In this article algebraic multigrid as preconditioners are designed, with biorthogonal wavelets, as intergrid operators for the Krylov subspace iterative methods. Construction of hierarchy of matrices in algebraic mul...In this article algebraic multigrid as preconditioners are designed, with biorthogonal wavelets, as intergrid operators for the Krylov subspace iterative methods. Construction of hierarchy of matrices in algebraic multigrid context is based on lowpass filter version of Wavelet Transform. The robustness and efficiency of this new approach is tested by applying it to large sparse, unsymmetric and ill-conditioned matrices from Tim Davis collection of sparse matrices. Proposed preconditioners have potential in reducing cputime, operator complexity and storage space of algebraic multigrid V-cycle and meet the desired accuracy of solution compared with that of orthogonal wavelets.展开更多
The paper investigates the robustness and parallel scaling properties of a novel physical factorization preconditioner with algebraic multigrid subsolves in the iterative solution of a cell-centered finite volume disc...The paper investigates the robustness and parallel scaling properties of a novel physical factorization preconditioner with algebraic multigrid subsolves in the iterative solution of a cell-centered finite volume discretization of the threedimensional multi-group radiation diffusion equations.The key idea is to take advantage of a particular kind of block factorization of the resulting system matrix and approximate the left-hand block matrix selectively spurred by parallel processing considerations.The spectral property of the preconditioned matrix is then analyzed.The practical strategy is considered sequentially and in parallel.Finally,numerical results illustrate the numerical robustness,computational efficiency and parallel strong and weak scalabilities over the real-world structured and unstructured coupled problems,showing its competitiveness with many existing block preconditioners.展开更多
Consider an AMG for the linear system Au=f. Up to now, only the uniform convergence of two-level AMG is proved for symmetric and positive definite L-matrices with weak diagonal dominance. Using the new form (1), we ex...Consider an AMG for the linear system Au=f. Up to now, only the uniform convergence of two-level AMG is proved for symmetric and positive definite L-matrices with weak diagonal dominance. Using the new form (1), we extend the results in [1] to the case that A is a general symmetric and positive definite matrix with weak diagonal dominance. In the following, we shall use the same notations as in [1].展开更多
We present and analyze a robust preconditioned conjugate gradient method for the higher order Lagrangian finite element systems of a class of elliptic problems. An auxiliary linear element stiffness matrix is chosen t...We present and analyze a robust preconditioned conjugate gradient method for the higher order Lagrangian finite element systems of a class of elliptic problems. An auxiliary linear element stiffness matrix is chosen to be the preconditioner for higher order finite elements. Then an algebraic multigrid method of linear finite element is applied for solving the preconditioner. The optimal condition number which is independent of the mesh size is obtained. Numerical experiments confirm the efficiency of the algorithm.展开更多
In this paper,we discuss an algebraic multigrid(AMG)method for nearly incompressible elasticity problems in two-dimensions.First,a two-level method is proposed by analyzing the relationship between the linear finite e...In this paper,we discuss an algebraic multigrid(AMG)method for nearly incompressible elasticity problems in two-dimensions.First,a two-level method is proposed by analyzing the relationship between the linear finite element space and the quartic finite element space.By choosing different smoothers,we obtain two types of two-level methods,namely TL-GS and TL-BGS.The theoretical analysis and numerical results show that the convergence rates of TL-GS and TL-BGS are independent of the mesh size and the Young’s modulus,and the convergence of the latter is greatly improved on the order p.However the convergence of both methods still depends on the Poisson’s ratio.To fix this,we obtain a coarse level matrix with less rigidity based on selective reduced integration(SRI)method and get some types of two-level methods by combining different smoothers.With the existing AMG method used as a solver on the first coarse level,an AMG method can be finally obtained.Numerical results show that the resulting AMG method has better efficiency for nearly incompressible elasticity problems.展开更多
This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology,including compatible relaxation and algebraic dis-tances for defining effective coarsening strategies,the least s...This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology,including compatible relaxation and algebraic dis-tances for defining effective coarsening strategies,the least squares method for com-puting accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in the least squares process.We review some recent re-search in the development,analysis and application of bootstrap algebraic multigrid and point to open problems in these areas.Results from our previous research as well as some new results for some model diffusion problems with highly oscillatory diffusion coefficient are presented to illustrate the basic components of the BAMG algorithm.展开更多
We present an algebraic version of an iterative multigrid method for obstacle problems,called projected algebraic multigrid(PAMG)here.We show that classical algebraic multigrid algorithms can easily be extended to dea...We present an algebraic version of an iterative multigrid method for obstacle problems,called projected algebraic multigrid(PAMG)here.We show that classical algebraic multigrid algorithms can easily be extended to deal with this kind of problem.This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising,for example,in financial engineering.展开更多
This article is to discuss the bilinear and linear immersed finite element(IFE)solutions generated from the algebraic multigrid solver for both stationary and moving interface problems.For the numerical methods based ...This article is to discuss the bilinear and linear immersed finite element(IFE)solutions generated from the algebraic multigrid solver for both stationary and moving interface problems.For the numerical methods based on finite difference formulation and a structured mesh independent of the interface,the stiffness matrix of the linear system is usually not symmetric positive-definite,which demands extra efforts to design efficient multigrid methods.On the other hand,the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite.Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems.The numerical examples demonstrate the features of the proposed algorithms,including the optimal convergence in both L 2 and semi-H1 norms of the IFE-AMG solutions,the high efficiency with proper choice of the components and parameters of AMG,the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems,and the relationship between the cost and the moving interface location.展开更多
Parallel algebraic multigrid methods in gyrokinetic turbulence simulations are presented.Discretized equations of the elliptic operator−■^(2)u+αu=f(with bothα=0 and α≠0)are ubiquitous in magnetically confined fus...Parallel algebraic multigrid methods in gyrokinetic turbulence simulations are presented.Discretized equations of the elliptic operator−■^(2)u+αu=f(with bothα=0 and α≠0)are ubiquitous in magnetically confined fusion plasma applications.Whenαis equal to zero a“pure”Laplacian or Poisson equation results and whenαis greater than zero a so called Helmholtz equation is produced.Taking a gyrokinetic turbulence simulation model as a testbed,we investigate the performance characteristics of basic classes of linear solvers(direct,one-level iterative,and multilevel iterative methods)on 2D unstructured finite element method(FEM)problems for both the Poisson and the Helmholtz equations.展开更多
文摘In this article algebraic multigrid as preconditioners are designed, with biorthogonal wavelets, as intergrid operators for the Krylov subspace iterative methods. Construction of hierarchy of matrices in algebraic multigrid context is based on lowpass filter version of Wavelet Transform. The robustness and efficiency of this new approach is tested by applying it to large sparse, unsymmetric and ill-conditioned matrices from Tim Davis collection of sparse matrices. Proposed preconditioners have potential in reducing cputime, operator complexity and storage space of algebraic multigrid V-cycle and meet the desired accuracy of solution compared with that of orthogonal wavelets.
基金supported by the National Natural Science Foundation of China(Grant 11971414)Hunan National Applied Mathematics Center(Grant 2020ZYT003)the Research Foundation of Education Bureau of Hunan(Grant 21B0162).
文摘The paper investigates the robustness and parallel scaling properties of a novel physical factorization preconditioner with algebraic multigrid subsolves in the iterative solution of a cell-centered finite volume discretization of the threedimensional multi-group radiation diffusion equations.The key idea is to take advantage of a particular kind of block factorization of the resulting system matrix and approximate the left-hand block matrix selectively spurred by parallel processing considerations.The spectral property of the preconditioned matrix is then analyzed.The practical strategy is considered sequentially and in parallel.Finally,numerical results illustrate the numerical robustness,computational efficiency and parallel strong and weak scalabilities over the real-world structured and unstructured coupled problems,showing its competitiveness with many existing block preconditioners.
基金Project supported by the National Natural Science Foundation of China.
文摘Consider an AMG for the linear system Au=f. Up to now, only the uniform convergence of two-level AMG is proved for symmetric and positive definite L-matrices with weak diagonal dominance. Using the new form (1), we extend the results in [1] to the case that A is a general symmetric and positive definite matrix with weak diagonal dominance. In the following, we shall use the same notations as in [1].
文摘We present and analyze a robust preconditioned conjugate gradient method for the higher order Lagrangian finite element systems of a class of elliptic problems. An auxiliary linear element stiffness matrix is chosen to be the preconditioner for higher order finite elements. Then an algebraic multigrid method of linear finite element is applied for solving the preconditioner. The optimal condition number which is independent of the mesh size is obtained. Numerical experiments confirm the efficiency of the algorithm.
基金supported in part by NSF-10771178 and NSF-10672138 in Chinathe Basic Research Program of China under the grant 2005CB321702+1 种基金the Key Project of Chinese Ministry of Education and the Scientific Research Fund of Hunan Provincial Education Department(208093,07A068)the Provincial Natural Science Foundation of Hunan(07JJ6004)。
文摘In this paper,we discuss an algebraic multigrid(AMG)method for nearly incompressible elasticity problems in two-dimensions.First,a two-level method is proposed by analyzing the relationship between the linear finite element space and the quartic finite element space.By choosing different smoothers,we obtain two types of two-level methods,namely TL-GS and TL-BGS.The theoretical analysis and numerical results show that the convergence rates of TL-GS and TL-BGS are independent of the mesh size and the Young’s modulus,and the convergence of the latter is greatly improved on the order p.However the convergence of both methods still depends on the Poisson’s ratio.To fix this,we obtain a coarse level matrix with less rigidity based on selective reduced integration(SRI)method and get some types of two-level methods by combining different smoothers.With the existing AMG method used as a solver on the first coarse level,an AMG method can be finally obtained.Numerical results show that the resulting AMG method has better efficiency for nearly incompressible elasticity problems.
文摘This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology,including compatible relaxation and algebraic dis-tances for defining effective coarsening strategies,the least squares method for com-puting accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in the least squares process.We review some recent re-search in the development,analysis and application of bootstrap algebraic multigrid and point to open problems in these areas.Results from our previous research as well as some new results for some model diffusion problems with highly oscillatory diffusion coefficient are presented to illustrate the basic components of the BAMG algorithm.
文摘We present an algebraic version of an iterative multigrid method for obstacle problems,called projected algebraic multigrid(PAMG)here.We show that classical algebraic multigrid algorithms can easily be extended to deal with this kind of problem.This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising,for example,in financial engineering.
基金supported by DOE grant DE-FE0009843National Natural Science Foundation of China(11175052)GRF of HKSAR#501012 and NSERC(Canada).
文摘This article is to discuss the bilinear and linear immersed finite element(IFE)solutions generated from the algebraic multigrid solver for both stationary and moving interface problems.For the numerical methods based on finite difference formulation and a structured mesh independent of the interface,the stiffness matrix of the linear system is usually not symmetric positive-definite,which demands extra efforts to design efficient multigrid methods.On the other hand,the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite.Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems.The numerical examples demonstrate the features of the proposed algorithms,including the optimal convergence in both L 2 and semi-H1 norms of the IFE-AMG solutions,the high efficiency with proper choice of the components and parameters of AMG,the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems,and the relationship between the cost and the moving interface location.
文摘Parallel algebraic multigrid methods in gyrokinetic turbulence simulations are presented.Discretized equations of the elliptic operator−■^(2)u+αu=f(with bothα=0 and α≠0)are ubiquitous in magnetically confined fusion plasma applications.Whenαis equal to zero a“pure”Laplacian or Poisson equation results and whenαis greater than zero a so called Helmholtz equation is produced.Taking a gyrokinetic turbulence simulation model as a testbed,we investigate the performance characteristics of basic classes of linear solvers(direct,one-level iterative,and multilevel iterative methods)on 2D unstructured finite element method(FEM)problems for both the Poisson and the Helmholtz equations.
