Mesenchymal stem/stromal cells are potential optimal cell sources for stem cell therapies,and pretreatment has proven to enhance cell vitality and function.In a recent publication,Li et al explored a new combination o...Mesenchymal stem/stromal cells are potential optimal cell sources for stem cell therapies,and pretreatment has proven to enhance cell vitality and function.In a recent publication,Li et al explored a new combination of pretreatment condi-tions.Here,we present an editorial to comment on their work and provide our view on mesenchymal stem/stromal cell precondition.展开更多
For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such a...For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.展开更多
In this paper we give an overview of the present state of fast solvers for the solution of the incompressible Navier-Stokes equations discretized by the finite element method and linearized by Newton or Picard's m...In this paper we give an overview of the present state of fast solvers for the solution of the incompressible Navier-Stokes equations discretized by the finite element method and linearized by Newton or Picard's method.It is shown that block preconditioners form an excellent approach for the solution,however if the grids are not to fine preconditioning with a Saddle point ILU matrix(SILU) may be an attractive alternative. The applicability of all methods to stabilized elements is investigated.In case of the stand-alone Stokes equations special preconditioners increase the efficiency considerably.展开更多
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
Explicit Exact and Approximate Inverse Preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is ...Explicit Exact and Approximate Inverse Preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is discussed. The second order iterative methods behave quite similar to first order methods and the development of efficient preconditioners for solving the original linear system is a decisive factor for making the second order iterative methods superior to the first order iterative methods. Adaptive preconditioned Conjugate Gradient methods using explicit approximate preconditioners for solving efficiently large sparse systems of algebraic equations are also presented. The generalized Approximate Inverse Matrix techniques can be efficiently used in conjunction with explicit iterative schemes leading to effective composite semi-direct solution methods for solving large linear systems of algebraic equations.展开更多
Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential ...Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.展开更多
In this paper,we develop subspace correction preconditioners for discontinuous Galerkin(DG)discretizations of elliptic problems with hp-refnement.These preconditioners are based on the decomposition of the DG fnite el...In this paper,we develop subspace correction preconditioners for discontinuous Galerkin(DG)discretizations of elliptic problems with hp-refnement.These preconditioners are based on the decomposition of the DG fnite element space into a conforming subspace,and a set of small nonconforming edge spaces.The conforming subspace is preconditioned using a matrix-free low-order refned technique,which in this work,we extend to the hprefnement context using a variational restriction approach.The condition number of the resulting linear system is independent of the granularity of the mesh h,and the degree of the polynomial approximation p.The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees.Numerical examples are shown on several test cases involving adaptively and randomly refned meshes,using both the symmetric interior penalty method and the second method of Bassi and Rebay(BR2).展开更多
In this paper,we design an efficient domain decomposition(DD)preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems.By proper equivalent alge...In this paper,we design an efficient domain decomposition(DD)preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems.By proper equivalent algebraic operations,the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonel matrix efficiently.Actually,the first block of this block-diagonal matrix corresponds to a multiscale H(div)problem,and thus,the direct inverse of this block is unpractical and unstable for the large-scale problem.To remedy this issue,a two-level overlapping DD preconditioner is proposed for this//(div)problem.Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis(e.g.,Raviart-Thomas element)on the coarse grid.The condition number of our preconditioned DD method for this multiscale H(div)system is bounded by C(1+务)(1+log4(^)),where 6 denotes the width of overlapping region,and H,h are the typical sizes of the subdomain and fine mesh.Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.展开更多
A computational fluid dynamics(CFD)solver for a GPU/CPU heterogeneous architecture parallel computing platform is developed to simulate incompressible flows on billion-level grid points.To solve the Poisson equation,t...A computational fluid dynamics(CFD)solver for a GPU/CPU heterogeneous architecture parallel computing platform is developed to simulate incompressible flows on billion-level grid points.To solve the Poisson equation,the conjugate gradient method is used as a basic solver,and a Chebyshev method in combination with a Jacobi sub-preconditioner is used as a preconditioner.The developed CFD solver shows good performance on parallel efficiency,which exceeds 90%in the weak-scalability test when the number of grid points allocated to each GPU card is greater than 2083.In the acceleration test,it is found that running a simulation with 10403 grid points on 125 GPU cards accelerates by 203.6x over the same number of CPU cores.The developed solver is then tested in the context of a two-dimensional lid-driven cavity flow and three-dimensional Taylor-Green vortex flow.The results are consistent with previous results in the literature.展开更多
In this paper,a two-step semi-regularized Hermitian and skew-Hermitian splitting(SHSS)iteration method is constructed by introducing a regularization matrix in the(1,1)-block of the first iteration step,to solve the s...In this paper,a two-step semi-regularized Hermitian and skew-Hermitian splitting(SHSS)iteration method is constructed by introducing a regularization matrix in the(1,1)-block of the first iteration step,to solve the saddle-point linear system.By carefully selecting two different regularization matrices,two kinds of SHSS preconditioners are proposed to accelerate the convergence rates of the Krylov subspace iteration methods.Theoretical analysis about the eigenvalue distribution demonstrates that the proposed SHSS preconditioners can make the eigenvalues of the corresponding preconditioned matrices be clustered around 1 and uniformly bounded away from 0.The eigenvector distribution and the upper bound on the degree of the minimal polynomial of the SHSS-preconditioned matrices indicate that the SHSS-preconditioned Krylov subspace iterative methods can converge to the true solution within finite steps in exact arithmetic.In addition,the numerical example derived from the optimal control problem shows that the SHSS preconditioners can significantly improve the convergence speeds of the Krylov subspace iteration methods,and their convergence rates are independent of the discrete mesh size.展开更多
This paper proposes a two-parameter block triangular splitting(TPTS)preconditioner for the general block two-by-two linear systems.The eigenvalues of the corresponding preconditioned matrix are proved to cluster aroun...This paper proposes a two-parameter block triangular splitting(TPTS)preconditioner for the general block two-by-two linear systems.The eigenvalues of the corresponding preconditioned matrix are proved to cluster around 0 or 1 under mild conditions.The limited numerical results show that the TPTS preconditioner is more efficient than the classic block-diagonal and block-triangular preconditioners when applied to the flexible generalized minimal residual(FGMRES)method.展开更多
Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibi...Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibit a saddle point structure. To achieve this task, a Newton-based root-finding algorithm is usually employed which in turn necessitates to solve a saddle point system at every Newton iteration. The involved linear systems being large scale and ill-conditioned, effective linear solvers must be implemented. Here, we develop and test several methods for solving the saddle point systems, considering in particular the LU factorization, as direct approach, and the preconditioned generalized minimal residual (ΡGMRES) solver, an iterative approach. We apply the various solvers within the root-finding algorithm for Flow over backward facing step systems. The particularity of Flow over backward facing step system is an interesting case for studying the performance and solution strategy of a turbulence model. In this case, the flow is subjected to a sudden increase of cross-sectional area, resulting in a separation of flow starting at the point of expansion, making the system of differential equations particularly stiff. We assess the performance of the direct and iterative solvers in terms of computational time, numbers of Newton iterations and time steps.展开更多
文摘Mesenchymal stem/stromal cells are potential optimal cell sources for stem cell therapies,and pretreatment has proven to enhance cell vitality and function.In a recent publication,Li et al explored a new combination of pretreatment condi-tions.Here,we present an editorial to comment on their work and provide our view on mesenchymal stem/stromal cell precondition.
文摘For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.
文摘In this paper we give an overview of the present state of fast solvers for the solution of the incompressible Navier-Stokes equations discretized by the finite element method and linearized by Newton or Picard's method.It is shown that block preconditioners form an excellent approach for the solution,however if the grids are not to fine preconditioning with a Saddle point ILU matrix(SILU) may be an attractive alternative. The applicability of all methods to stabilized elements is investigated.In case of the stand-alone Stokes equations special preconditioners increase the efficiency considerably.
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.
文摘Explicit Exact and Approximate Inverse Preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is discussed. The second order iterative methods behave quite similar to first order methods and the development of efficient preconditioners for solving the original linear system is a decisive factor for making the second order iterative methods superior to the first order iterative methods. Adaptive preconditioned Conjugate Gradient methods using explicit approximate preconditioners for solving efficiently large sparse systems of algebraic equations are also presented. The generalized Approximate Inverse Matrix techniques can be efficiently used in conjunction with explicit iterative schemes leading to effective composite semi-direct solution methods for solving large linear systems of algebraic equations.
文摘Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.
基金This work was performed under the auspices of the U.S.Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and was partially supported by the LLNL-LDRD Program under Project No.20-ERD-002(LLNL-JRNL-814157).
文摘In this paper,we develop subspace correction preconditioners for discontinuous Galerkin(DG)discretizations of elliptic problems with hp-refnement.These preconditioners are based on the decomposition of the DG fnite element space into a conforming subspace,and a set of small nonconforming edge spaces.The conforming subspace is preconditioned using a matrix-free low-order refned technique,which in this work,we extend to the hprefnement context using a variational restriction approach.The condition number of the resulting linear system is independent of the granularity of the mesh h,and the degree of the polynomial approximation p.The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees.Numerical examples are shown on several test cases involving adaptively and randomly refned meshes,using both the symmetric interior penalty method and the second method of Bassi and Rebay(BR2).
文摘In this paper,we design an efficient domain decomposition(DD)preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems.By proper equivalent algebraic operations,the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonel matrix efficiently.Actually,the first block of this block-diagonal matrix corresponds to a multiscale H(div)problem,and thus,the direct inverse of this block is unpractical and unstable for the large-scale problem.To remedy this issue,a two-level overlapping DD preconditioner is proposed for this//(div)problem.Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis(e.g.,Raviart-Thomas element)on the coarse grid.The condition number of our preconditioned DD method for this multiscale H(div)system is bounded by C(1+务)(1+log4(^)),where 6 denotes the width of overlapping region,and H,h are the typical sizes of the subdomain and fine mesh.Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.
基金supported by the National Natural Science Foundation of China (NSFC)Basic Science Center Program for Multiscale Problems in Nonlinear Mechanics’(Grant No. 11988102)NSFC project (Grant No. 11972038)
文摘A computational fluid dynamics(CFD)solver for a GPU/CPU heterogeneous architecture parallel computing platform is developed to simulate incompressible flows on billion-level grid points.To solve the Poisson equation,the conjugate gradient method is used as a basic solver,and a Chebyshev method in combination with a Jacobi sub-preconditioner is used as a preconditioner.The developed CFD solver shows good performance on parallel efficiency,which exceeds 90%in the weak-scalability test when the number of grid points allocated to each GPU card is greater than 2083.In the acceleration test,it is found that running a simulation with 10403 grid points on 125 GPU cards accelerates by 203.6x over the same number of CPU cores.The developed solver is then tested in the context of a two-dimensional lid-driven cavity flow and three-dimensional Taylor-Green vortex flow.The results are consistent with previous results in the literature.
基金the National Natural Science Foundation of China(No.12001048)R&D Program of Beijing Municipal Education Commission(No.KM202011232019),China.
文摘In this paper,a two-step semi-regularized Hermitian and skew-Hermitian splitting(SHSS)iteration method is constructed by introducing a regularization matrix in the(1,1)-block of the first iteration step,to solve the saddle-point linear system.By carefully selecting two different regularization matrices,two kinds of SHSS preconditioners are proposed to accelerate the convergence rates of the Krylov subspace iteration methods.Theoretical analysis about the eigenvalue distribution demonstrates that the proposed SHSS preconditioners can make the eigenvalues of the corresponding preconditioned matrices be clustered around 1 and uniformly bounded away from 0.The eigenvector distribution and the upper bound on the degree of the minimal polynomial of the SHSS-preconditioned matrices indicate that the SHSS-preconditioned Krylov subspace iterative methods can converge to the true solution within finite steps in exact arithmetic.In addition,the numerical example derived from the optimal control problem shows that the SHSS preconditioners can significantly improve the convergence speeds of the Krylov subspace iteration methods,and their convergence rates are independent of the discrete mesh size.
基金the National Natural Science Foundation of China under Grant Nos.61273311 and 61803247.
文摘This paper proposes a two-parameter block triangular splitting(TPTS)preconditioner for the general block two-by-two linear systems.The eigenvalues of the corresponding preconditioned matrix are proved to cluster around 0 or 1 under mild conditions.The limited numerical results show that the TPTS preconditioner is more efficient than the classic block-diagonal and block-triangular preconditioners when applied to the flexible generalized minimal residual(FGMRES)method.
文摘Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibit a saddle point structure. To achieve this task, a Newton-based root-finding algorithm is usually employed which in turn necessitates to solve a saddle point system at every Newton iteration. The involved linear systems being large scale and ill-conditioned, effective linear solvers must be implemented. Here, we develop and test several methods for solving the saddle point systems, considering in particular the LU factorization, as direct approach, and the preconditioned generalized minimal residual (ΡGMRES) solver, an iterative approach. We apply the various solvers within the root-finding algorithm for Flow over backward facing step systems. The particularity of Flow over backward facing step system is an interesting case for studying the performance and solution strategy of a turbulence model. In this case, the flow is subjected to a sudden increase of cross-sectional area, resulting in a separation of flow starting at the point of expansion, making the system of differential equations particularly stiff. We assess the performance of the direct and iterative solvers in terms of computational time, numbers of Newton iterations and time steps.
