Analysis of tensor approximation schemes for continuous functions

In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate dimension weight

Sparse finite elements for elliptic problems with stochastic loading

Summary.: We formulate elliptic boundary value problems with stochastic loading in a bounded domain D⊂ℝ d . We show well-posedness of the problem in stochastic Sobolev spaces and we derive a deterministic elliptic PDE in D×D for the spatial correlation of the random solution. We show well-posedness and regularity results for this PDE in a scale of weighted Sobolev spaces with mixed highest order derivatives. Discretization with sparse tensor products of any hierarchic finite element (FE) spaces in D yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data. Multilevel preconditioning in D×D allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equatio

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

We compare the cost complexities of two approximation schemes for functions f∈Hp(Ω1×Ω2) which live on the product domain Ω1×Ω2 of sufficiently smooth domains Ω1⊂ℝn1 and Ω2⊂ℝn2, namely the singular value/Karhunen-Lòeve 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 Ω1 and Ω2. Then, the sparse grid approximation essentially needs only (ε−q), with q=max{n1,n2}/r, unknowns to reach a prescribed accuracy ε, provided that the smoothness of f satisfies p≥r((n1+n2)/max{n1,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 ((n1+n2)/max{n1,n2}), then the sparse grid approach gives essentially the rate (ε−q) with q=(n1+n2)/p, while, for the singular value decomposition, we can only prove the rate (ε−q) with q=(2 min{r,p}min{n1,n2} +2p max{n1,n2}){(2p−min{n1,n2}) 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 practic

Sparse finite elements for elliptic problems with stochastic loading

ISSN:0029-599XISSN:0945-324

An adaptive multiscale finite element method

This work is devoted to an adaptive multiscale finite element method (MsFEM) for solving elliptic problems with rapidly oscillating coeefficients. Starting from a general version of the MsFEM with oversampling, we de- rive an a posteriori estimate for the H1-error between the exact solution of the problem and a corresponding MsFEM approximation. Our esti- mate holds without any assumptions on scale separation or on the type of the heterogeneity. The estimator splits into different contributions which account for the coarse grid error, the fine grid error and the oversampling error. Based on the error estimate we construct an adaptive algorithm that is validated in numerical experiments