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On the Complexity of Parabolic Initial Value Problems with Variable Drift Petras, Knut Ritter, Klaus Partial differential equations parabolic problems Smolyak method optimal methods 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Knut Petras and Klaus Ritter 2005 Is Part Of Dagstuhl Seminar Proceedings, Volume 4401, Algorithms and Complexity for Continuous Problems (2005) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.04401.10 urn:nbn:de:0030-drops-1495 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04401.10 eng https://creativecommons.org/licenses/by/4.0/legalcode