Analytic regularity and best N-term approximation of high dimensional parametric initial value problems

Abstract

We consider nonlinear systems of ordinary differential equations (ODEs) on a Banach state space S over R or C, where the right hand side depends affinely linear on a parameter vector y = (yj)j_1, normalized such that |yj | _ 1. Under suitable analyticity assumptions on the ODEs, we prove that the solution {X(t; y) : 0 _ t _ T} of the corresponding IVP depends holomorphically on the parameter vector y, as a mapping from the infinite- dimensional parameter domain U = (−1, 1)N into a suitable function space in [0, T] × S. Such affined parameter dependence of the ODE arises, among others, in mass action models in computational biology (see, e.g. [11]). Using this regularity result, we prove summability theorems for coefficient sequences of polynomial chaos (pc) expansions of the parametric solutions {X(•; y)}y2U with respect to tensor product orthogonal polynomial bases of L2(U). We give sufficient conditions on the ODEs for N-term truncations of these expansions to converge on the entire parameter space with efficiency (i.e. accuracy versus complexity) being independent of the number of parameters viz. the dimension of the parameter space U.