Two-level preconditioning for $$h$$-version boundary element approximation of hypersingular operator with GenEO

Marchand, Pierre; Claeys, Xavier; Jolivet, Pierre; Nataf, Frédéric; Tournier, Pierre-Henri

doi:10.1007/s00211-020-01149-5

Cited by 12 publications

(10 citation statements)

References 28 publications

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“…The problems there are reduced to the interfaces between subdomains but the name GenEO was kept since these interfaces in some sense constitute an overlap between subdomains. The family of GenEO coarse spaces has grown since with e.g., the contributions [20] for Optimized Schwarz and [32] in the context of boundary element methods. In this article the coarse spaces will be referred to as coarse spaces of the GenEO family as their construction follows the same procedure as the two original GenEO coarse spaces: [44,45].…”

Section: Introductionmentioning

confidence: 99%

An abstract theory of domain decomposition methods with coarse spaces of the GenEO family

Spillane

2021

Preprint

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Two-level domain decomposition methods are preconditioned Krylov solvers. What separates one and two-level domain decomposition method is the presence of a coarse space in the latter. The abstract Schwarz framework is a formalism that allows to define and study a large variety of two-level methods. The objective of this article is to define, in the abstract Schwarz framework, a family of coarse spaces called the GenEO coarse spaces (for Generalized Eigenvalues in the Overlaps). This is a generalization of existing methods for particular choices of domain decomposition methods. Bounds for the condition numbers of the preconditioned operators are proved that are independent of the parameters in the problem (e.g., any coefficients in an underlying PDE or the number of subdomains). The coarse spaces are computed by finding low or high frequency spaces of some well chosen generalized eigenvalue problems in each subdomain. Contents

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Section: Introductionmentioning

confidence: 99%

An abstract theory of domain decomposition methods with coarse spaces of the GenEO family

Spillane

2021

Preprint

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“…Thus, the choice of R 0 is key to obtaining a wellconditioned system and building efficient two-level Schwarz preconditioners. Twolevel Schwarz preconditioners have been used to solve a large class of systems arising from a range of engineering applications (see, for example, [22,26,28,30,40,43] and references therein).…”

Section: Introduction To Domain Decompositionmentioning

confidence: 99%

A Robust Algebraic Domain Decomposition Preconditioner for Sparse Normal Equations

Daas¹,

Jolivet²,

Scott³

2021

Preprint

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Solving the normal equations corresponding to large sparse linear least-squares problems is an important and challenging problem. For very large problems, an iterative solver is needed and, in general, a preconditioner is required to achieve good convergence. In recent years, a number of preconditioners have been proposed. These are largely serial and reported results demonstrate that none of the commonly used preconditioners for the normal equations matrix is capable of solving all sparse least-squares problems. Our interest is thus in designing new preconditioners for the normal equations that are efficient, robust, and can be implemented in parallel. Our proposed preconditioners can be constructed efficiently and algebraically without any knowledge of the problem and without any assumption on the least-squares matrix except that it is sparse. We exploit the structure of the symmetric positive definite normal equations matrix and use the concept of algebraic local symmetric positive semi-definite splittings to introduce two-level Schwarz preconditioners for least-squares problems. The condition number of the preconditioned normal equations is shown to be theoretically bounded independently of the number of subdomains in the splitting. This upper bound can be adjusted using a single parameter τ that the user can specify. We discuss how the new preconditioners can be implemented on top of the PETSc library using only 150 lines of Fortran, C, or Python code. Problems arising from practical applications are used to compare the performance of the proposed new preconditioner with that of other preconditioners.

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“…Over the last decade, the range of symmetric positive definite problems for which two-level domain decomposition preconditioners can be made scalable and robust by a good choice of the coarse space has become very large by the development of so-called spectral coarse spaces. The following list gives an overview of some of these contributions: [29,30,13,15,16,14,39,40,17,1,21] for Additive Schwarz, [46] for Additive Average Schwarz, [41,1] for BDD and FETI, [26,24,10,23,11,32,25,47] for BDDC and/or FETI-DP, [20] for Optimized Schwarz and [28] in the context of boundary element methods.…”

Section: Introductionmentioning

confidence: 99%

Fully algebraic domain decomposition preconditioners with adaptive spectral bounds

Gouarin,

Spillane

2021

Preprint

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In this article a new family of preconditioners is introduced for symmetric positive definite linear systems. The new preconditioners, called the AWG preconditioners (for Algebraic-Woodbury-GenEO) are constructed algebraically. By this, we mean that only the knowledge of the matrix A for which the linear system is being solved is required. Thanks to the GenEO spectral coarse space technique, the condition number of the preconditioned operator is bounded theoretically from above. This upper bound can be made smaller by enriching the coarse space with more spectral modes.The novelty is that, unlike in previous work on the GenEO coarse spaces, no knowledge of a partially non-assembled form of A is required. Indeed, the spectral coarse space technique is not applied directly to A but to a low-rank modification of A of which a suitable nonassembled form is known by construction. The extra cost is a second (and to this day rather expensive) coarse solve in the preconditioner. One of the AWG preconditioners has already been presented in the short preprint [38]. This article is the first full presentation of the larger family of AWG preconditioners. It includes proofs of the spectral bounds as well as numerical illustrations.

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Two-level preconditioning for $$h$$-version boundary element approximation of hypersingular operator with GenEO

Cited by 12 publications

References 28 publications

An abstract theory of domain decomposition methods with coarse spaces of the GenEO family

An abstract theory of domain decomposition methods with coarse spaces of the GenEO family

A Robust Algebraic Domain Decomposition Preconditioner for Sparse Normal Equations

Fully algebraic domain decomposition preconditioners with adaptive spectral bounds

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