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

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.