A Robust Multilevel Approximate Inverse Preconditioner for Symmetric Positive Definite Matrices

Franceschini, A.; Magri, Victor Antonio Paludetto; Ferronato, Massimiliano; Janna, Carlo

doi:10.1137/16m1109503

Cited by 9 publications

(7 citation statements)

References 27 publications

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“…which shows that reaching the minimum of each [F AF T ] ii with \scrW \scrS \scrB becoming dense is equivalent to choosing \widetil F = W T ideal . Using an approach similar to the one presented in [26], we approximate the ideal prolongator by running Algorithm 3 with configuration parameters k p , \rho p , and \epsilon p , where in place of computing the row vectors g i , we calculate the column vectors w i of matrix W . In this way, at each step of the procedure we compute, for the current pattern, a minimizer of tr(P T AP ) and select the most promising entries to enlarge W .…”

Section: Prolongation Operatorsmentioning

confidence: 99%

“…where v T f,i is the ith row of V f . Even if we suppose that the F/C partition is such that V c has full rank, in practical applications the number of test vectors is much smaller than the problem dimension, n t \ll n c , and systems (26) are overdetermined. For this Algorithm 4.…”

Section: Least Squares Corrected Abfmentioning

confidence: 99%

“…Usually, to limit the number of non-zeros in W , only strong connections are considered. However, the connectivity of the SoC matrix may vary significantly, especially in difficult problems, with the consequence that an a priori selected nonzero pattern may give rise to both overdetermined and underdetermined row systems (26). The first occurrence causes unnecessary large operator complexity, while the latter prevents the construction of an effective prolongation as the target range cannot be represented locally.…”

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confidence: 99%

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A Novel Algebraic Multigrid Approach Based on Adaptive Smoothing and Prolongation for Ill-Conditioned Systems

Magri¹,

Franceschini²,

Janna³

2019

SIAM J. Sci. Comput.

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The numerical simulation of modern engineering problems can easily incorporate millions or even billions of unknowns. In several applications, sparse linear systems with symmetric positive definite matrices need to be solved, and algebraic multigrid (AMG) methods represent common choices for the role of iterative solvers or preconditioners. The reason for their popularity relies on the fast convergence that these methods provide even in the solution of large size problems, which is a consequence of the AMG ability to reduce particular error components across their multilevel hierarchy. Despite carrying the name ``algebraic,"" most of these methods still make assumptions on additional information other than the global assembled matrix, such as the knowledge of the operator's near kernel, which limits their applicability as black-box solvers. In this work, we introduce a novel AMG approach featuring the adaptive factored sparse approximate inverse (aFSAI) method as a flexible smoother, as well as three new approaches to adaptively compute the prolongation operator. We assess the performance of the proposed AMG through the solution of a set of model problems along with real-world engineering test cases. Moreover, we perform comparisons to other methods such as the aFSAI and BoomerAMG preconditioners, showing that our new method proves to be superior to the first strategy and comparable to the second one, if not better, as in the solution of linear elasticity models.

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Section: Prolongation Operatorsmentioning

confidence: 99%

Section: Least Squares Corrected Abfmentioning

confidence: 99%

mentioning

confidence: 99%

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A Novel Algebraic Multigrid Approach Based on Adaptive Smoothing and Prolongation for Ill-Conditioned Systems

Magri¹,

Franceschini²,

Janna³

2019

SIAM J. Sci. Comput.

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“…Another solution to this problem consists of writing an approximation of the Schur complement S i as the summation of SPD matrices instead of a subtraction, as presented in the current work. This idea, recently presented in our previous work, 23 ensures a greater robustness at the cost of a higher computational setup effort.…”

Section: Block Tridiagonal Fsaimentioning

confidence: 99%

Multilevel approaches for FSAI preconditioning

Magri

Franceschini

Ferronato

et al. 2018

Numerical Linear Algebra App

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Summary Factorized sparse approximate inverse (FSAI) preconditioners are robust algorithms for symmetric positive matrices, which are particularly attractive in a parallel computational environment because of their inherent and almost perfect scalability. Their parallel degree is even redundant with respect to the actual capabilities of the current computational architectures. In this work, we present two new approaches for FSAI preconditioners with the aim of improving the algorithm effectiveness by adding some sequentiality to the native formulation. The first one, denoted as block tridiagonal FSAI, is based on a block tridiagonal factorization strategy, whereas the second one, domain decomposition FSAI, is built by reordering the matrix graph according to a multilevel k‐way partitioning method followed by a bandwidth minimization algorithm. We test these preconditioners by solving a set of symmetric positive definite problems arising from different engineering applications. The results are evaluated in terms of performance, scalability, and robustness, showing that both strategies lead to faster convergent schemes regarding the number of iterations and total computational time in comparison with the native FSAI with no significant loss in the algorithmic parallel degree.

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“…Multilevel variants have been also proposed. 28,[32][33][34] Multilevel techniques improve performance and convergence behavior while reducing memory requirements induced by increased fill-in.…”

Section: Introductionmentioning

confidence: 99%

Incomplete inverse matrices

Filelis‐Papadopoulos

2021

Numerical Linear Algebra App

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The solution of large sparse linear systems is required in many scientific fields such as computational fluid dynamics, computational electromagnetism, computational finance, etc. The computation of the solution of these systems is performed with preconditioned iterative methods, which rely on effective preconditioning schemes. A new class of approximate inverses is proposed, namely incomplete inverse matrices, which are computed using a recursive Schur complement-based approach. This class of approximate inverses is based on a priori knowledge of a sparsity pattern. In order to have finer control over the density of the proposed approximate inverse, especially in the case of three-dimensional problems, on-the-fly filtration is used, resulting in substantial reduction in the number of nonzero elements. Implementation details and analysis for computing the proposed scheme are given. Numerical results depicting the effectiveness and applicability of the proposed scheme are also provided. K E Y W O R D Sapproximate inverse matrix, filtering, preconditioned iterative methods, sparsity patterns

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A Robust Multilevel Approximate Inverse Preconditioner for Symmetric Positive Definite Matrices

Cited by 9 publications

References 27 publications

A Novel Algebraic Multigrid Approach Based on Adaptive Smoothing and Prolongation for Ill-Conditioned Systems

A Novel Algebraic Multigrid Approach Based on Adaptive Smoothing and Prolongation for Ill-Conditioned Systems

Multilevel approaches for FSAI preconditioning

Incomplete inverse matrices

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