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

Julio E Castrillón-Candás1, Mark Kon1

  • 1Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston MA 02215.

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Summary

This study introduces uncertainty quantification and numerical analysis for efficient stochastic Newton iterate evaluation. The novel sparse grid method significantly accelerates computations for power flow problems, outperforming Monte Carlo simulations.

Keywords:
Approximation TheoryComplex AnalysisNewton-Kantorovich TheoremNon-linear Stochastic NewtworksPower FlowSparse GridsUncertainty Quantification

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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.