On the decay rate of the singular values of bivariate functions

Abstract

© 2018 Society for Industrial and Applied Mathematics. In this work, we establish a new truncation error estimate of the singular value decomposition (SVD) for a class of Sobolev smooth bivariate functions κ∈L2(Ω,Hs(D)), s≥0, and κ ∈ L2(Ω,Hs(D)) with D ⊂ Rd, where Hs(D):= Ws,2(D) and Hs(D):= {v ∈ L2(D): (−∆)s/2v ∈ L2(D)} with −∆ being the negative Laplacian on D coupled with specific boundary conditions. To be precise, we show the order O(M−s/d) for the truncation error of the SVD series expansion after the Mth term. This is achieved by deriving the sharp decay rate O(n−1−2s/d) for the square of the nth largest singular value of the associated integral operator, which improves on known results in the literature. We then use this error estimate to analyze an algorithm for solving a class of elliptic PDEs with random coefficient in the multiquery context, which employs the Karhunen–Loève approximation of the stochastic diffusion coefficient to truncate the model.