Sparse Approximation Algorithms for High Dimensional Parametric Initial Value Problems

We consider the efficient numerical approximation on nonlinear systems of initial value Ordinary Differential Equations (ODEs) on Banach state spaces $\mathcal{S}$ over $\mathbb{R}$ or $\mathbb{C}$. We assume the right hand side depends $in$ $affine$ $fashion$ on a vector $y =(y_j)_{j \geq 1}$ of possibly countably many parameters, normalized such that $|y_j| \leq 1$. Such affine parameter dependence of the ODE arises, among others, in mass action models in computational biology and in stochiometry with uncertain reaction rate constants. We review results from [19] on $N$-term approximation rates for the parametric solutions, i.e.summability theorems for coefficient sequences of generalized polynomial chaos (gpc) expansions of the parametric solutions {$X(\cdot ; y$)}$_{y \in U}$ with respect to tensorized 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 parameterspace with efficiency (i.e. accuracy versus complexity) being independent of the number of parameters viz. the dimension of the parameter space $U$. We investigate a heuristic adaptive approach for computing sparse, approximate representations of the {$X(t;y) : 0 \leq t \leq T$} $\subset S$. We increase efficiency by relating the accuracy of the adaptive initial value ODE solver to the estimated detail operator in the Smolyak formula. We also report tests which indicate that the proposed algorithms and the analyticity results of [19] hold for more general, nonaffine analytic dependence on parameters