Talk 9731 data/Fall

Applied Math And Analysis Seminar
Monday, December 5, 2016, 4:30pm, 119 Physics
Peng Chen (University of Texas, Austin)
Sparse Quadrature for High-Dimensional Integration with Gaussian Measure: Breaking the Curse of Dimensionality

High-dimensional integration problems, which suffer from curse of dimensionality, are faced in many computational applications, such as uncertainty quantification, Bayesian inverse problems, PDE-constrained stochastic optimization, computational finance, etc. Recent work has made great progress in theories and algorithms for treating uniform measure. However, high-dimensional integration with Gaussian measure, commonly used in these fields, is less studied. In this talk I will present a sparse quadrature that breaks the curse of dimensionality for high/infinite-dimensional integration with Gaussian measure. Both a priori and a goal-oriented adaptive construction algorithms for the sparse quadrature are proposed by tensorization of univariate quadratures in a carefully selected (admissible) index set. Several univariate quadrature rules, including Gauss--Hermite, transformed Gauss--Kronrod--Patterson, and Genz--Keister are investigated. The best-$N$ term algebraic convergence rate $N^{-s}$is obtained under certain assumption on the regularity of the parametric map with respect to the Gaussian distributed parameters. The rate $s$ is shown to be dependent only on a sparsity parameter that controls the regularity and independent of the number of active parameter dimensions. Examples of nonlinear parametric functions and parametric partial differential equations (PDE) are provided to illustrate the regularity assumption. Finally, I will present numerical experiments on the integration of parametric function, parametric PDE, and parametric Bayesian inversion for the demonstration of the dimension-independent convergence of the sparse quadrature errors. The convergence is shown to be much faster than that of Monte Carlo quadrature errors for the test problems with sufficient sparsity.

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