On the decay rate of the singular values of bivariate functions

Abstract

In this work, we establish a new truncation error estimate of the singular value decomposition (SVD) for a class of Sobolev smooth bivariate functions <em>κ</em> ∈ <em>L²</em>(Ω, <em>H^s</em>(<em>D</em>)), <em>s</em> ≥ 0 and <em>κ</em> ∈ <em>L²</em>(Ω, <em>&#7714;^s</em>(<em>D</em>)) with <em>D</em> ⊂ <em>&#8477; ^d</em>, where <em>H^s</em>(<em>D</em>) := <em>W^s,2</em>(<em>D</em>) and <em>&#7714;^s</em>(<em>D</em>) := {<em>&nu;</em> ∈ <em>L²</em>(<em>D</em>) : (−∆)^s/2<em>&nu;</em> ∈ <em>L²</em>(<em>D</em>)} with −∆ being the negative Laplacian on <em>D</em> coupled with specific boundary conditions. To be precise, we show the order <em>O</em>(<em>M ^−s/d</em>) for the truncation error of the SVD series expansion after the <em>M</em> -th term. This is achieved by deriving the sharp decay rate <em>O</em>(<em>n^{−1−2s&frasl;d}</em>) for the square of the <em>n</em>-th 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 multi-query context, which employs the Karhunen-Loève approximation of the stochastic diffusion coefficient to truncate the model.