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

Abstract

We study the worst-case $\varepsilon$-complexity of nonlinear initial-value problems $u^{(k)}(x)=g\left(x,u(x),u^{\prime }(x),\dots ,u^{(q)}(x)\right)$, $x\in [a,b]$, $0⩽q<k$, 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 $\mathrm{\Theta }\left((1/\varepsilon {)}^{1/r}\right)$, which is independent of k and q. By defining an algorithm using integral information, we show that the complexity is $O\left((1/\varepsilon {)}^{1/(r+k-q)}\right)$ if linear information is used. Hence, linear information is more powerful than standard information. For $q=0$ for instance, the complexity decreases from $\mathrm{\Theta }\left((1/\varepsilon {)}^{1/r}\right)$ to $O\left((1/\varepsilon {)}^{1/(r+k)}\right)$. We also give a lower bound on the $\varepsilon$-complexity for linear information. We show that the complexity is $\mathrm{\Omega }\left((1/\varepsilon {)}^{1/(r+k)}\right)$, which means that upper and lower bounds match for $q=0$. The gap for the remaining values of q is an open problem.