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

Authors Klaus Ritter, Thomas Müller-Gronbach

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DagSemProc.04401.6.pdf
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Keywords
  • Stochastic heat equation
  • Non-uniform time discretization
  • minimal errors
  • upper and lower bounds

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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.

Cite As Get BibTex

Klaus Ritter and Thomas Müller-Gronbach. Lower Bounds and Non-Uniform Time Discretization for Approximation of Stochastic Heat Equations. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 4401, pp. 1-37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005) https://doi.org/10.4230/DagSemProc.04401.6

Author Details

Klaus Ritter
Thomas Müller-Gronbach
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