Smoothed aggregation algebraic multigrid for stochastic PDE problems with layered materials

2014 ◽  
Vol 21 (2) ◽  
pp. 239-255 ◽  
Author(s):  
Marian Brezina ◽  
Alireza Doostan ◽  
Tom Manteuffel ◽  
Steve McCormick ◽  
John Ruge
Keyword(s):  
Layered Materials ◽  
Algebraic Multigrid ◽  
Stochastic Pde ◽  
Smoothed Aggregation

scholarly journals Algebraic multigrid methods

Acta Numerica ◽  
2017 ◽  
Vol 26 ◽  
pp. 591-721 ◽  
Author(s):  
Jinchao Xu ◽  
Ludmil Zikatanov
Keyword(s):  
Minimization Problem ◽  
Large Scale ◽  
Multigrid Method ◽  
Approximate Solutions ◽  
Multigrid Methods ◽  
Algebraic Multigrid ◽  
Unified Framework ◽  
Smoothed Aggregation ◽  
Algebraic Multigrid Methods ◽  
Coarse Space

This paper provides an overview of AMG methods for solving large-scale systems of equations, such as those from discretizations of partial differential equations. AMG is often understood as the acronym of ‘algebraic multigrid’, but it can also be understood as ‘abstract multigrid’. Indeed, we demonstrate in this paper how and why an algebraic multigrid method can be better understood at a more abstract level. In the literature, there are many different algebraic multigrid methods that have been developed from different perspectives. In this paper we try to develop a unified framework and theory that can be used to derive and analyse different algebraic multigrid methods in a coherent manner. Given a smoother$R$for a matrix$A$, such as Gauss–Seidel or Jacobi, we prove that the optimal coarse space of dimension$n_{c}$is the span of the eigenvectors corresponding to the first$n_{c}$eigenvectors$\bar{R}A$(with$\bar{R}=R+R^{T}-R^{T}AR$). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with$\bar{R}A$, and demonstrate that coarse spaces of most existing AMG methods can be viewed as approximate solutions of this trace-minimization problem. Furthermore, we provide a general approach to the construction of quasi-optimal coarse spaces, and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG and spectral AMGe.

scholarly journals An improved convergence analysis of smoothed aggregation algebraic multigrid

2011 ◽  
Vol 19 (3) ◽  
pp. 441-469 ◽  
Author(s):  
Marian Brezina ◽  
Petr Vaněk ◽  
Panayot S. Vassilevski
Keyword(s):  
Convergence Analysis ◽  
Algebraic Multigrid ◽  
Smoothed Aggregation

Convergence of algebraic multigrid based on smoothed aggregation

2001 ◽  
Vol 88 (3) ◽  
pp. 559-579 ◽  
Author(s):  
Petr Van\vek ◽  
Marian Brezina ◽  
Jan Mandel
Keyword(s):  
Algebraic Multigrid ◽  
Smoothed Aggregation

Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems

Computing ◽  
1996 ◽  
Vol 56 (3) ◽  
pp. 179-196 ◽  
Author(s):  
P. Vaněk ◽  
J. Mandel ◽  
M. Brezina
Keyword(s):  
Fourth Order ◽  
Elliptic Problems ◽  
Algebraic Multigrid ◽  
Smoothed Aggregation

scholarly journals Comparison of Algebraic Multigrid Preconditioners for Solving Helmholtz Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Dandan Chen ◽  
Ting-Zhu Huang ◽  
Liang Li
Keyword(s):  
Krylov Subspace ◽  
Spectral Estimation ◽  
Solution Process ◽  
Krylov Subspace Methods ◽  
Good Choice ◽  
Algebraic Multigrid ◽  
Subspace Methods ◽  
Helmholtz Equations ◽  
Smoothed Aggregation ◽  
Aggregation Technique

An algebraic multigrid (AMG) with aggregation technique to coarsen is applied to construct a better preconditioner for solving Helmholtz equations in this paper. The solution process consists of constructing the preconditioner by AMG and solving the preconditioned Helmholtz problems by Krylov subspace methods. In the setup process of AMG, we employ the double pairwise aggregation (DPA) scheme firstly proposed by Y. Notay (2006) as the coarsening method. We compare it with the smoothed aggregation algebraic multigrid and meanwhile show shifted Laplacian preconditioners. According to numerical results, we find that DPA algorithm is a good choice in AMG for Helmholtz equations in reducing time and memory. Spectral estimation of system preconditioned by the three methods and the influence of second-order and fourth-order accurate discretizations on the three techniques are also considered.

Finite Element Resistivity Forward Modeling Using Algebraic Multigrid Preconditioned Conjugate Gradient Method

2012 ◽  
Vol 249-250 ◽  
pp. 792-797
Author(s):  
Gui Hong Zou ◽  
Hua Qing Liang
Keyword(s):  
Finite Element ◽  
Tensor Product ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Forward Modeling ◽  
Algebraic Multigrid ◽  
Preconditioned Conjugate Gradient ◽  
Uniform Mesh ◽  
Smoothed Aggregation

An algebraic multigrid by smoothed aggregation preconditioned conjugate gradient method is developed to solve the liner system arising from 3-D direct current finite element resistivity forward modeling. The algorithm combines the efficiency of algebraic multigrid method and the stability of conjugate gradient method. Algebraic multigrid by smoothed aggregation keep in high-efficiency while simulation using local quasi-uniform mesh and its convergence effect will reduce while numerical modeling using anisotropic stretched grids. However tensor product non-equidistant mesh, a kind of anisotropic stretched grids, is often used in 3-D direct current resistivity forward modeling. In order to improve this situation, a factor is added to guide correct aggregation. Consequently, a typical example is used to prove that the improvement is the right. Finally, it is natural to conclude that the algorithm suggested in this paper is efficient and robust whether simulation using local quasi-uniform mesh or tensor product non-equidistant mesh

High Resolution Electron Microscopy of internal interfaces in layered materials

1990 ◽  
Vol 48 (4) ◽  
pp. 320-321
Author(s):  
Yoichi Ishida ◽  
Hideki Ichinose ◽  
Yutaka Takahashi ◽  
Jin-yeh Wang
Keyword(s):  
Grain Boundaries ◽  
Mirror Symmetry ◽  
Functional Materials ◽  
High Resolution Electron Microscopy ◽  
Layered Materials ◽  
Plane Boundary ◽  
Chemical Information ◽  
Layer Plane ◽  
High Tc ◽  
Cubic Metals

Layered materials draw attention in recent years in response to the world-wide drive to discover new functional materials. High-Tc superconducting oxide is one example. Internal interfaces in such layered materials differ significantly from those of cubic metals. They are often parallel to the layer of the neighboring crystals in sintered samples(layer plane boundary), while periodically ordered interfaces with the two neighboring crystals in mirror symmetry to each other are relatively rare. Consequently, the atomistic features of the interface differ significantly from those of cubic metals. In this paper grain boundaries in sintered high-Tc superconducting oxides, joined interfaces between engineering ceramics with metals, and polytype interfaces in vapor-deposited bicrystal are examined to collect atomic information of the interfaces in layered materials. The analysis proved that they are not neccessarily more complicated than that of simple grain boundaries in cubic metals. The interfaces are majorly layer plane type which is parallel to the compound layer. Secondly, chemical information is often available, which helps the interpretation of the interface atomic structure.

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