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SPARSE HIERARCHICAL PRECONDITIONERS USING PIECEWISE SMOOTH APPROXIMATIONS OF EIGENVECTORS.

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Title: SPARSE HIERARCHICAL PRECONDITIONERS USING PIECEWISE SMOOTH APPROXIMATIONS OF EIGENVECTORS.
Authors: KLOCKIEWICZ, BAZYLI, DARVE, ERIC
Source: SIAM Journal on Scientific Computing; 2020, Vol. 42 Issue 6, pA3907-A3931, 25p
Subject Terms: SMOOTHNESS of functions, ELLIPTIC equations, FLOW simulations, LINEAR equations, RIGID bodies
Abstract: When solving linear systems arising from PDE discretizations, iterative methods (such as conjugate gradient (CG), GMRES, or MINRES) are often the only practical choice. To converge in a small number of iterations, however, they have to be coupled with an efficient preconditioner. One approach to preconditioning is provided by the hierarchical approximate factorization methods. However, to guarantee sufficient accuracy on the eigenvectors corresponding to the smallest eigenvalues, these methods typically have to be performed at very stringent accuracies, making the preconditioner expensive to apply. On the other hand, for a large class of problems, including many elliptic equations, the eigenvectors corresponding to small eigenvalues are smooth functions of the PDE grid. In this paper, we describe a hierarchical approximate factorization approach which focuses on improving accuracy on the smooth eigenvectors. The improved accuracy is achieved by preserving the action of the factorized matrix on piecewise polynomial functions of the grid. Based on the factorization, we propose a family of sparse preconditioners with Ϭ(n) or Ϭ (n log n) construction complexities. Our methods exhibit rapid convergence of CG in benchmarks run on large elliptic problems, arising for example in flow or mechanical simulations. In the case of the linear elasticity equation the preconditioners are exact on the near-kernel rigid body modes. [ABSTRACT FROM AUTHOR]
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Database: Complementary Index
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Abstract:When solving linear systems arising from PDE discretizations, iterative methods (such as conjugate gradient (CG), GMRES, or MINRES) are often the only practical choice. To converge in a small number of iterations, however, they have to be coupled with an efficient preconditioner. One approach to preconditioning is provided by the hierarchical approximate factorization methods. However, to guarantee sufficient accuracy on the eigenvectors corresponding to the smallest eigenvalues, these methods typically have to be performed at very stringent accuracies, making the preconditioner expensive to apply. On the other hand, for a large class of problems, including many elliptic equations, the eigenvectors corresponding to small eigenvalues are smooth functions of the PDE grid. In this paper, we describe a hierarchical approximate factorization approach which focuses on improving accuracy on the smooth eigenvectors. The improved accuracy is achieved by preserving the action of the factorized matrix on piecewise polynomial functions of the grid. Based on the factorization, we propose a family of sparse preconditioners with Ϭ(n) or Ϭ (n log n) construction complexities. Our methods exhibit rapid convergence of CG in benchmarks run on large elliptic problems, arising for example in flow or mechanical simulations. In the case of the linear elasticity equation the preconditioners are exact on the near-kernel rigid body modes. [ABSTRACT FROM AUTHOR]
ISSN:10648275
DOI:10.1137/20M1315683