On the complexity of parabolic initial-value problems with variable drift

  • Authors:

  • Knut Petras;Klaus Ritter

  • Affiliations:

  • Institut für Angewandte Mathematik, Technische Universität Braunschweig, Pockelsstrasse 14, 38106 Braunschweig, Germany;Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany

  • Venue:
  • Journal of Complexity - Special issue: Algorithms and complexity for continuous problems Schloss Dagstuhl, Germany, September 2004
  • Year:
  • 2006

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Abstract

We study the intrinsic difficulty of solving linear parabolic initial-value problems numerically at a single point. We present a worst-case analysis for deterministic as well as for randomized (or Monte Carlo) algorithms, assuming that the drift coefficients and the potential vary in given function spaces. We use fundamental solutions (parametrix method) for equations with unbounded coefficients to relate the initial-value problem to multivariate integration and weighted approximation problems. Hereby we derive lower and upper bounds for the minimal errors. The upper bounds are achieved by algorithms that use Smolyak formulas and, in the randomized case, variance reduction. We apply our general results to equations with coefficients from Holder classes, and here, in many cases, the upper and lower bounds almost coincide and our algorithms are almost optimal.