Error Estimates for the ANOVA Method with Polynomial Chaos Interpolation: Tensor Product Functions

Authors:
Zhongqiang Zhang;Minseok Choi;George Em Karniadakis
Affiliations:
zhongqiang-zhang@brown.edu and minseok_choi@brown.edu and gk@dam.brown.edu;-;-
Venue:
SIAM Journal on Scientific Computing
Year:
2012

Citing 13
Cited 1

When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?

Journal of Complexity
Chebyshev--Legendre Spectral Viscosity Method for Nonlinear Conservation Laws

SIAM Journal on Numerical Analysis
Strong tractability of multivariate integration using quasi-Monte Carlo algorithms

Mathematics of Computation
The effective dimension and quasi-Monte Carlo integration

Journal of Complexity
Sufficient conditions for fast quasi-Monte Carlo convergence

Journal of Complexity
Liberating the weights

Journal of Complexity
Why Are High-Dimensional Finance Problems Often of Low Effective Dimension?

SIAM Journal on Scientific Computing
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations

Journal of Computational Physics
Efficient Weighted Lattice Rules with Applications to Finance

SIAM Journal on Scientific Computing
The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications

Journal of Computational Physics
Multi-element probabilistic collocation method in high dimensions

Journal of Computational Physics
Dimension-wise integration of high-dimensional functions with applications to finance

Journal of Complexity
Adaptive ANOVA decomposition of stochastic incompressible and compressible flows

Journal of Computational Physics

Adaptive ANOVA decomposition of stochastic incompressible and compressible flows

Journal of Computational Physics

Quantified Score

Hi-index	0.01

Visualization

Abstract

We focus on the analysis of variance (ANOVA) method for high dimensional function approximation using Jacobi polynomial chaos to represent the terms of the expansion. First, we develop a weight theory inspired by quasi-Monte Carlo theory to identify which functions have low effective dimension using the ANOVA expansion in different norms. We then present estimates for the truncation error in the ANOVA expansion and for the interpolation error using multielement polynomial chaos in the weighted Korobov spaces over the unit hypercube. We consider both the standard ANOVA expansion using the Lebesgue measure and the anchored ANOVA expansion using the Dirac measure. The optimality of different sets of anchor points is also examined through numerical examples.