Randomized Sampling for Basis Function Construction in Generalized Finite Element Methods

DOI10.1137/18M1166432zbMath1446.65162arXiv1801.06938OpenAlexW3037853466MaRDI QIDQ5117157

Ke Chen, Qin Li, Stephen J. Wright, Jian-feng Lu

Publication date: 20 August 2020

Published in: Multiscale Modeling & Simulation (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1801.06938

zbMATH Keywords

generalized eigenvalue problemelliptic equationgeneralized finite element methodsrandomized samplingrandom singular value decomposition

Mathematics Subject Classification ID

Sampling theory, sample surveys (62D05) Numerical computation of eigenvalues and eigenvectors of matrices (65F15) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30)

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Cites Work

• Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions
• Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework. I: Total energy calculation
• Sparse operator compression of higher-order elliptic operators with rough coefficients
• The heterogeneous multiscale methods
• A multiscale finite element method for elliptic problems in composite materials and porous media
• Uncertain loading and quantifying maximum energy concentration within composite structures
• Numerical methods for multiscale elliptic problems
• Randomized Oversampling for Generalized Multiscale Finite Element Methods
• Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization
• Hierarchical Matrices: Algorithms and Analysis
• Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems
• Domain Decomposition Preconditioners for Multiscale Flows in High Contrast Media: Reduced Dimension Coarse Spaces
• Analysis of the heterogeneous multiscale method for elliptic homogenization problems
• Localization of elliptic multiscale problems
• Extensions of Lipschitz mappings into a Hilbert space
• Metric-based upscaling
• Why Finite Element Discretizations Can Be Factored by Triangular Hierarchical Matrices
• Asymptotic analysis of transport processes
• Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients
• THE PARTITION OF UNITY METHOD
• Evaluation of Local Multiscale Approximation Spaces for Partition of Unity Methods
• Convergence of a Nonconforming Multiscale Finite Element Method
• Bayesian Numerical Homogenization
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