi and Hairer, 2001). The model simulation time is substantial for this study given that the EMO algorithms will have to evaluate thousands of simulations while automatically calibrating model parameters.…”

Section: Integrated Surface-subsurface Model Descriptionmentioning

confidence: 99%

How effective and efficient are multiobjective evolutionary algorithms at hydrologic model calibration?

Tang

Reed

Wagener

2006

Hydrol. Earth Syst. Sci.

177

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Abstract. This study provides a comprehensive assessment of state-of-the-art evolutionary multiobjective optimization (EMO) tools' relative effectiveness in calibrating hydrologic models. The relative computational efficiency, accuracy, and ease-of-use of the following EMO algorithms are tested: Epsilon Dominance Nondominated Sorted Genetic Algorithm-II (ε-NSGAII), the Multiobjective Shuffled Complex Evolution Metropolis algorithm (MOSCEM-UA), and the Strength Pareto Evolutionary Algorithm 2 (SPEA2). This study uses three test cases to compare the algorithms' performances: (1) a standardized test function suite from the computer science literature, (2) a benchmark hydrologic calibration test case for the Leaf River near Collins, Mississippi, and (3) a computationally intensive integrated surface-subsurface model application in the Shale Hills watershed in Pennsylvania. One challenge and contribution of this work is the development of a methodology for comprehensively comparing EMO algorithms that have different search operators and randomization techniques. Overall, SPEA2 attained competitive to superior results for most of the problems tested in this study. The primary strengths of the SPEA2 algorithm lie in its search reliability and its diversity preservation operator. The biggest challenge in maximizing the performance of SPEA2 lies in specifying an effective archive size without a priori knowledge of the Pareto set. In practice, this would require significant trial-and-error analysis, which is problematic for more complex, computationally intensive calibration applications. ε-NSGAII appears to be superior to MOSCEM-UA and competitive with SPEA2 for hydrologic model calibration. ε-NSGAII's primary strength lies in its ease-of-use due to its dynamic population sizing and archiving which lead to rapid convergence to very high quality solutions with minimal user input. MOSCEM-UA is best suited for hydrologic model calibration applications that have small parameter sets and small Correspondence to: P. Reed (preed@engr.psu.edu) model evaluation times. In general, it would be expected that MOSCEM-UA's performance would be met or exceeded by either SPEA2 or ε-NSGAII.

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“…This is the general form of a differential-algebraic delay equation. Codes that are written for such systems (like RADAR5 [GH01,GH08]) can therefore be applied to neutral state-dependent delay equations.…”

mentioning

confidence: 99%

Asymptotic Expansions for Regularized State-Dependent Neutral Delay Equations

Guglielmi¹,

Hairer²

2012

SIAM J. Math. Anal.

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Abstract. Singularly perturbed delay differential equations arising from the regularization of state dependent neutral delay equations are considered. Asymptotic expansions of their solutions are constructed and their limit for ε → 0 + is studied. Due to discontinuities in the derivative of the solution of the neutral delay equation and the presence of different time scales when crossing breaking points, new difficulties have to be managed. A two-dimensional dynamical system is presented which characterizes whether classical or weak solutions are approximated by the regularized problem. A new type of expansion (in powers of √ ε) turns out to be necessary for the study of the transition from weak to classical solutions. The techniques of this article can also be applied to the study of general singularly perturbed delay equations.

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Implementing Radau IIA Methods for Stiff Delay Differential Equations

Cited by 109 publications

References 11 publications

An efficient unified approach for the numerical solution of delay differential equations

An efficient unified approach for the numerical solution of delay differential equations

How effective and efficient are multiobjective evolutionary algorithms at hydrologic model calibration?

Asymptotic Expansions for Regularized State-Dependent Neutral Delay Equations

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