Information-based complexity
The real number model in numerical analysis
Journal of Complexity
Explicit cost bounds of algorithms for multivariate tensor product problems
Journal of Complexity
Complexity of weighted approximation over R
Journal of Approximation Theory
A new algorithm and worst case complexity for Feynman-Kac path integration
Journal of Computational Physics
Applicability of Smolyak's algorithms to certain banach spaces of multivariate functions
Journal of Complexity
Worst case complexity of multivariate Feynman-Kac path integration
Journal of Complexity
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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 Hölder classes, and here, in many cases, the upper and lower bounds almost coincide and our algorithms are almost optimal.