On the optimal robust solution of IVPs with noisy information

Abstract

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