Multiscale-Spectral GFEM and optimal oversampling

Babuška, Ivo; Lipton, Robert; Sinz, Paul; Stuebner, Michael

doi:10.1016/j.cma.2020.112960

Cited by 28 publications

(22 citation statements)

References 43 publications

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“…The authors proved quasi-optimality of the final multiscale space in the sense that Galerkin approximations to symmetric elliptic problems converge exponentially fast depending on the dimension of the space. The approach was later generalized to other applications, for which we refer to Smetana and Patera (2016), Babuška, Lipton, Sinz and Stuebner (2020) and the references therein. As an alternative to the spectral construction of basis functions, it is also possible to use randomized sampling algorithms to approximate the optimal subspaces with provable nearly optimal convergence rate; see Buhr and Smetana (2018).…”

Section: History Of Spectral Approachesmentioning

confidence: 99%

Numerical homogenization beyond scale separation

Altmann¹,

Henning

Peterseim³

2021

Acta Numerica

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Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at reducing complex large-scale problems to simplified numerical models valid on some target scale of interest, thereby accounting for the impact of features on smaller scales that are otherwise not resolved. While constructive approaches in the mathematical theory of homogenization are restricted to problems with a clear scale separation, modern numerical homogenization methods can accurately handle problems with a continuum of scales. This paper reviews such approaches embedded in a historical context and provides a unified variational framework for their design and numerical analysis. Apart from prototypical elliptic model problems, the class of partial differential equations covered here includes wave scattering in heterogeneous media and serves as a template for more general multi-physics problems.

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Section: History Of Spectral Approachesmentioning

confidence: 99%

Numerical homogenization beyond scale separation

Altmann¹,

Henning

Peterseim³

2021

Acta Numerica

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“…Furthermore, we employ an adaptive algorithm that is driven by a probabilistic a posteriori error estimator. The output of the algorithm is an approximation space Λ n rand that satisfies the property 7 where the accuracy tol and the failure probability ε algofail are prescribed by the user. For further details we refer to section SM2 and to [12] where methods from randomized linear algebra [27] have been used to approximate the optimal local approximation spaces in the elliptic setting.…”

Section: Approximation Of the Optimal Local Approximation Spacesmentioning

confidence: 99%

“…Concerning (real-world) applications the optimal local approximation spaces are employed, for instance, for the construction of digital twins [34,38] and in the context of data assimilation [59]. Other options for approximating the optimal local reduced spaces besides the random sampling technique [12] employed here are proposed in [7,13].…”

Section: Introductionmentioning

confidence: 99%

Optimal local approximation spaces for parabolic problems

Schleuß¹,

Smetana²

2020

Preprint

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We propose local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed in multiscale and domain decomposition methods. The diffusion coefficient can be arbitrarily rough in space and time. To construct local approximation spaces we consider a compact transfer operator that acts on the space of local solutions and covers the full time dimension. The optimal local spaces are then given by the left singular vectors of the transfer operator. To proof compactness of the latter we combine a suitable parabolic Caccioppoli inequality with the compactness theorem of Aubin-Lions. In contrast to the elliptic setting [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406] we need an additional regularity result to combine the two results. Furthermore, we employ the generalized finite element method to couple local spaces and construct an approximation of the global solution. Since our approach yields reduced space-time bases, the computation of the global approximation does not require a time stepping method and is thus computationally efficient. Moreover, we derive rigorous local and global a priori error bounds. In detail, we bound the global approximation error in a graph norm by the local errors in the L 2 (H 1 )-norm, noting that the space the transfer operator maps to is equipped with this norm. Numerical experiments demonstrate an exponential decay of the singular values of the transfer operator and the local and global approximation errors for problems with high contrast or multiscale structure regarding space and time.

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“…In [2], based on PUM, an optimal local basis is constructed for elliptic equations with rough coefficients, which achieves O(H) accuracy in the energy norm using only O(log d+1 (1/H)) number of basis functions in each local domain. Moreover, by using the right-hand side information, a nearly exponentially decaying error with respect to the number of basis functions can be achieved; see also [1,3]. We will borrow some techniques in [2] to prove the nearly exponential convergence of our method.…”

mentioning

confidence: 99%

Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions

Chen¹,

Hou²,

Wang³

2021

Multiscale Model. Simul.

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In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main results is that we prove that the proposed multiscale method achieves nearly exponential convergence in the approximation error with respect to the computational degrees of freedom. Our strategy is to perform an energy orthogonal decomposition of the solution space into a coarse scale component comprising a-harmonic functions in each element of the mesh, and a fine scale component named the bubble part that can be computed locally and efficiently. The coarse scale component depends entirely on function values on edges. Our approximation on each edge is made in the Lions--Magenes space H 1/2 00 (e), which we will demonstrate to be a natural and powerful choice. We construct edge basis functions using local oversampling and singular value decomposition. When local information of the right-hand side is adaptively incorporated into the edge basis functions, we prove a nearly exponential convergence rate of the approximation error. Numerical experiments validate and extend our theoretical analysis; in particular, we observe no obvious degradation in accuracy for high-contrast media problems.

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Multiscale-Spectral GFEM and optimal oversampling

Cited by 28 publications

References 43 publications

Numerical homogenization beyond scale separation

Numerical homogenization beyond scale separation

Optimal local approximation spaces for parabolic problems

Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions

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