Complexity of Initial Value Problems in Banach Spaces

We study the complexity of initial value problems for Banach space valued ordinary differential equations in the randomized setting. The right-hand side is assumed to be $r$-smooth, the $r$-th derivatives being $\rho$-Hölder continuous. We develop and analyze a randomized algorithm. Furthermore, we prove lower bounds and thus obtain complexity estimates. They are related to the type of the underlying Banach space. We also consider the deterministic setting. The results extend previous ones for the finite dimensional case from [2, 9, 10].