Communications in Applied Mathematics and Computational Science

Theoretically optimal inexact spectral deferred correction methods

Martin Weiser and Sunayana Ghosh

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Abstract

In several initial value problems with particularly expensive right-hand side evaluation or implicit step computation, there is a tradeoff between accuracy and computational effort. We consider inexact spectral deferred correction (SDC) methods for solving such initial value problems. SDC methods are interpreted as fixed-point iterations and, due to their corrective iterative nature, allow one to exploit the accuracy-work tradeoff for a reduction of the total computational effort. First we derive error models bounding the total error in terms of the evaluation errors. Then we define work models describing the computational effort in terms of the evaluation accuracy. Combining both, a theoretically optimal local tolerance selection is worked out by minimizing the total work subject to achieving the requested tolerance. The properties of optimal local tolerances and the predicted efficiency gain compared to simpler heuristics, and reasonable practical performance, are illustrated with simple numerical examples.

Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 13, Number 1 (2018), 53-86.

Dates
Received: 14 February 2017
Revised: 30 October 2017
Accepted: 30 October 2017
First available in Project Euclid: 28 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.camcos/1522202439

Digital Object Identifier
doi:10.2140/camcos.2018.13.53

Mathematical Reviews number (MathSciNet)
MR3778320

Zentralblatt MATH identifier
06864866

Subjects
Primary: 65L05: Initial value problems 65L20: Stability and convergence of numerical methods 65L70: Error bounds 65M70: Spectral, collocation and related methods

Keywords
spectral deferred corrections initial value problems error propagation adaptive control of tolerances inexact work models accuracy models

Citation

Weiser, Martin; Ghosh, Sunayana. Theoretically optimal inexact spectral deferred correction methods. Commun. Appl. Math. Comput. Sci. 13 (2018), no. 1, 53--86. doi:10.2140/camcos.2018.13.53. https://projecteuclid.org/euclid.camcos/1522202439


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