Local Fourier analysis for multigrid with overlapping smoothers applied to systems of PDEs

MacLachlan, Scott; Oosterlee, Cornelis W.

doi:10.1002/nla.762

Cited by 63 publications

(106 citation statements)

References 49 publications

(68 reference statements)

Supporting

Mentioning

104

Contrasting

Order By: Relevance

“…The first set on the right-hand size has zero measure due to the definition of C in (20). Thus it remains to show that the second set also has zero measure.…”

Section: Coarseningmentioning

confidence: 98%

See 1 more Smart Citation

Fourier Analysis of Periodic Stencils in Multigrid Methods

Bolten¹,

Rittich²

2018

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. Many applications require the numerical solution of a partial differential equation (PDE), leading to large and sparse linear systems. Often a multigrid method can solve these systems efficiently. To adapt a multigrid method to a given problem, local Fourier analysis (LFA) can be used. It provides quantitative predictions about the behavior of the components of a multigrid method. In this paper we generalize LFA to handle what we call periodic stencils. An operator given by a periodic stencil has a block Fourier symbol representation. It gives a way to compute the spectral radius and norm of the operator. Furthermore block Fourier symbols can be used to find out how an operator acts on smooth/oscillatory input and whether its output will be smooth/oscillatory. This information can then be used to construct efficient smoothers and coarse grid corrections. We consider a particular PDE with jumping coefficients and show that it leads to a periodic stencil. LFA shows that the Jacobi method is a suitable smoother for this problem and an operator dependent interpolation is better than linear interpolation, as suggested by numerical experiments described in the literature. If an operator is given by an ordinary stencil, then block smoothers yield periodic stencils if the blocks correspond to rectangles in the domain. LFA shows that the block Jacobi and the red-black block Jacobi method efficiently reduce more frequencies than their pointwise versions. Further, it yields that a block smoother used in combination with aggressive coarsening can to some degree compensate for the reduced convergence rate caused by aggressive coarsening.

show abstract

“…The first set on the right-hand size has zero measure due to the definition of C in (20). Thus it remains to show that the second set also has zero measure.…”

Section: Coarseningmentioning

confidence: 98%

“…Consider the set {x ∈ X : r(a(x)) > C − ε}. We want to show that this set has measure zero, which would be a contradiction to the definition of C in (20). To this extent define the set M := {λ ∈ C : C − ε ≤ |λ| ≤ C} .…”

Section: Coarseningmentioning

confidence: 99%

Fourier Analysis of Periodic Stencils in Multigrid Methods

Bolten¹,

Rittich²

2018

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…This tool was applied to linear finite element discretizations for poroelasticity problems. Later, in [22], the analysis for such overlapping block smoothers is performed on rectangular grids for finite element discretizations of the grad-div, curl-curl and Stokes equations. In [31], we present and extend this analysis to general discretizations on triangular grids, including some special techniques for the case of edge-based discretizations.…”

Section: Local Fourier Analysis For Vanka Smoother Based Multigridmentioning

confidence: 99%

Poroelasticity problem: numerical difficulties and efficient multigrid solution

Rodrigo

2015

SeMA

View full text Add to dashboard Cite

This work contains some of the more relevant results obtained by the author regarding the numerical solution of the Biot's consolidation problem. The emphasis here is on the stable discretization and the highly efficient solution of the resulting algebraic system of equations, which is of saddle point type. On the one hand, a stabilized linear finite element scheme providing oscillation-free solutions for this model is proposed and theoretically analyzed. On the other hand, a monolithic multigrid method is considered for the solution of the resulting system of equations after discretization by using the stabilized scheme. Since this system is of saddle point type, special smoothers of "Vanka"-type have to be considered. This multigrid method is designed with the help of an special local Fourier analysis that takes into account the specific characteristics of the considered block-relaxations. Results from this analysis are presented and compared with those experimentally computed.

show abstract

“…The convergence rate can be predicted by Local Fourier Analysis (LFA), a mathematical tool that analyzes a given iterative method by freezing coefficients and neglecting boundary conditions. The LFA is used widely in the multigrid community [28,41]. We have begun to extend the 5 technique to deal with block-smoothers and aggressive coarsening in addition to the standard LFA techniques [6].…”

Section: Algorithmic Engineeringmentioning

confidence: 99%

ExaStencils: Advanced Stencil-Code Engineering

Lengauer

Apel

Bolten

et al. 2014

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. Project ExaStencils pursues a radically new approach to stencil-code engineering. Present-day stencil codes are implemented in general-purpose programming languages, such as Fortran, C, or Java, or derivates thereof, and harnesses for parallelism, such as OpenMP, OpenCL or MPI. ExaStencils favors a much more domain-specific approach with languages at several layers of abstraction, the most abstract being the mathematical formulation, the most concrete the optimized target code. At every layer, the corresponding language expresses not only computational directives but also domain knowledge of the problem and platform to be leveraged for optimization. This approach will enable a highly automated code generation at all layers and has been demonstrated successfully before in the U.S. projects FFTW and SPIRAL for certain linear transforms. The Challenges of Exascale ComputingThe performance of supercomputers is on the way from petascale to exascale. Software technology for high-performance computing has been struggling to keep up with the advances in computing power, from terascale in 1996 to petascale in 2009 on to exascale, now being only a factor of 30 away and predicted for the end of the present decade. So far, traditional host languages, such as Fortran and C, being equipped with harnesses for parallelism, such as MPI and OpenMP, have taken most of the burden, and they are being developed further with some new abstractions, notably the partitioned global address space (PGAS) memory model [1] [10]. Yet, the sequential host languages remain generalpurpose: Fortran or C or, if object orientation is desired, C ++ or Java.The step from petascale to exascale performance challenges present-day software technology much more than the advances from gigascale to terascale and terascale to petascale have. The reason is the explicit treatment of the massive parallelism inside one node of a high-performance cluster cannot be avoided any longer. That is, the cluster nodes must be manycores with high numbers of cores. The reorientation of the computer market from single cores to multicores and manycores has been observed with concern [29]. In the high-performance market, the situation is somewhat alleviated by the fact that the additional cycles that large numbers of cores provide are actually being yearned for. But, the question of how to exploit them with efficient and robust software remains.While the potential for massive parallelism on and off the chip is the single most serious challenge to exascale software technology, other challenges take on a high priority and are frequently being mentioned, such as power conservation, fault tolerance and heterogeneity of the execution platform [2]. At best, one would strive for performance portability, i.e., the ability to switch the software with ease from one platform, when it is being decommissioned, to the next, while maintaining highest performance. ExaStencils Application Domain: Stencil CodesStencil codes have extremely high significance and value for a good-sized c...

show abstract

Local Fourier analysis for multigrid with overlapping smoothers applied to systems of PDEs

Cited by 63 publications

References 49 publications

Fourier Analysis of Periodic Stencils in Multigrid Methods

Fourier Analysis of Periodic Stencils in Multigrid Methods

Poroelasticity problem: numerical difficulties and efficient multigrid solution

ExaStencils: Advanced Stencil-Code Engineering

Contact Info

Product

Resources

About