# Analytic regularity and stochastic collocation of high dimensional Newton iterates

## TL;DR

This paper develops a complex analytic regularity framework for high-dimensional stochastic Newton iterates, enabling efficient computation of stochastic moments using sparse grids, with applications to power flow problems.

## Contribution

It introduces a new regularity theory for stochastic Newton methods and demonstrates their efficient computation via sparse grids, with proven convergence rates.

## Key findings

- Convergence rates are subexponential or algebraic.
- Sparse grids effectively compute low probability events.
- Numerical experiments confirm theoretical predictions.

## Abstract

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 stochastic moments. 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 39 bus New England power system model with large stochastic loads are consistent with the theoretical convergence rates.