Computational complexity of smooth differential equations

Abstract

The computational complexity of the solution h to the ordinary differential equation h(0) = 0, h′ (t) = g(t, h(t)) under various assumptions on the function g has been investigated in hope of understanding the intrinsic hardness of solving the equation numerically. Kawamura showed in 2010 that the solution h can be PSPACE-hard even if g is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of g and obtain the following results: the solution h can still be PSPACE-hard if g is assumed to be of class C 1; for each k ≥ 2, the solution h can be hard for the counting hierarchy if g is of class C k . © 2012 Springer-Verlag.