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

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

doi:10.1137/17m1161178

Cited by 23 publications

(16 citation statements)

References 53 publications

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“…This is particularly true for structural problems where damping highest frequencies often requires the use of weights or Chebyshev polynomials [52,53]. In [49], the aFSAI [29] is proposed as smoother and its effectiveness is assessed on an extensive set of numerical experiments. aFSAI is designed for SPD matrices and, as smoother, takes the following form:…”

Section: Adaptive Smoother Computationmentioning

confidence: 99%

“…The last key component of our AMG method is the construction of suitable prolongation and restriction operators. Following the idea proposed in [42] and successively refined in [49], we choose to build an interpolation operator fitting as close as possible the set of test vectors computed in the early setup stage. More precisely, the prolongation weights β j are computed in order to minimize the interpolation residual:…”

Section: Adaptive Prolongationmentioning

confidence: 99%

“…where the n tv × n i matrix V c,i collects the n i vectors v k for k ∈ J i . The main feature of the DPLS algorithm described in [49] is the dynamic construction during setup of the prolongation pattern, i.e., the sets J i for i ∈ C.…”

Section: Adaptive Prolongationmentioning

confidence: 99%

“…• C f s : FSAI complexity, i.e., an average value of the aFSAI density over the multigrid hierarchy, as defined in [49];…”

Section: Default Parameters Studymentioning

confidence: 99%

“…This work presents an extension of the adaptive Smoothing and Prolongation based Algebraic Multigrid method (aSP-AMG), proposed by [49], with the aim of specifically improving its performance for the solution of linear elasticity problems. Such method follows the path of bootstrap and adaptive AMG which is to build an approximation of the near-null space components of the problem at hand automatically.…”

Section: Introductionmentioning

confidence: 99%

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A robust adaptive algebraic multigrid linear solver for structural mechanics

Franceschini

Magri

Mazzucco

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.

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Section: Adaptive Smoother Computationmentioning

confidence: 99%

Section: Adaptive Prolongationmentioning

confidence: 99%

Section: Adaptive Prolongationmentioning

confidence: 99%

“…• C f s : FSAI complexity, i.e., an average value of the aFSAI density over the multigrid hierarchy, as defined in [49];…”

Section: Default Parameters Studymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A robust adaptive algebraic multigrid linear solver for structural mechanics

Franceschini

Magri

Mazzucco

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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Parallel Newton–Chebyshev polynomial preconditioners for the conjugate gradient method

Bergamaschi

Martínez

2021

Comp and Math Methods

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In this note, we exploit polynomial preconditioners for the conjugate gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix equation X −1 = A and the Chebyshev polynomials for preconditioning. We propose a simple modification of one parameter which avoids clustering of extremal eigenvalues in order to speed-up convergence.We provide results on very large matrices (up to 8.6 billion unknowns) in a parallel environment showing the efficiency of the proposed class of preconditioners.

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Algebraic multigrid for systems of elliptic boundary‐value problems

Lee

2020

Numerical Linear Algebra App

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Summary This article develops an algebraic multigrid (AMG) method for solving systems of elliptic boundary‐value problems. It is well known that multigrid for systems of elliptic equations faces many challenges that do not arise for most scalar equations. These challenges include strong intervariable couplings, multidimensional and possibly large near‐nullspaces, analytically unknown near‐nullspaces, delicate selection of coarse degrees of freedom (CDOFs), and complex construction of intergrid operators. In this article, we consider only the selection of CDOFs and the construction of the interpolation operator. The selection is an extension of the Ruge–Stuben algorithm using a new strength of connection measure taken between nodal degrees of freedom, that is, between all degrees of freedom located at a gridpoint to all degrees of freedom at another gridpoint. This measure is based on a local correlation matrix generated for a set of smoothed test vectors derived from a relaxation‐based procedure. With this measure, selection of the CDOFs is then determined by the number of strongly correlated connections at each node, with the selection processed by a Ruge–Stuben coloring scheme. Having selected the CDOFs, the interpolation operator is constructed using a bootstrap AMG (BAMG) procedure. We apply the BAMG procedure either over the smoothed test vectors to obtain an intervariable interpolation scheme or over the like‐variable components of the smoothed test vectors to obtain an intravariable interpolation scheme. Moreover, comparing the correlation measured between the intravariable couplings with the correlation between all couplings, a mixed intravariable and intervariable interpolation scheme is developed. We further examine an indirect BAMG method that explicitly uses the coefficients of the system operator in constructing the interpolation weights. Finally, based on a weak approximation criterion, we consider a simple scheme to adapt the order of the interpolation (i.e., adapt the caliber or maximum number of coarse‐grid points that a fine‐grid point can interpolate from) over the computational domain.

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

Cited by 23 publications

References 53 publications

A robust adaptive algebraic multigrid linear solver for structural mechanics

A robust adaptive algebraic multigrid linear solver for structural mechanics

Parallel Newton–Chebyshev polynomial preconditioners for the conjugate gradient method

Algebraic multigrid for systems of elliptic boundary‐value problems

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