DagSemProc.04401.10.pdf
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On the Complexity of Parabolic Initial Value Problems with Variable Drift
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Dagstuhl Seminar Proceedings, Volume 4401
Series: Dagstuhl Seminar Proceedings (DagSemProc) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2005-04-19
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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
https://doi.org/10.4230/DagSemProc.04401.10
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.
Keywords
- Partial differential equations
- parabolic problems
- Smolyak method
- optimal methods
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