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 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
In this paper image with horizontal motion blur, vertical motion blur and angled motion blur are considered. We construct several difference schemes to the highly nonlinear term △↓.（△↓u/√｜△↓｜^2＋β） of the ...In this paper image with horizontal motion blur, vertical motion blur and angled motion blur are considered. We construct several difference schemes to the highly nonlinear term △↓.（△↓u/√｜△↓｜^2＋β） of the total variation-based image motion deblurring problem. The large nonlinear system is linearized by fixed point iteration method. An algebraic multigrid method with Krylov subspace acceleration is used to solve the corresponding linear equations as in [7]. The algorithms can restore the image very well. We give some numerical experiments to demonstrate that our difference schemes are efficient and robust.展开更多
We study a class of preconditioners to solve large-scale linear systems arising from fully implicit reservoir simulation. These methods are discussed in the framework of the auxiliary space preconditioning method for ...We study a class of preconditioners to solve large-scale linear systems arising from fully implicit reservoir simulation. These methods are discussed in the framework of the auxiliary space preconditioning method for generality. Unlike in the case of classical algebraic preconditioning methods, we take several analytical and physical considerations into account. In addition, we choose appropriate auxiliary problems to design the robust solvers herein. More importantly, our methods are user-friendly and general enough to be easily ported to existing petroleum reservoir simulators. We test the efficiency and robustness of the proposed method by applying them to a couple of benchmark problems and real-world reservoir problems. The numerical results show that our methods are both efficient and robust for large reservoir models.展开更多
In this paper we review a number of auxiliary space based preconditioners for the second order definite and semi-definite Maxwell problems discretized with the lowest order Nedelec finite elements. We discuss the para...In this paper we review a number of auxiliary space based preconditioners for the second order definite and semi-definite Maxwell problems discretized with the lowest order Nedelec finite elements. We discuss the parallel implementation of the most promising of these methods, the ones derived from the recent Hiptmair-Xu （HX） auxiliary space decomposition [Hiptmair and Xu, SIAM J. Numer. Anal., 45 （2007）, pp. 2483-2509]. An extensive set of numerical experiments demonstrate the scalability of our implementation on large-scale H（curl） problems.展开更多
We have developed efficient numerical algorithms for solving 3D steadystate Poisson-Nernst-Planck(PNP)equations with excess chemical potentials described by the classical density functional theory(cDFT).The coupled PN...We have developed efficient numerical algorithms for solving 3D steadystate Poisson-Nernst-Planck(PNP)equations with excess chemical potentials described by the classical density functional theory(cDFT).The coupled PNP equations are discretized by a finite difference scheme and solved iteratively using the Gummel method with relaxation.The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation.Then,the algebraic multigrid method is applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations.A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed,which reduces computational complexity from O(N2)to O(NlogN),where N is the number of grid points.Integrals involving the Dirac delta function are evaluated directly by coordinate transformation,which yields more accurate results compared to applying numerical quadrature to an approximated delta function.Numerical results for ion and electron transport in solid electrolyte for lithiumion(Li-ion)batteries are shown to be in good agreement with the experimental data and the results from previous studies.展开更多
We concentrate on the parallel,fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetrypreserving finite volume element discretization of the unstea...We concentrate on the parallel,fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetrypreserving finite volume element discretization of the unsteady three-temperature radiation diffusion equations in high dimensions.In this article,motivated by[M.J.Gander,S.Loisel,D.B.Szyld,SIAM J.Matrix Anal.Appl.33(2012)653–680]and[S.Nardean,M.Ferronato,A.S.Abushaikha,J.Comput.Phys.442(2021)110513],we aim to develop the additive and multiplicative Schwarz preconditioners subdividing the physical quantities rather than the underlying domain,and consider their sequential and parallel implementations using a simplified explicit decoupling factor approximation and algebraic multigrid subsolves to address such linear systems.Robustness,computational efficiencies and parallel scalabilities of the proposed approaches are numerically tested in a number of representative real-world capsule implosion benchmarks.展开更多
The concrete aggregate model is considered as a type of weakly discontinuous problem consisting of three phases:aggregates which randomly distributed in different shapes,cement paste and internal transition zone(ITZ)....The concrete aggregate model is considered as a type of weakly discontinuous problem consisting of three phases:aggregates which randomly distributed in different shapes,cement paste and internal transition zone(ITZ).Because of different shapes of aggregate and thin ITZs,a huge number of elements are often used in the finite element(FEM)analysis.In order to ensure the accuracy of the numerical solutions near the interfaces,we need to use higher-order elements.The widely used FEM softwares such as ANSYS and ABAQUS all provide the option of quadratic elements.However,they have much higher computational complexity than the linear elements.The corresponding coefficient matrix of the system of equations is a highly ill-conditioned matrix due to the large difference between three phase materials,and the convergence rate of the commonly used solving methods will deteriorate.In this paper,two types of simple and efficient preconditioners are proposed for the system of equations of the concrete aggregate models on unstructured triangle meshes by using the resulting hierarchical structure and the properties of the diagonal block matrices.The main computational cost of these preconditioners is how to efficiently solve the system of equations by using linear elements,and thus we can provide some efficient and robust solvers by calling the existing geometric-based algebraic multigrid(GAMG)methods.Since the hierarchical basis functions are used,we need not present those algebraic criterions to judge the relationships between the unknown variables and the geometric node types,and the grid transfer operators are also trivial.This makes it easy to find the linear element matrix derived directly from the fine level matrix,and thus the overall efficiency is greatly improved.The numerical results have verified the efficiency of the resulting preconditioned conjugate gradient(PCG)methods which are applied to the solution of several typical aggregate models.展开更多
Image restoration is a fundamental problem in image processing. Blind image restoration has a great value in its practical application. However, it is not an easy problem to solve due to its complexity and difficulty....Image restoration is a fundamental problem in image processing. Blind image restoration has a great value in its practical application. However, it is not an easy problem to solve due to its complexity and difficulty. In this paper, we combine our robust algorithm for known blur operator with an alternating minimization implicit iterative scheme to deal with blind deconvolution problem, recover the image and identify the point spread function（PSF）. The only assumption needed is satisfy the practical physical sense. Numerical experiments demonstrate that this minimization algorithm is efficient and robust over a wide range of PSF and have almost the same results compared with known PSF algorithm.展开更多
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.
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.
文摘In this paper image with horizontal motion blur, vertical motion blur and angled motion blur are considered. We construct several difference schemes to the highly nonlinear term △↓.（△↓u/√｜△↓｜^2＋β） of the total variation-based image motion deblurring problem. The large nonlinear system is linearized by fixed point iteration method. An algebraic multigrid method with Krylov subspace acceleration is used to solve the corresponding linear equations as in [7]. The algorithms can restore the image very well. We give some numerical experiments to demonstrate that our difference schemes are efficient and robust.
基金supported by Petro-China Joint Research Funding(Grant No.12HT1050002654)National Science Foundation of USA(Grant No.DMS-1217142)+1 种基金the Dean’s Startup FundAcademy of Mathematics and System Sciences and the State High Tech Development Plan of China(863 Program)(GrantNo.2012AA01A309)
文摘We study a class of preconditioners to solve large-scale linear systems arising from fully implicit reservoir simulation. These methods are discussed in the framework of the auxiliary space preconditioning method for generality. Unlike in the case of classical algebraic preconditioning methods, we take several analytical and physical considerations into account. In addition, we choose appropriate auxiliary problems to design the robust solvers herein. More importantly, our methods are user-friendly and general enough to be easily ported to existing petroleum reservoir simulators. We test the efficiency and robustness of the proposed method by applying them to a couple of benchmark problems and real-world reservoir problems. The numerical results show that our methods are both efficient and robust for large reservoir models.
基金This work performed under the auspices of the U.S.Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.UCRL-JRNL-237306
文摘In this paper we review a number of auxiliary space based preconditioners for the second order definite and semi-definite Maxwell problems discretized with the lowest order Nedelec finite elements. We discuss the parallel implementation of the most promising of these methods, the ones derived from the recent Hiptmair-Xu （HX） auxiliary space decomposition [Hiptmair and Xu, SIAM J. Numer. Anal., 45 （2007）, pp. 2483-2509]. An extensive set of numerical experiments demonstrate the scalability of our implementation on large-scale H（curl） problems.
基金the Materials Synthesis and Simulation across Scales(MS3)Initiative(Laboratory Directed Research and Development(LDRD)Program)at Pacific Northwest National Laboratory(PNNL).Work by GL was supported by the U.S.Department of Energy(DOE)Office of Science’s Advanced Scientific Computing Research Applied Mathematics program and work by BZ by Early Career Award Initiative(LDRD Program)at PNNL.PNNL is operated by Battelle for the DOE under Contract DE-AC05-76RL01830.
文摘We have developed efficient numerical algorithms for solving 3D steadystate Poisson-Nernst-Planck(PNP)equations with excess chemical potentials described by the classical density functional theory(cDFT).The coupled PNP equations are discretized by a finite difference scheme and solved iteratively using the Gummel method with relaxation.The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation.Then,the algebraic multigrid method is applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations.A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed,which reduces computational complexity from O(N2)to O(NlogN),where N is the number of grid points.Integrals involving the Dirac delta function are evaluated directly by coordinate transformation,which yields more accurate results compared to applying numerical quadrature to an approximated delta function.Numerical results for ion and electron transport in solid electrolyte for lithiumion(Li-ion)batteries are shown to be in good agreement with the experimental data and the results from previous studies.
基金financially supported by Hunan National Applied Mathematics Center(2020ZYT003)National Natural Science Foundation of China(11971414,62102167)+1 种基金Research Foundation of Education Bureau of Hunan(21B0162)Guangdong Basic and Applied Basic Research Foundation(2020A1515110364).
文摘We concentrate on the parallel,fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetrypreserving finite volume element discretization of the unsteady three-temperature radiation diffusion equations in high dimensions.In this article,motivated by[M.J.Gander,S.Loisel,D.B.Szyld,SIAM J.Matrix Anal.Appl.33(2012)653–680]and[S.Nardean,M.Ferronato,A.S.Abushaikha,J.Comput.Phys.442(2021)110513],we aim to develop the additive and multiplicative Schwarz preconditioners subdividing the physical quantities rather than the underlying domain,and consider their sequential and parallel implementations using a simplified explicit decoupling factor approximation and algebraic multigrid subsolves to address such linear systems.Robustness,computational efficiencies and parallel scalabilities of the proposed approaches are numerically tested in a number of representative real-world capsule implosion benchmarks.
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.11601462)the Hunan Provincial Natural Science Foundation of China(Grant No.14JJ2063)the Scientific Research Fund of Hunan Provincial Education Department(Grant No.15A183).
文摘The concrete aggregate model is considered as a type of weakly discontinuous problem consisting of three phases:aggregates which randomly distributed in different shapes,cement paste and internal transition zone(ITZ).Because of different shapes of aggregate and thin ITZs,a huge number of elements are often used in the finite element(FEM)analysis.In order to ensure the accuracy of the numerical solutions near the interfaces,we need to use higher-order elements.The widely used FEM softwares such as ANSYS and ABAQUS all provide the option of quadratic elements.However,they have much higher computational complexity than the linear elements.The corresponding coefficient matrix of the system of equations is a highly ill-conditioned matrix due to the large difference between three phase materials,and the convergence rate of the commonly used solving methods will deteriorate.In this paper,two types of simple and efficient preconditioners are proposed for the system of equations of the concrete aggregate models on unstructured triangle meshes by using the resulting hierarchical structure and the properties of the diagonal block matrices.The main computational cost of these preconditioners is how to efficiently solve the system of equations by using linear elements,and thus we can provide some efficient and robust solvers by calling the existing geometric-based algebraic multigrid(GAMG)methods.Since the hierarchical basis functions are used,we need not present those algebraic criterions to judge the relationships between the unknown variables and the geometric node types,and the grid transfer operators are also trivial.This makes it easy to find the linear element matrix derived directly from the fine level matrix,and thus the overall efficiency is greatly improved.The numerical results have verified the efficiency of the resulting preconditioned conjugate gradient(PCG)methods which are applied to the solution of several typical aggregate models.
文摘Image restoration is a fundamental problem in image processing. Blind image restoration has a great value in its practical application. However, it is not an easy problem to solve due to its complexity and difficulty. In this paper, we combine our robust algorithm for known blur operator with an alternating minimization implicit iterative scheme to deal with blind deconvolution problem, recover the image and identify the point spread function（PSF）. The only assumption needed is satisfy the practical physical sense. Numerical experiments demonstrate that this minimization algorithm is efficient and robust over a wide range of PSF and have almost the same results compared with known PSF algorithm.
