Analytic regularity and stochastic collocation of high-dimensional Newton iterates.

Area of Science:

• Numerical Analysis
• Uncertainty Quantification
• Computational Mathematics

Background:

• Stochastic high dimensional problems present computational challenges.
• Efficient evaluation of Newton iterates is crucial for solving these problems.
• Existing methods like Monte Carlo can be computationally expensive.

Purpose of the Study:

• To develop an efficient method for evaluating stochastic high dimensional Newton iterates.
• To apply uncertainty quantification and numerical analysis concepts.
• To accelerate computations in problems like power flow analysis.

Main Methods:

• Development of complex analytic regularity theory for solutions with respect to random variables.
• Justification and application of sparse grids for statistical measure computation.
• Derivation of convergence rates with respect to random perturbations.

Main Results:

• Subexponential or algebraic convergence rates are achieved.
• Sparse grids are effective for computing low probability events with high confidence.
• Numerical experiments on a 39 bus power system demonstrate consistency with theoretical rates.

Conclusions:

• The proposed method offers significant speedups (at least 10^11 times faster) compared to Monte Carlo for the power flow problem.
• The approach provides accurate computation of statistical measures for stochastic systems.
• This work enables more efficient analysis of complex systems with uncertainties.