Complexity of certain nonlinear two-point BVPs with Neumann boundary conditions

We study the solution of two-point boundary-value problems for second order ODEs with boundary conditions imposed on the first derivative of the solution. The right-hand side function g is assumed to be r times (r≥1) continuously differentiable with the rth derivative being a Hölder function with exponent ϱ∈(0,1]. The boundary conditions are defined through a continuously differentiable function f. We define an algorithm for solving the problem with error of order m−(r+ϱ) and cost of order mlogm evaluations of g and f and arithmetic operations, where m∈N. We prove that this algorithm is optimal up to the logarithmic factor in the cost. This yields that the worst-case ε-complexity of the problem (i.e., the minimal cost of solving the problem with the worst-case error at most ε>0) is essentially Θ((1/ε)1/(r+ϱ)), up to a log1/ε factor in the upper bound. The same bounds hold for r+ϱ≥2 even if we additionally assume convexity of g. For r=1, ϱ∈(0,1] and convex functions g, the information ε-complexity is shown to be Θ((1/ε)1/2).