Complexity of parametric initial value problems in Banach spaces

Abstract

We study the approximate solution of initial value problems for parameter dependent finite or infinite systems of scalar ordinary differential equations (ODEs). Both the deterministic and the randomized setting is considered, with input data from various smoothness classes. We study deterministic and Monte Carlo multilevel algorithms and derive convergence rates. Moreover, we prove their optimality by showing matching (in some limit cases up to logarithmic factors) lower bounds and settle this way the complexity. Comparisons between the deterministic and randomized setting are given, as well.