High-Dimensional Finite Elements for Elliptic Problems with Multiple Scales

Hoáng, Viêt Há; Schwab, Christoph

doi:10.1137/030601077

Cited by 92 publications

(103 citation statements)

References 17 publications

Supporting

Mentioning

100

Contrasting

Order By: Relevance

“…We mention only mathematical finance (pricing of derivative contracts on baskets of n assets under stochastic volatility models with d m − 1 ≥ 0 'hidden' volatility drivers for the mth risky asset with the case d m = 1 corresponding to deterministic volatility; see, e.g., [HMS05]), multiscale problems (elliptic homogenization problems with n separated length scales); see, e.g., [HS05], stochastic PDEs (the computation of n-point correlation functions for random solutions); see, e.g., [vPS06].…”

Section: Pde's On (High Dimensional) Product Domainsmentioning

confidence: 99%

Adaptive wavelet algorithms for elliptic PDE's on product domains

Schwab

Stevenson

2008

Math. Comp.

Self Cite

View full text Add to dashboard Cite

Abstract. With standard isotropic approximation by (piecewise) polynomials of fixed order in a domain D ⊂ R d , the convergence rate in terms of the number N of degrees of freedom is inversely proportional to the space dimension d. This so-called curse of dimensionality can be circumvented by applying sparse tensor product approximation, when certain high order mixed derivatives of the approximated function happen to be bounded in L 2 . It was shown by Nitsche (2006) that this regularity constraint can be dramatically reduced by considering best N -term approximation from tensor product wavelet bases. When the function is the solution of some well-posed operator equation, dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms, assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible. Applying piecewise smooth wavelets, we verify this compressibility for general, non-separable elliptic PDEs in tensor domains. Applications of the general theory developed include adaptive Galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust, i.e., independent of the scale parameters, resp. of the size of the anisotropy.

show abstract

Section: Pde's On (High Dimensional) Product Domainsmentioning

confidence: 99%

Adaptive wavelet algorithms for elliptic PDE's on product domains

Schwab

Stevenson

2008

Math. Comp.

Self Cite

View full text Add to dashboard Cite

show abstract

“…After unfolding (Cioranescu et al, 2008), it gives rise to the product of the macroscopic physical domain and the periodic microscopic domain of the cell problem; see Matache (2002). For multiple scales, a general product appears here which still can be written as the product of two domains, one containing, for example, the macroscopic scale and the other consisting of the product of the domains of the different microscales (Hoang & Schwab, 2005).…”

Section: Introductionmentioning

confidence: 99%

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

Griebel

Harbrecht

2013

IMA Journal of Numerical Analysis

View full text Add to dashboard Cite

We compare the cost complexities of two approximation schemes for functions f ∈ H p (Ω 1 × Ω 2 ) which live on the product domain Ω 1 × Ω 2 of sufficiently smooth domains Ω 1 ⊂ R n 1 and Ω 2 ⊂ R n 2 , 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 Ω 1 and Ω 2 . Then, the sparse grid approximation essentially needs only O(ε −q ), with q = max{n 1 , n 2 }/r, unknowns to reach a prescribed accuracy ε, provided that the smoothness of f satisfies p r((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 < r((n 1 + n 2 )/max{n 1 , n 2 }), then the sparse grid approach gives essentially the rate O(ε −q ) with q = (n 1 + n 2 )/p, while, for the singular value decomposition, we can only prove the rate O(ε −q ) with q = (2 min{r, p} min{n 1 , n 2 } + 2p max{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.

show abstract

“…The convergence analysis of these methods can be made using the same analytical framework -the difference between the two approaches arising during the discretization. Another more global approach has been introduced by Schwab and Hoang in [41], inspired by…”

Section: −ˆQmentioning

confidence: 99%

Numerical homogenization: survey, new results, and perspectives

Gloria

2012

ESAIM: Proc.

View full text Add to dashboard Cite

Abstract. These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which H-converges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar's correctors. Hence the convergence analysis of these methods relies on the H-convergence theory. When one is interested in convergence rates, the story is different. In particular one first needs to make additional structure assumptions on the heterogeneities (say periodicity for instance). In that case, a crucial tool is the spectral interpretation of the corrector equation by Papanicolaou and Varadhan. Spectral analysis does not only allow to obtain convergence rates, but also to devise efficient new approximation methods. For both qualitative and quantitative properties, the development and the analysis of numerical homogenization methods rely on seminal concepts of the homogenization theory. These notes contain some new results.Résumé. Ces notes de cours dressent unétat de l'art des méthodes d'homogénéisation numérique pour leséquations elliptiques linéaires. Le fil conducteur choisi est l'analyse. La plupart des méthodes d'homogénéisation numérique s'interprète comme des discrétisations (plus ou moins différentes) d'une même famille de problèmes continus approchés qui H-converge vers le problème homogénéisé. De même, le concept de correcteur numérique s'interprète comme une approximation des correcteurs introduits par Tartar. Ainsi l'analyse de convergence repose essentiellement sur la théorie de la H-convergence. Si on s'intéresse aux estimations quantitatives d'erreur, il faut faire des hypothèses supplémentaires de structure sur les hétérogénéités (périodicité par exemple). Dans ce cas, un outil important est l'interprétation spectrale de l'équation du correcteur introduite par Papanicolaou et Varadhan, qui permet non seulement de démontrer des résultats quantitatifs, mais aussi de développer des méthodes numériques efficaces. Qu'il s'agisse de propriétés qualitatives ou quantitatives, le développement et l'analyse de méthodes d'homogénéisation numérique reposent sur des concepts fondateurs de la théorie de l'homogénéisation. Ces notes contiennent quelques résultats nouveaux. Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx

show abstract

High-Dimensional Finite Elements for Elliptic Problems with Multiple Scales

Cited by 92 publications

References 17 publications

Adaptive wavelet algorithms for elliptic PDE's on product domains

Adaptive wavelet algorithms for elliptic PDE's on product domains

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

Numerical homogenization: survey, new results, and perspectives

Contact Info

Product

Resources

About