Approximation of bi-variate functions: singular value decomposition versus sparse grids

Abstract

We compare the cost complexities of two approximation schemes for functions $fin H^p(Omega_1times Omega_2)$ which live on the product domain $Omega_1timesOmega_2$ of sufficiently smooth domains $Omega_1subsetmathbb{R}^{n_1}$ and $Omega_2subsetmathbb{R}^{n_2}$, namely the singular value / Karhunen-L`oeve decomposition and the sparse grid representation. Here we assume that suitable finite element methods with associated {em fixed} order $r$ of accuracy are given on the domains $Omega_1$ and $Omega_2$. Then, the sparse grid approximation essentially needs only $mathcal{O}(varepsilon^{-q})$ with $q=frac{max{n_1,n_2}}{r}$ unknowns to reach a prescribed accuracy $varepsilon$ provided that the smoothness of $f$ satisfies $pge rfrac{n_1+n_2}{max{n_1,n_2}}$, which is an almost optimal rate. The singular value decomposition produces this rate only if $f$ is analytical since otherwise the decay of the singular values is not fast enough. If $p>rfrac{n_1+n_2}{max{n_1,n_2}}$, then the sparse grid approach gives essentially the rate $mathcal{O}(varepsilon^{-q})$ with $q=frac{n_1+n_2}{p}$ while, for the singular value decomposition, we can only prove the rate $mathcal{O}(varepsilon^{-q})$ with $q=frac{2min{r,p}min{n_1,n_2}+2pmax{n_1,n_2}}{(2p-min{n_1,n_2})min{r,p}}$. We derive the resulting complexities, compare the two approaches and present numerical results which demonstrate that these rates are also achieved in numerical practice. $p frac{n_1+n_2}{max{n_1,n_2}}$, then the sparse grid approach gives essentially the rate $mathcal{O} (varepsilon^{-q})$ with $q=frac{n_1+n_2}{p}$ while, for the singular value decomposition, we can only prove the rate $mathcal{O}(varepsilon^{-q})$ with $q=frac{2min{r,p}min{n_1,n_2} +2pmax{n_1,n_2}}{(2p-min{n_1,n_2})min{r,p}}$. We derive the resulting complexities, compare the two approaches and present numerical results which demonstrate that these rates are also achieved in numerical practi