On the Complexity of Parabolic Initial Value Problems with Variable Drift

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 H

• Type: Conference
• Year: 2004
• Where: DAGSTUHL
• Authors: Knut Petras, Klaus Ritter
• Keywords: Coefficients, DAGSTUHL 2004, DAGSTUHL 2007, Initial Value Problem, Upper Bounds