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On the Complexity of Parabolic Initial Value Problems with Variable Drift

Authors Knut Petras, Klaus Ritter

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DagSemProc.04401.10.pdf
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Keywords
  • Partial differential equations
  • parabolic problems
  • Smolyak method
  • optimal methods

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Abstract

We consider linear parabolic initial value 
problems of second order in several dimensions. 
The initial condition is supposed to be fixed 
and we investigate the comutational complexity if 
the coefficients of the parabolic equations
may vary in certain function spaces. Using the 
parametrix method (or Neumann series), we prove 
that lower bounds for the error of numerical 
methods are related to lower bounds for 
integration problems. On the other hand, 
approximating the Neumann series with Smolyak's 
method, we show that the problem is not much 
harder than a certain approximation problem. For 
Hölder classes on compact sets, e.g., lower and 
upper bounds are close together, such that we have 
an almost optimal method.

Cite As Get BibTex

Knut Petras and Klaus Ritter. On the Complexity of Parabolic Initial Value Problems with Variable Drift. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 4401, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005) https://doi.org/10.4230/DagSemProc.04401.10

Author Details

Knut Petras
Klaus Ritter
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