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Analytic regularity and stochastic collocation of high-dimensional Newton iterates. (English) Zbl 1462.65054
Summary: 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 to 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 with the Monte Carlo method, our approach is at least $$10^{11}$$ times faster for the same accuracy.
##### MSC:
 65H10 Numerical computation of solutions to systems of equations 60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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