Complexity of initial-value problems for ordinary differential equations of order k

• Authors: Marek Szczesny
• Venue: Journal of Complexity
• Year: 2006

We study the worst-case ε-complexity of nonlinear initial-value problems u(k)(x)=g (x, u (x), u'(x),..., u(q)(x)), x ∈ [a, b], 0 ≤qk, with given initial conditions. We assume that function g has r(r ≥ 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 Θ ((1/ε)1/r), which is independent of k and q. By defining an algorithm using integral information, we show that the complexity is O((1/ε)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 Θ((1/ε)1/r) to O((1/ε)1/(r+k)). We also give a lower bound on the ε-complexity for linear information. We show that the complexity is Ω((1/ε)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.