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A Robust Two-Level Schwarz Preconditioner For Sparse Matrices

Hussam Al Daas, Pierre Jolivet, Frédéric Nataf, Pierre-Henri Tournier

TL;DR

A fully algebraic two-level additive Schwarz preconditioner for general sparse large-scale matrices and compares the proposed preconditioners to the state-of-the-art domain decomposition preconditioners are compared.

Abstract

This paper introduces a fully algebraic two-level additive Schwarz preconditioner for general sparse large-scale matrices. The preconditioner is analyzed for symmetric positive definite (SPD) matrices. For those matrices, the coarse space is constructed based on approximating two local subspaces in each subdomain. These subspaces are obtained by approximating a number of eigenvectors corresponding to dominant eigenvalues of two judiciously posed generalized eigenvalue problems. The number of eigenvectors can be chosen to control the condition number. For general sparse matrices, the coarse space is constructed by approximating the image of a local operator that can be defined from information in the coefficient matrix. The connection between the coarse spaces for SPD and general matrices is also discussed. Numerical experiments show the great effectiveness of the proposed preconditioners on matrices arising from a wide range of applications. The set of matrices includes SPD, symmetric indefinite, nonsymmetric, and saddle-point matrices. In addition, we compare the proposed preconditioners to the state-of-the-art domain decomposition preconditioners.

A Robust Two-Level Schwarz Preconditioner For Sparse Matrices

TL;DR

A fully algebraic two-level additive Schwarz preconditioner for general sparse large-scale matrices and compares the proposed preconditioners to the state-of-the-art domain decomposition preconditioners are compared.

Abstract

This paper introduces a fully algebraic two-level additive Schwarz preconditioner for general sparse large-scale matrices. The preconditioner is analyzed for symmetric positive definite (SPD) matrices. For those matrices, the coarse space is constructed based on approximating two local subspaces in each subdomain. These subspaces are obtained by approximating a number of eigenvectors corresponding to dominant eigenvalues of two judiciously posed generalized eigenvalue problems. The number of eigenvectors can be chosen to control the condition number. For general sparse matrices, the coarse space is constructed by approximating the image of a local operator that can be defined from information in the coefficient matrix. The connection between the coarse spaces for SPD and general matrices is also discussed. Numerical experiments show the great effectiveness of the proposed preconditioners on matrices arising from a wide range of applications. The set of matrices includes SPD, symmetric indefinite, nonsymmetric, and saddle-point matrices. In addition, we compare the proposed preconditioners to the state-of-the-art domain decomposition preconditioners.
Paper Structure (25 sections, 9 theorems, 72 equations, 9 figures, 1 table)

This paper contains 25 sections, 9 theorems, 72 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Subdomain-permutation
  5. Local matrix and one-level Schwarz
  6. Local SPSD splitting and local projections
  7. Local SPSD splitting
  8. Local harmonic projection
  9. Lifting projection
  10. Two-level theory
  11. The coarse space and decomposition spaces
  12. The spectral coarse space
  13. The decomposition spaces
  14. The bilinear form
  15. The interpolation operator
  16. ...and 10 more sections

Key Result

Lemma 3.1

\newlabellemma:spsd_splitting0 Let $\widetilde{A}_i$ be defined such that where Then, $\widetilde{A}_i$ is a local SPSD splitting of $A$, i.e., $\widetilde{A}_i$ and $A-\widetilde{A}_i$ are SPSD matrices.

Figures (9)

  • Figure 1: For a partitioning with $N=256$ subdomains of a three-dimensional diffusion (resp. Stokes) problem on the left-hand side (resp. right-hand side), the first $200$ eigenvalues $\lambda$ (resp. singular values $\sigma$) of the generalized eigenvalue problem $\Pi_i^\top D_iA_{ii}D_i\Pi u = \lambda^2 A_{ii}u$ (resp. $\Pi_i^\top D_i D_i\Pi_i u = \sigma^2 u$) are plotted for three values of overlapping layers $\delta = 1$, $3$, and $5$; showing values only for subdomains $i=1$, $65$, $129$, and $193$.
  • Figure 2: Bilaplace 2D (1002001 unknowns)
  • Figure 3: Diffusion 3D (531441 unknowns)
  • Figure 4: Elasticity 2D (1084202 unknowns)
  • Figure 5: Elasticity 3D (521823 unknowns)
  • ...and 4 more figures

Theorems & Definitions (21)

  • Lemma 3.1
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Definition 3.3: Local harmonic operator
  • Lemma 3.4
  • Proof 3
  • Theorem 3.5
  • Proof 4
  • Remark 4.1
  • ...and 11 more