Complexity of initial-value problems for ordinary differential equations of order k
We study the worst-case E-complexity of nonlinear initial-value problems u((k)) (x)=g (x, u(x), u'(x),..., u((q)) (x)), x is an element of [a, b], 0  = 1) continuous bounded partial derivatives. We consider two types of information about g: standard information defined by values of g or its partial derivatives, and linear information defined by the values of linear functionals on g. For standard information, we show that the worst-case complexity is Theta ((1/epsilon) (1/r)), which is independent of k and q. By defining an algorithm using integral information, we show that the complexity is O((1/epsilon)(1/(r+k-q))) if linear information is used. Hence, linear information is more powerful than standard information. For q 0 for instance, the complexity decreases from Theta ((1/epsilon)(1/r)) to O ((1/epsilon)(1/(r+k))). We also give a lower bound on the E-complexity for linear information. We show that the complexity is Omega ((1/epsilon)(1/(r+k))), which means that upper and lower bounds match for q = 0. The gap for the remaining values of q is an open problem. (c) 2006 Elsevier Inc. All rights reserved.
standard information linear information worst-case complexity integral information kth order initial-value problems