On the optimal robust solution of IVPs with noisy information

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We investigate the optimal solution of systems of initial-value problems with smooth right-hand side functions f from a Holder class Fregr,?$F^{r,\varrho }_{\text {reg}}$, where r ? 0 is the number of continuous derivatives of f, and ? ? (0, 1] is the Holder exponent of rth partial derivatives. We consider algorithms that use n evaluations of f, the ith evaluation being corrupted by a noise ?i of deterministic or random nature. For ? ? 0, in the deterministic case the noise ?i is a bounded vector, ??i?≤?. In the random case, it is a vector-valued random variable bounded in average, (E(??i?q))1/q ≤ ?, q ? [1, + ?). We point out an algorithm whose Lp error (p ? [0, + ?]) is O(n ? (r + ?) + ?), independently of the noise distribution. We observe that the level n ? (r + ?) + ? cannot be improved in a class of information evaluations and algorithms. For ? > 0, and a certain model of ?-dependent cost, we establish optimal values of n(?) and ?(?) that should be used in order to get the error at most ? with minimal cost.