The use of integral information in the solution of a two-point boundary value problem

Abstract

We study the worst-case ε-complexity of a two-point boundary value problem $u^{\prime\prime}(x)=f(x)u(x)$, $x \in [0,T]$, $u(0)=c$, $u^{\prime}(T)=0$, where $c,T \in \mathbb{R}$ ($c \neq 0$, $T \gt 0$) and $f$ is a nonnegative function with $r$ ($r\geq 0$) continuous bounded derivatives. We prove an upper bound on the complexity for linear information showing that a speed-up by two orders of magnitude can be obtained compared to standard information. We define an algorithm based on integral information and analyze its error, which provides an upper bound on the $\varepsilon$-complexity.