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.

Summary.We consider minimal-error algorithms for solving systems ofWe show how to increase the order of an algorithm by one, using additionally integrals of f. We define the Taylor-integral algorithm which has the error of order n -~r+l) which is minimal among all algorithms which use n linear or nonlinear smooth functionals of f, in the class of bounded functions f with bounded partial derivatives up to order r.We show that the Taylor algorithm has the error of order n r which is minimal among all algorithms which use n evaluations of f and/or its partial derivatives.

Quantum algorithms and complexity have recently been studied not only for discrete, but also for some numerical problems. Most attention has been paid so far to the integration and approximation problems, for which a speed-up is shown in many important cases by quantum computers with respect to deterministic and randomized algorithms on a classical computer. In this paper, we deal with the randomized and quantum complexity of initial-value problems. For this nonlinear problem, we show that both randomized and quantum algorithms yield a speed-up over deterministic algorithms. Upper bounds on the complexity in the randomized and quantum settings are shown by constructing algorithms with a suitable cost, where the construction is based on integral information. Lower bounds result from the respective bounds for the integration problem. r 2004 Elsevier Inc. All rights reserved.

We study the problem, initiated by Kacewicz [Randomized and quantum algorithms yield a speed-up for initial-value problems, J. Complexity 20 (2004) 821-834; see also http://arXiv.org/abs/quant-ph/ 0311148], of finding randomized and quantum complexity of initial-value problems. We showed in Kacewicz (2004) that a speed-up in both settings over the worst-case deterministic complexity is possible. In the present paper we prove, by defining new algorithms, that further improvement in upper bounds on the randomized and quantum complexity can be achieved. In the Hölder class of righthand side functions with r continuous bounded partial derivatives, with rth derivative being a Hölder function with exponent , the ε-complexity is shown to be O (1/ε) 1/(r+ +1/3) in the randomized setting, and O (1/ε) 1/(r+ +1/2) on a quantum computer (up to logarithmic factors). This is an improvement for the general problem over the results from Kacewicz (2004). The gap still remaining between upper and lower bounds on the complexity is further discussed for a special problem. We consider scalar autonomous problems, with the aim of computing the solution at the end point of the interval of integration. For this problem, we fill up the gap by establishing (essentially) matching upper and lower complexity bounds. We show that the complexity in this case is(1/ε) 1/(r+ +1/2) in the randomized setting, and (1/ε) 1/(r+ +1) in the quantum setting (again up to logarithmic factors). Hence, this problem is essentially as hard as the integration problem.

Restricted complexity estimation is a major topic in control-oriented identification. Conditional algorithms are used to identify linear finite-dimensional models of complex systems, the aim being to minimize the worst-case identification error. High computational complexity of optimal solutions suggests employing suboptimal estimation algorithms. The paper studies different classes of conditional estimators and provides results that assess the reliability level of suboptimal algorithms