The randomized complexity of initial value problems

We study the complexity of randomized solution of initial value problems for systems of ordinary differential equations (ODE). The input data are assumed to be γγ-smooth (γ=r+ργ=r+ρ: the r  th derivatives satisfy a ρρ-Hölder condition). Recently, the following almost sharp estimate of the order of the n  th minimal error was given by Kacewicz [Almost optimal solution of initial-value problems by randomized and quantum algorithms, J. Complexity 22 (2006) 676–690, see also 〈〈http://arXiv.org/abs/quant-ph/0510045〉〉]:c1n-γ-1/2⩽enran⩽c2(ε)n-γ-1/2+ε,with an arbitrary ε>0ε>0. We present a Taylor Monte Carlo method and show that it has error rate n-γ-1/2n-γ-1/2, this way establishing the exact order of the randomized nth minimal error.