Analytic regularity and stochastic collocation of high-dimensional Newton iterates

In this paper we introduce concepts from uncertainty quantification (UQ) and numerical analysis for the efficient evaluation of stochastic high dimensional Newton iterates. In particular, we develop complex analytic regularity theory of the solution with respect to the random variables. This justifies the application of sparse grids for the computation of statistical measures. Convergence rates are derived and are shown to be subexponential or algebraic with respect to the number of realizations of random perturbations. Due the accuracy of the method, sparse grids are well suited for computing low probability events with high confidence. We apply our method to the power flow problem. Numerical experiments on the non-trivial 39 bus New England power system model with large stochastic loads are consistent with the theoretical convergence rates. Moreover, compared to the Monte Carlo method our approach is at least 10¹¹ times faster for the same accuracy.