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 $ \kappa {\,\in\,} L^2(\Omega,H^s(D))$, $s{\,\geq\,} 0$, and $\kappa\in L^2(\Omega,\dot{H}^s(D))$ with $D \subset\mathbb{R}^d$, where $H^s(D):=W^{s,2}(D)$ and $\dot H^s(D):=\{v\in L^2(D): (-\Delta)^{s/2}v\in L^2(D)\}$ with $-\Delta$ being the negative Laplacian on $D$ coupled with specific boundary conditions. To be precise, we show the order $\mathcal{O}(M^{-s/d})$ for the truncation error of the SVD series expansion after the $M$th term. This is achieved by deriving the sharp decay rate $\mathcal{O}(n^{-1-{2s}/{d}})$ for the square of the $n$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 multiquery context, which employs the Karhunen--Loève approximation of the stochastic diffusion coefficient to truncate the model.