On the decay rate of the singular values of bivariate functions

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\,} L2(\Omega,Hs(D))$, $s{\,\geq\,} 0$, and $\kappa\in L2(\Omega,\dot{H}s(D))$ with $D \subset\mathbb{R}d$, where $Hs(D):=W{s,2}(D)$ and $\dot Hs(D):=\{v\in L2(D): (-\Delta){s/2}v\in L2(D)\}$ 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.