We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. We define for this purpose a sequence of new algorithms, whose error/cost properties improve from step to step. This leads to new upper complexity bounds, which differ from known lower bounds only by an arbitrarily small positive parameter in the exponent, and a logarithmic factor. In both randomized and quantum settings, initial-value problems turn out to be essentially as difficult as scalar integration.
We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These algorithms yield new upper complexity bounds, which differ from known lower bounds by only an arbitrarily small positive parameter in the exponent, and a logarithmic factor. In both the randomized and quantum settings, initial-value problems turn out to be essentially as difficult as scalar integration.
The complexity of initial-value problems is well studied for systems of equations of first order. In this paper, we study the
ε-complexity for initial-value problems for scalar equations of higher order. We consider two models of computation, the randomized model and the quantum model. We construct almost optimal algorithms adjusted to scalar equations of higher order, without passing to systems of first order equations. The analysis of these algorithms allows us to establish upper complexity bounds. We also show (almost) matching lower complexity bounds. The
ε-complexity in the randomized and quantum setting depends on the regularity of the right-hand side function, but is independent of the order of equation. Comparing the obtained bounds with results known in the deterministic case, we see that randomized algorithms give us a speed-up by
1/2, and quantum algorithms by
1 in the exponent. Hence, the speed-up does not depend on the order of equation, and is the same as for the systems of equations of first order. We also include results of some numerical experiments which confirm theoretical results.