TY - JOUR
AU1 - Griebel, Michael
AU2 - Harbrecht, Helmut
AB - We compare the cost complexities of two approximation schemes for functions fHp(12) which live on the product domain 12 of sufficiently smooth domains 1n1 and 2n2, namely the singular value/KarhunenLeve decomposition and the sparse grid representation. Here, we assume that suitable finite element methods with associated fixed order r of accuracy are given on the domains 1 and 2. Then, the sparse grid approximation essentially needs only (q), with qmaxn1,n2/r, unknowns to reach a prescribed accuracy , provided that the smoothness of f satisfies pr((n1n2)/maxn1,n2), 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<r ((n1n2)/maxn1,n2), then the sparse grid approach gives essentially the rate  (q) with q(n1n2)/p, while, for the singular value decomposition, we can only prove the rate (q) with q(2 minr,pminn1,n2 2p maxn1,n2)(2pminn1,n2) minr,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.
TI - Approximation of bi-variate functions: singular value decomposition versus sparse grids
JF - IMA Journal of Numerical Analysis
DO - 10.1093/imanum/drs047
DA - 2014-01-29
UR - https://www.deepdyve.com/lp/oxford-university-press/approximation-of-bi-variate-functions-singular-value-decomposition-KA4TdShKCd
SP - 28
EP - 54
VL - 34
IS - 1
DP - DeepDyve
ER -