Computation of eigenvalues by numerical upscaling

Målqvist, Axel; Peterseim, Daniel

doi:10.1007/s00211-014-0665-6

Cited by 51 publications

(72 citation statements)

References 36 publications

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“…• SubInfo provides a list with different restriction operators for each sub-domain (actually our implementation provides this information via a generator). Each entry has to provide the following details: -SubInfo.R Restriction operator mapping from the fine space restriction to the patch, see equation (11). -SubInfo.RH Restriction operator mapping from the coarse space restriction to the patch, see equation (12).…”

Section: Numerical Examplesmentioning

confidence: 99%

Efficient implementation of the localized orthogonal decomposition method

Engwer

Henning

Målqvist

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales. We show how the method can be implemented in a fairly standard Finite Element framework and discuss its realization for different types of problems, such as linear elliptic problems with rough coefficients and linear eigenvalue problems. the field of multiscale partial differential equations including rough polyharmonic splines [41], iterative numerical homogenization [42], and gamblets [43]. While previous works focused on the numerical analysis of the method, this paper aims at the detailed explanation of how the method can be algorithmically realized. We give detailed explanations on how the method works on an algebraic level. The results may as well be useful for implementing related multiscale methods. PreliminariesIn this section we recall the Localized Orthogonal Decomposition (LOD) for finite element spaces. The decomposition is always with respect to a linear elliptic part of the differential operator. Computational domain and boundaryFor the rest of the paper, we consider a bounded polygonal domain Ω ⊂ R d . The boundary ∂Ω is divided into two parts Γ D and Γ N . On Γ D we prescribe a Dirichlet boundary condition and on Γ N we prescribe a Neumann boundary condition. We have Γ D ∪ Γ N = ∂Ω and we assume Γ D = ∅. With that, we define the spacewhere v |Γ D = 0 is understood in the sense of traces. Elliptic differential operatorSubsequently we consider the following differential operator. Let κ ∈ L ∞ (Ω, R d×d ) denote a matrix-valued, symmetric, possibly highly varying and heterogeneous coefficient with uniform spectral bounds γ min > 0 and γ max ≥ γ min , σ(κ(x)) ⊂ [γ min , γ max ] for almost all x ∈ Ω.This coefficient defines a scalar product A(·, ·) on H 1 Γ D (Ω) that is given by Meshes and spacesWe wish to discretize a problem that is associated with A(·, ·). Then the discretization is constrained by the diffusion coefficient κ, in the sense that variations of κ must be resolved by the computational mesh. We call such a discretization a fine scale discretization. In addition to this, we have a second discretization on a coarse scale. The coarse mesh is arbitrary and no more related to A(·, ·). It contains elements of maximum diameter H > 0. The fine mesh consists of elements of maximum diameter h < H. Let T H , T h denote the corresponding simplicial or quadrilateral subdivisions of Ω into (closed) conforming shape regular simplicial elements or conforming shape regular quadrilateral elements, i.e.,Ω =We assume that T h is a regular, possibly non-uniform, mesh refinement of T H . Furthermore we also assume that T H and T h are shape-regular in the sense that there exists a positive constant c 0 such that max maxand regular in the sense that any two elements are either disjoint, share exactly one face, share exactly one edge, or share exactly one ve...

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Section: Numerical Examplesmentioning

confidence: 99%

Efficient implementation of the localized orthogonal decomposition method

Engwer

Henning

Målqvist

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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“…Remark 4.1. It is shown in [40] that for approximate eigenvalues with respect to the LOD coarse spaces on scale H, a post-processing step can improve the eigenvalue error from H 4 to H 6 . The post-processing step is a correction with exact solve on the finest level.…”

Section: Spe10mentioning

confidence: 99%

“…Although high-dimensional eigenvalue problems are ubiquitous in physical sciences, data and imaging sciences, and machine learning, the class of eigensolvers is not as diverse as that of linear solvers (which comprises many efficient algorithms such as geometric and algebraic multigrid [13,23], approximate Gaussian elimination [36], etc.). In particular, eigenvalue problems may involve operators with nonseparable multiple scales, and the nonlinear interplay between those coupled scales and the eigenvalue problem poses significant challenges for numerical analysis and scientific computing [4,16,64,40].…”

Section: Introductionmentioning

confidence: 99%

“…Recently, two-level [62,40] and multilevel [37,38,39,60,61] correction methods have been proposed to reduce the complexity of solving eigenpairs associated with low eigenvalues by first solving a coarse mesh/scale approximation, which can then be corrected by solving linear systems (corresponding to linearized eigenvalue problems) on a hierarchy of finer meshes/scales. Although the multilevel correction approach has been extended to multigrid methods for linear and nonlinear eigenvalue problems [18,37,38,39,27,60,61], the regularity estimates required for linear complexity do not hold for PDEs with rough coefficients and a naive application of the correction approach to multiscale eigenvalue problems may converge very slowly.…”

Section: Introductionmentioning

confidence: 99%

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Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction

Xie¹,

Zhang²,

Owhadi³

2019

SIAM J. Numer. Anal.

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We present a method for the fast computation of the eigenpairs of a bijective positive symmetric linear operator L. The method is based on a combination of operator adapted wavelets (gamblets) with hierarchical subspace correction. First, gamblets provide a raw but fast approximation of the eigensubspaces of L by block-diagonalizing L into sparse and well-conditioned blocks. Next, the hierarchical subspace correction method, computes the eigenpairs associated with the Galerkin restriction of L to a coarse (low dimensional) gamblet subspace, and then, corrects those eigenpairs by solving a hierarchy of linear problems in the finer gamblet subspaces (from coarse to fine, using multigrid iteration). The proposed algorithm is robust to the presence of multiple (a continuum of) scales and is shown to be of near-linear complexity when L is an (arbitrary local, e.g. differential) operator mapping H s 0 (Ω) to H −s (Ω) (e.g. an elliptic PDE with rough coefficients).

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“…One of the main advantages is that no assumptions on scale separation or periodicity of the coefficient are needed. Recently, this technique has been applied to several other problems, for instance, semilinear elliptic equations [12], boundary value problems [11], eigenvalue problems [18], linear and semilinear parabolic equations [16], and the linear wave equation [1].…”

Section: Introductionmentioning

confidence: 99%

A generalized finite element method for linear thermoelasticity

Målqvist

Persson

2017

ESAIM: M2AN

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Abstract. We propose and analyze a generalized finite element method designed for linear quasistatic thermoelastic systems with spatial multiscale coefficients. The method is based on the local orthogonal decomposition technique introduced by Målqvist and Peterseim (Math. Comp., 83(290): 2583Comp., 83(290): -2603Comp., 83(290): , 2014. We prove convergence of optimal order, independent of the derivatives of the coefficients, in the spatial H 1 -norm. The theoretical results are confirmed by numerical examples.

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Computation of eigenvalues by numerical upscaling

Cited by 51 publications

References 36 publications

Efficient implementation of the localized orthogonal decomposition method

Efficient implementation of the localized orthogonal decomposition method

Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction

A generalized finite element method for linear thermoelasticity

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