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URN: urn:nbn:de:0030-drops-1518
URL: http://drops.dagstuhl.de/opus/volltexte/2005/151/
Ritter, Klaus ; Müller-Gronbach, Thomas

Lower Bounds and Non-Uniform Time Discretization for Approximation of Stochastic Heat Equations

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Abstract

We study algorithms for approximation of the mild solution of stochastic heat equations on the spatial domain ]0,1[^d. The error of an algorithm is defined in L_2-sense. We derive lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W. For equations with additive noise we derive matching upper bounds and we construct asymptotically optimal algorithms. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise. In the latter case the use of non-uniform time discretizations is crucial.

BibTeX - Entry

@InProceedings{ritter_et_al:DSP:2005:151,
  author =	{Klaus Ritter and Thomas M{\"u}ller-Gronbach},
  title =	{Lower Bounds and Non-Uniform Time Discretization for Approximation of Stochastic Heat Equations},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  year =	{2005},
  editor =	{Thomas M{\"u}ller-Gronbach and Erich Novak and Knut Petras and Joseph F. Traub},
  number =	{04401},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2005/151},
  annote =	{Keywords: Stochastic heat equation , Non-uniform time discretization , minimal errors , upper and lower bounds}
}

Keywords: Stochastic heat equation , Non-uniform time discretization , minimal errors , upper and lower bounds
Seminar: 04401 - Algorithms and Complexity for Continuous Problems
Issue date: 2005
Date of publication: 19.04.2005

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