Local Fourier Analysis of Balancing Domain Decomposition By Constraints Algorithms

Brown, Jed; He, Yunhui; MacLachlan, Scott

doi:10.1137/18m1191373

Cited by 11 publications

(5 citation statements)

References 41 publications

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“…Similar work has been done for several other common coupled systems, such as poroelasticity [12,28,30] or Stokes-Darcy flow [29]. With insights gained from this and similar work, LFA has also been applied in much broader settings, such as the optimization of algorithmic parameters in balancing domain decomposition by constraints preconditioners [5]. While the use of a Fourier ansatz inherently limits LFA to a class of homogeneous (or periodic [1,23]) discretized operators (block-structured with (multilevel) Toeplitz blocks), recent work has shown that LFA can be applied to increasingly complex and challenging classes of both problems and solution algorithms, limited only by one's ability to optimize parameters within the LFA symbols.…”

mentioning

confidence: 80%

“…Generalizations of (2.6) are also possible, such as to minimize the condition number of a preconditioned system corresponding to other solution approaches. For example, the authors of [5] use LFA to study the condition numbers of the preconditioned system in one type of domain decomposition method, where a different minimax problem arises. Similarly, in [43], the optimization of E(p, θ) is considered in the highly non-normal case.…”

Section: Minimax Problem In Lfamentioning

confidence: 99%

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Tuning Multigrid Methods with Robust Optimization

Brown,

He,

MacLachlan

et al. 2020

Preprint

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Local Fourier analysis is a useful tool for predicting and analyzing the performance of many efficient algorithms for the solution of discretized PDEs, such as multigrid and domain decomposition methods. The crucial aspect of local Fourier analysis is that it can be used to minimize an estimate of the spectral radius of a stationary iteration, or the condition number of a preconditioned system, in terms of a symbol representation of the algorithm. In practice, this is a "minimax" problem, minimizing with respect to solver parameters the appropriate measure of work, which involves maximizing over the Fourier frequency. Often, several algorithmic parameters may be determined by local Fourier analysis in order to obtain efficient algorithms. Analytical solutions to minimax problems are rarely possible beyond simple problems; the status quo in local Fourier analysis involves grid sampling, which is prohibitively expensive in high dimensions. In this paper, we propose and explore optimization algorithms to solve these problems efficiently. Several examples, with known and unknown analytical solutions, are presented to show the effectiveness of these approaches.

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confidence: 80%

Section: Minimax Problem In Lfamentioning

confidence: 99%

Tuning Multigrid Methods with Robust Optimization

Brown,

He,

MacLachlan

et al. 2020

Preprint

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“…. By using macro-elements, [21,9], our LFA of p-multigrid can be extended to LFA of h-multigrid. For 1D analysis, a macro-element is single element comprising of a pair of sub-elements with separate quadrature spaces, which is equivalent to partially assembling the finite element operator on two element subdomains.…”

Section: Extension To H-multigridmentioning

confidence: 99%

“…Although [31] discussed LFA of p-multigrid for the discontinuous Galerkin method, this formulation cannot be extended to the continuous Galerkin method. Our LFA formulation for p-multigrid with high-order finite element discretizations for the continuous Galerkin method can be generalized to reproduce previous work on h-multigrid for high-order finite elements [16] and can be extended to LFA for hmultigrid of finite difference discretizations that can be represented via finite element discretizations as well as LFA of block smoothers [22,9] We investigate the performance of Jacobi and Chebyshev semi-iterative smoothers for p-multigrid with aggressive coarsening for the scalar Laplacian in one and two dimensions, and we validate our LFA of p-multigrid with the scalar Laplacian in three dimensions against numerical experiments. Our analysis demonstrates that the performance of p-multigrid with these two smoothers degrades as we coarsen more aggressively.…”

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confidence: 99%

Local Fourier Analysis of P-Multigrid for High-Order Finite Element Operators

Thompson¹,

Brown²,

He³

2021

Preprint

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Multigrid methods are popular for solving linear systems derived from discretizing PDEs. Local Fourier Analysis (LFA) is a technique for investigating and tuning multigrid methods. P-multigrid is popular for high-order or spectral finite element methods, especially on unstructured meshes. In this paper, we introduce LFAToolkit.jl, a new Julia package for LFA of high-order finite element methods. LFAToolkit.jl analyzes preconditioning techniques for arbitrary systems of second order PDEs and supports mixed finite element methods. Specifically, we develop LFA of p-multigrid with arbitrary second-order PDEs using high-order finite element discretizations and examine the performance of Jacobi and Chebyshev smoothing for two-grid schemes with aggressive p-coarsening. A natural extension of this LFA framework is the analysis of h-multigrid for finite element discretizations or finite difference discretizations that can be represented in the language of finite elements. With this extension, we can replicate previous work on the LFA of h-multigrid for arbitrary order discretizations using a convenient and extensible abstraction. Examples in one, two, and three dimensions are presented to validate our LFA of p-multigrid for the Laplacian and linear elasticity.

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“…On the other hand, LFA could also be applied to different types of discretization schemes as well, including the continuous Galerkin finite‐element, 2,15 the discontinuous Galerkin, 16,17 the finite‐difference, 18,19 and the finite‐volume methods 20 . Recently, LFA has been designed to combine with/improve the periodic stencil operator, 21 the domain decomposition method, 22 and multicolored contexts 23,24 …”

Section: Introductionmentioning

confidence: 99%

Independence of placement for local Fourier analysis

2021

Numerical Linear Algebra App

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Local Fourier analysis (LFA) serves a prominent role in the prediction of the convergence factor of multigrid methods for discretizations of PDEs. However, few discussions about the implementation of LFA for complicated discretizations of PDEs exist, such as higher order finite elements, as staggered meshes could lead to complex LFA representations of the grid‐transfer operator. In this work, we prove that the LFA representation for d‐dimensional PDEs is independent of the placement of degrees of freedom (DoFs). Intuitively speaking, the seeding of the unknowns has no direct effect on the LFA presentation, instead, it serves as a unitary transformation between different representations. Thus, different LFA representations for a given PDE have the same spectrum and norm. Furthermore, we provide a uniform representation in terms of the location of the unknowns, named simple representation, where we allocate all of the DoFs at nodes, resulting in a simple and unified way to compute the symbols of the discrete operators, especially for the grid‐transfer operators. This simple representation can contribute to the generalization of the implementation of LFA for different types of discretizations and different problems, especially for higher order discretization methods.

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Local Fourier Analysis of Balancing Domain Decomposition By Constraints Algorithms

Cited by 11 publications

References 41 publications

Tuning Multigrid Methods with Robust Optimization

Tuning Multigrid Methods with Robust Optimization

Local Fourier Analysis of P-Multigrid for High-Order Finite Element Operators

Independence of placement for local Fourier analysis

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