Sparse Adaptive Approximation of High Dimensional Parametric Initial Value Problems

Abstract

We consider non-linear systems of ordinary differential equations (ODEs) on a state space $\mathcal{S}$. We consider the general setting when $\mathcal{S}$ is a Banach space over ℝ or ℂ. We assume the right-hand side depends in an affine fashion on a sequence y=(y _j ) _j≥1 of possibly countably many parameters, normalized such that |y _j |≤1. Under suitable analyticity assumptions on the ODEs, we prove that the parametric solution $\{X(t;y): 0\leq t\leq T\}\subset\mathcal{S}$ 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 on $[0,T]\times \mathcal{S}$. While such affine parameter dependence of the ODE seems quite special it is of great importance for applications as it arises, among others, in mass–action models in computational biology (see, e.g., Horn and Jackson in Arch. Ration. Mech. Anal. 47:81–116, 1972) and in stoichiometry with uncertain reaction rate constants. Using our analytic regularity result, we prove summability theorems for coefficient sequences of generalized polynomial chaos (gpc) expansions of the parametric solutions {X(⋅;y)} _y∈U with respect to tensor product orthogonal polynomial bases of L ²(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. Computational aspects of such approximations will be addressed in Hansen et al. (Proceedings of the 5th International Conference on High Performance Scientific Computing, 2013).