A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this...A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this paper is to propose some efficient and accurate numerical methods for the computational solution of one-dimensional continuous basic models for the growth and control of brain tumors. After computing the analytical solution, we construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization. Then, we investigate the convergence behavior of Conjugate gradient and generalized minimum residual as Krylov subspace methods to solve the tridiagonal toeplitz matrix system derived.展开更多
In this study,a one-dimensional two-phase flow four-equation model was developed to simulate the water faucet problem.The performance of six different Krylov subspace methods,namely the generalized minimal residual(GM...In this study,a one-dimensional two-phase flow four-equation model was developed to simulate the water faucet problem.The performance of six different Krylov subspace methods,namely the generalized minimal residual(GMRES),transpose-free quasi-minimal residual,quasi-minimal residual,conjugate gradient squared,biconjugate gradient stabilized,and biconjugate gradient,was evaluated with and without the application of an incomplete LU(ILU)factorization preconditioner for solving the water faucet problem.The simulation results indicate that using the ILU preconditioner with the Krylov subspace methods produces better convergence performance than that without the ILU preconditioner.Only the GMRES demonstrated an acceptable convergence performance under the Krylov subspace methods without the preconditioner.The velocity and pressure distribution in the water faucet problem could be determined using the Krylov subspace methods with an ILU preconditioner,while GMRES could determine it without the need for a preconditioner.However,there are significant advantages of using an ILU preconditioner with the GMRES in terms of efficiency.The different Krylov subspace methods showed similar performance in terms of computational efficiency under the application of the ILU preconditioner.展开更多
Krylov subspace projection methods are known to be highly efficient for solving large linear systems. Many different versions arise from different choices to the left and right subspaces. These methods were classified...Krylov subspace projection methods are known to be highly efficient for solving large linear systems. Many different versions arise from different choices to the left and right subspaces. These methods were classified into two groups in terms of the different forms of matrix H-m, the main properties in applications and the new versions of these two types of methods were briefly reviewed, then one of the most efficient versions, GMRES method was applied to oil reservoir simulation. The block Pseudo-Elimination method was used to generate the preconditioned matrix. Numerical results show much better performance of this preconditioned techniques and the GMRES method than that of preconditioned ORTHMIN method, which is now in use in oil reservoir simulation. Finally, some limitations of Krylov subspace methods and some potential improvements to this type of methods are further presented.展开更多
In this paper we reconsider the range-restricted GMRES (RRGMRES) method for solving nonsymmetric linear systems. We first review an important result for the usual GMRES method. Then we give an example to show that the...In this paper we reconsider the range-restricted GMRES (RRGMRES) method for solving nonsymmetric linear systems. We first review an important result for the usual GMRES method. Then we give an example to show that the range-restricted GMRES method does not admit such a result. Finally, we give a modified result for the range-restricted GMRES method. We point out that the modified version can be used to show that the range-restricted GMRES method is also a regularization method for solving linear ill-posed problems.展开更多
An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors , where , N being very large. Such sequences arise, for example, in the solution o...An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors , where , N being very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences converge more quickly is to apply to them vector extrapolation methods. Two types of methods exist in the literature: polynomial type methods and epsilon algorithms. In most applications, the polynomial type methods have proved to be superior convergence accelerators. Three polynomial type methods are known, and these are the minimal polynomial extrapolation (MPE), the reduced rank extrapolation (RRE), and the modified minimal polynomial extrapolation (MMPE). In this work, we develop yet another polynomial type method, which is based on the singular value decomposition, as well as the ideas that lead to MPE. We denote this new method by SVD-MPE. We also design a numerically stable algorithm for its implementation, whose computational cost and storage requirements are minimal. Finally, we illustrate the use of SVD-MPE with numerical examples.展开更多
Recently,some authors(Shen and Shi,2016)studied the generalized shiftsplitting(GSS)iteration method for singular saddle point problem with nonsymmetric positive definite(1,1)-block and symmetric positive semidefinite(...Recently,some authors(Shen and Shi,2016)studied the generalized shiftsplitting(GSS)iteration method for singular saddle point problem with nonsymmetric positive definite(1,1)-block and symmetric positive semidefinite(2,2)-block.In this paper,we further apply the GSS iteration method to solve singular saddle point problem with nonsymmetric positive semidefinite(1,1)-block and symmetric positive semidefinite(2,2)-block,prove the semi-convergence of the GSS iteration method and analyze the spectral properties of the corresponding preconditioned matrix.Numerical experiment is given to indicate that the GSS iteration method with appropriate iteration parameters is effective and competitive for practical use.展开更多
It was proposed that a robust and efficient parallelizable preconditioner for solving general sparse linear systems of equations, in which the use of sparse approximate inverse (AINV) techniques in a multi-level block...It was proposed that a robust and efficient parallelizable preconditioner for solving general sparse linear systems of equations, in which the use of sparse approximate inverse (AINV) techniques in a multi-level block ILU (BILUM) preconditioner were investigated. The resulting preconditioner retains robustness of BILUM preconditioner and has two advantages over the standard BILUM preconditioner: the ability to control sparsity and increased parallelism. Numerical experiments are used to show the effectiveness and efficiency of the new preconditioner.展开更多
An iterative solution of linear systems is studied,which arises from the discretization of a wire antennas attached with dielectric objects by the hybrid finite-element method and the method of moment (hybrid FEM-MoM...An iterative solution of linear systems is studied,which arises from the discretization of a wire antennas attached with dielectric objects by the hybrid finite-element method and the method of moment (hybrid FEM-MoM).It is efficient to model such electromagnetic problems by hybrid FEM-MoM,since it takes both the advantages of FEM's and MoM's ability.But the resulted linear systems are complicated,and it is hard to be solved by Krylov subspace methods alone,so a two-level preconditioning technique will be studied and applied to accelerate the convergence rate of the Krylov subspace methods.Numerical results show the effectiveness of the proposed two-level preconditioning technique.展开更多
In this paper, we propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b where is a singular and in general non-hermitian matrix that has an arb...In this paper, we propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b where is a singular and in general non-hermitian matrix that has an arbitrary index. DQMR algorithm for singular systems is analogous to QMR algorithm for non-singular systems. We compare this algorithm with DGMRES by numerical experiments.展开更多
The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach....The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(z) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sine diseretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.展开更多
An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitt...An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitting iteration methods. We consider the saddle-point linear systems with singular or semidefinite (1, 1) blocks. Moreover, this method is applied to precondition the GMRES. Numerical results have confirmed the effectiveness of the method and showed that the new method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse saddle-point linear systems.展开更多
This paper gives the truncated version of the generalized minimum backward error algorithm(GMBACK)—the incomplete generalized minimum backward perturbation algorithm(IGMBACK)for large nonsymmetric linear systems.It i...This paper gives the truncated version of the generalized minimum backward error algorithm(GMBACK)—the incomplete generalized minimum backward perturbation algorithm(IGMBACK)for large nonsymmetric linear systems.It is based on an incomplete orthogonalization of the Krylov vectors in question,and gives an approximate or quasi-minimum backward perturbation solution over the Krylov subspace.Theoretical properties of IGMBACK including finite termination,existence and uniqueness are discussed in details,and practical implementation issues associated with the IGMBACK algorithm are considered.Numerical experiments show that,the IGMBACK method is usually more efficient than GMBACK and GMRES,and IMBACK,GMBACK often have better convergence performance than GMRES.Specially,for sensitive matrices and right-hand sides being parallel to the left singular vectors corresponding to the smallest singular values of the coefficient matrices,GMRES does not necessarily converge,and IGMBACK,GMBACK usually converge and outperform GMRES.展开更多
A shift splitting concept is introduced and, correspondingly, a shift-splitting iteration scheme and a shift-splitting preconditioner are presented, for solving the large sparse system of linear equations of which the...A shift splitting concept is introduced and, correspondingly, a shift-splitting iteration scheme and a shift-splitting preconditioner are presented, for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned non-Hermitian positive definite matrix. The convergence property of the shift-splitting iteration method and the eigenvalue distribution of the shift-splitting preconditioned matrix are discussed in depth, and the best possible choice of the shift is investigated in detail. Numerical computations show that the shift-splitting preconditioner can induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the large sparse non-Hermitian positive definite systems of linear equations.展开更多
In this paper, a relaxed Hermitian and skew-Hermitian splitting (RHSS) preconditioner is proposed for saddle point problems from the element-free Galerkin (EFG) discretization method. The EFG method is one of the ...In this paper, a relaxed Hermitian and skew-Hermitian splitting (RHSS) preconditioner is proposed for saddle point problems from the element-free Galerkin (EFG) discretization method. The EFG method is one of the most widely used meshfree methods for solving partial differential equations. The RHSS preconditioner is constructed much closer to the coefficient matrix than the well-known HSS preconditioner, resulting in a RHSS fixed-point iteration. Convergence of the RHSS iteration is analyzed and an optimal parameter, which minimizes the spectral radius of the iteration matrix is described. Using the RHSS pre- conditioner to accelerate the convergence of some Krylov subspace methods (like GMRES) is also studied. Theoretical analyses show that the eigenvalues of the RHSS precondi- tioned matrix are real and located in a positive interval. Eigenvector distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are obtained. A practical parameter is suggested in implementing the RHSS preconditioner. Finally, some numerical experiments are illustrated to show the effectiveness of the new preconditioner.展开更多
In this paper, by exploiting the special block and sparse structure of the coefficient matrix, we present a new preconditioning strategy for solving large sparse linear systems arising in the time-dependent distribute...In this paper, by exploiting the special block and sparse structure of the coefficient matrix, we present a new preconditioning strategy for solving large sparse linear systems arising in the time-dependent distributed control problem involving the heat equation with two different functions. First a natural order-reduction is performed, and then the reduced- order linear system of equations is solved by the preconditioned MINRES algorithm with a new preconditioning techniques. The spectral properties of the preconditioned matrix are analyzed. Numerical results demonstrate that the preconditioning strategy for solving the large sparse systems discretized from the time-dependent problems is more effective for a wide range of mesh sizes and the value of the regularization parameter.展开更多
The integration of surface normals for the purpose of computing the shape of a surface in 3D space is a classic problem in computer vision. However,even nowadays it is still a challenging task to devise a method that ...The integration of surface normals for the purpose of computing the shape of a surface in 3D space is a classic problem in computer vision. However,even nowadays it is still a challenging task to devise a method that is flexible enough to work on non-trivial computational domains with high accuracy, robustness,and computational efficiency. By uniting a classic approach for surface normal integration with modern computational techniques, we construct a solver that fulfils these requirements. Building upon the Poisson integration model, we use an iterative Krylov subspace solver as a core step in tackling the task. While such a method can be very efficient, it may only show its full potential when combined with suitable numerical preconditioning and problem-specific initialisation. We perform a thorough numerical study in order to identify an appropriate preconditioner for this purpose.To provide suitable initialisation, we compute this initial state using a recently developed fast marching integrator. Detailed numerical experiments illustrate the benefits of this novel combination. In addition, we show on real-world photometric stereo datasets that the developed numerical framework is flexible enough to tackle modern computer vision applications.展开更多
This paper deals with fast and reliable numerical solution methods for the incompress- ible non-Newtonian Navier-Stokes equations. To handle the nonlinearity of the governing equations, the Picard and Newton methods a...This paper deals with fast and reliable numerical solution methods for the incompress- ible non-Newtonian Navier-Stokes equations. To handle the nonlinearity of the governing equations, the Picard and Newton methods are used to linearize these coupled partial dif- ferential equations. For space discretization we use the finite element method and utilize the two-by-two block structure of the matrices in the arising algebraic systems of equa- tions. The Krylov subspace iterative methods are chosen to solve the linearized discrete systems and the development of computationally and numerically efficient preconditioners for the two-by-two block matrices is the main concern in this paper. In non-Newtonian flows, the viscosity is not constant and its variation is an important factor that effects the performance of some already known preconditioning techniques. In this paper we examine the performance of several preconditioners for variable viscosity applications, and improve them further to be robust with respect to variations in viscosity.展开更多
We present simultaneous reduction algorithms for two(nonsymmetric)matrices A and B to upper Hessenberg and lower Hessenberg forms,respectively.One is through the simultaneous similarity reduction and the other is thro...We present simultaneous reduction algorithms for two(nonsymmetric)matrices A and B to upper Hessenberg and lower Hessenberg forms,respectively.One is through the simultaneous similarity reduction and the other is through a Lanczos–Arnoldi-type iteration.The algorithm that uses the Lanczos–Arnoldi-type iteration can be considered as a generalization of both the nonsymmetric Lanczos algorithm and the standard Arnoldi algorithm.We shall also apply our reduction to construct a model reduction for certain kind second-order single-input single-output system.It is proved that the model reduction has the desirable moment matching property.展开更多
We propose a new algorithm for the total variation based on image denoising problem. The split Bregman method is used to convert an unconstrained minimization denoising problem to a linear system in the outer iteratio...We propose a new algorithm for the total variation based on image denoising problem. The split Bregman method is used to convert an unconstrained minimization denoising problem to a linear system in the outer iteration. An algebraic multi-grid method is applied to solve the linear system in the inner iteration. Furthermore, Krylov subspace acceleration is adopted to improve convergence in the outer iteration. Numerical experiments demonstrate that this algorithm is efficient even for images with large signal-to-noise ratio.展开更多
In this paper, we propose an efficient combination model of the second-order ROF model and a simple fourth-order partial differential equation (PDE) for image denoising. The split Bregman method is used to convert t...In this paper, we propose an efficient combination model of the second-order ROF model and a simple fourth-order partial differential equation (PDE) for image denoising. The split Bregman method is used to convert the nonlinear combination model into a linear system in the outer iteration, and an algebraic multigrid method is applied to solve the linear system in the inner iteration. Furthermore, Krylov subspace acceleration is adopted to improve convergence in the outer iteration. At the same time, we prove that the model is strictly convex and exists a unique global minimizer. We have also conducted a variety of numerical experiments to analyze the parameter selection criteria and discuss the performance of ~he fourth-order PDE in the combination model. The results show that our model can reduce blocky effects and our algorithm is efficient and robust to solve the proposed model.展开更多
文摘A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this paper is to propose some efficient and accurate numerical methods for the computational solution of one-dimensional continuous basic models for the growth and control of brain tumors. After computing the analytical solution, we construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization. Then, we investigate the convergence behavior of Conjugate gradient and generalized minimum residual as Krylov subspace methods to solve the tridiagonal toeplitz matrix system derived.
文摘In this study,a one-dimensional two-phase flow four-equation model was developed to simulate the water faucet problem.The performance of six different Krylov subspace methods,namely the generalized minimal residual(GMRES),transpose-free quasi-minimal residual,quasi-minimal residual,conjugate gradient squared,biconjugate gradient stabilized,and biconjugate gradient,was evaluated with and without the application of an incomplete LU(ILU)factorization preconditioner for solving the water faucet problem.The simulation results indicate that using the ILU preconditioner with the Krylov subspace methods produces better convergence performance than that without the ILU preconditioner.Only the GMRES demonstrated an acceptable convergence performance under the Krylov subspace methods without the preconditioner.The velocity and pressure distribution in the water faucet problem could be determined using the Krylov subspace methods with an ILU preconditioner,while GMRES could determine it without the need for a preconditioner.However,there are significant advantages of using an ILU preconditioner with the GMRES in terms of efficiency.The different Krylov subspace methods showed similar performance in terms of computational efficiency under the application of the ILU preconditioner.
文摘Krylov subspace projection methods are known to be highly efficient for solving large linear systems. Many different versions arise from different choices to the left and right subspaces. These methods were classified into two groups in terms of the different forms of matrix H-m, the main properties in applications and the new versions of these two types of methods were briefly reviewed, then one of the most efficient versions, GMRES method was applied to oil reservoir simulation. The block Pseudo-Elimination method was used to generate the preconditioned matrix. Numerical results show much better performance of this preconditioned techniques and the GMRES method than that of preconditioned ORTHMIN method, which is now in use in oil reservoir simulation. Finally, some limitations of Krylov subspace methods and some potential improvements to this type of methods are further presented.
文摘In this paper we reconsider the range-restricted GMRES (RRGMRES) method for solving nonsymmetric linear systems. We first review an important result for the usual GMRES method. Then we give an example to show that the range-restricted GMRES method does not admit such a result. Finally, we give a modified result for the range-restricted GMRES method. We point out that the modified version can be used to show that the range-restricted GMRES method is also a regularization method for solving linear ill-posed problems.
文摘An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors , where , N being very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences converge more quickly is to apply to them vector extrapolation methods. Two types of methods exist in the literature: polynomial type methods and epsilon algorithms. In most applications, the polynomial type methods have proved to be superior convergence accelerators. Three polynomial type methods are known, and these are the minimal polynomial extrapolation (MPE), the reduced rank extrapolation (RRE), and the modified minimal polynomial extrapolation (MMPE). In this work, we develop yet another polynomial type method, which is based on the singular value decomposition, as well as the ideas that lead to MPE. We denote this new method by SVD-MPE. We also design a numerically stable algorithm for its implementation, whose computational cost and storage requirements are minimal. Finally, we illustrate the use of SVD-MPE with numerical examples.
基金Supported by Guangxi Science and Technology Department Specific Research Project of Guangxi for Research Bases and Talents(Grant No.GHIKE-AD23023001)Natural Science Foundation of Guangxi Minzu University(Grant No.2021KJQD01)Xiangsi Lake Young Scholars Innovation Team of Guangxi University for Nationalities(Grant No.2021RSCXSHQN05)。
文摘Recently,some authors(Shen and Shi,2016)studied the generalized shiftsplitting(GSS)iteration method for singular saddle point problem with nonsymmetric positive definite(1,1)-block and symmetric positive semidefinite(2,2)-block.In this paper,we further apply the GSS iteration method to solve singular saddle point problem with nonsymmetric positive semidefinite(1,1)-block and symmetric positive semidefinite(2,2)-block,prove the semi-convergence of the GSS iteration method and analyze the spectral properties of the corresponding preconditioned matrix.Numerical experiment is given to indicate that the GSS iteration method with appropriate iteration parameters is effective and competitive for practical use.
文摘It was proposed that a robust and efficient parallelizable preconditioner for solving general sparse linear systems of equations, in which the use of sparse approximate inverse (AINV) techniques in a multi-level block ILU (BILUM) preconditioner were investigated. The resulting preconditioner retains robustness of BILUM preconditioner and has two advantages over the standard BILUM preconditioner: the ability to control sparsity and increased parallelism. Numerical experiments are used to show the effectiveness and efficiency of the new preconditioner.
基金supported by the National Natural Science Foundation of China under Grand No. 10926190, 60973015 the Project of National Defense Key Lab under Grand No. 9140C6902030906
文摘An iterative solution of linear systems is studied,which arises from the discretization of a wire antennas attached with dielectric objects by the hybrid finite-element method and the method of moment (hybrid FEM-MoM).It is efficient to model such electromagnetic problems by hybrid FEM-MoM,since it takes both the advantages of FEM's and MoM's ability.But the resulted linear systems are complicated,and it is hard to be solved by Krylov subspace methods alone,so a two-level preconditioning technique will be studied and applied to accelerate the convergence rate of the Krylov subspace methods.Numerical results show the effectiveness of the proposed two-level preconditioning technique.
文摘In this paper, we propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b where is a singular and in general non-hermitian matrix that has an arbitrary index. DQMR algorithm for singular systems is analogous to QMR algorithm for non-singular systems. We compare this algorithm with DGMRES by numerical experiments.
文摘The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(z) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sine diseretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.
文摘An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitting iteration methods. We consider the saddle-point linear systems with singular or semidefinite (1, 1) blocks. Moreover, this method is applied to precondition the GMRES. Numerical results have confirmed the effectiveness of the method and showed that the new method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse saddle-point linear systems.
文摘This paper gives the truncated version of the generalized minimum backward error algorithm(GMBACK)—the incomplete generalized minimum backward perturbation algorithm(IGMBACK)for large nonsymmetric linear systems.It is based on an incomplete orthogonalization of the Krylov vectors in question,and gives an approximate or quasi-minimum backward perturbation solution over the Krylov subspace.Theoretical properties of IGMBACK including finite termination,existence and uniqueness are discussed in details,and practical implementation issues associated with the IGMBACK algorithm are considered.Numerical experiments show that,the IGMBACK method is usually more efficient than GMBACK and GMRES,and IMBACK,GMBACK often have better convergence performance than GMRES.Specially,for sensitive matrices and right-hand sides being parallel to the left singular vectors corresponding to the smallest singular values of the coefficient matrices,GMRES does not necessarily converge,and IGMBACK,GMBACK usually converge and outperform GMRES.
基金Research supported by The China NNSF 0utstanding Young Scientist Foundation (No.10525102), The National Natural Science Foundation (No.10471146), and The National Basic Research Program (No.2005CB321702), P.R. China.
文摘A shift splitting concept is introduced and, correspondingly, a shift-splitting iteration scheme and a shift-splitting preconditioner are presented, for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned non-Hermitian positive definite matrix. The convergence property of the shift-splitting iteration method and the eigenvalue distribution of the shift-splitting preconditioned matrix are discussed in depth, and the best possible choice of the shift is investigated in detail. Numerical computations show that the shift-splitting preconditioner can induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the large sparse non-Hermitian positive definite systems of linear equations.
基金Acknowledgments. The authors express their thanks to the referees for the comments and constructive suggestions, which were valuable in improving the quality of the manuscript. This work is supported by the National Natural Science Foundation of China(11172192) and the National Natural Science Pre-Research Foundation of Soochow University (SDY2011B01).
文摘In this paper, a relaxed Hermitian and skew-Hermitian splitting (RHSS) preconditioner is proposed for saddle point problems from the element-free Galerkin (EFG) discretization method. The EFG method is one of the most widely used meshfree methods for solving partial differential equations. The RHSS preconditioner is constructed much closer to the coefficient matrix than the well-known HSS preconditioner, resulting in a RHSS fixed-point iteration. Convergence of the RHSS iteration is analyzed and an optimal parameter, which minimizes the spectral radius of the iteration matrix is described. Using the RHSS pre- conditioner to accelerate the convergence of some Krylov subspace methods (like GMRES) is also studied. Theoretical analyses show that the eigenvalues of the RHSS precondi- tioned matrix are real and located in a positive interval. Eigenvector distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are obtained. A practical parameter is suggested in implementing the RHSS preconditioner. Finally, some numerical experiments are illustrated to show the effectiveness of the new preconditioner.
基金The work was supported by the National Natural Science Foundation of China (11271174). The authors would like to thank the referees for the comments and constructive suggestions, which are valuable in improving the quality of the manuscript.
文摘In this paper, by exploiting the special block and sparse structure of the coefficient matrix, we present a new preconditioning strategy for solving large sparse linear systems arising in the time-dependent distributed control problem involving the heat equation with two different functions. First a natural order-reduction is performed, and then the reduced- order linear system of equations is solved by the preconditioned MINRES algorithm with a new preconditioning techniques. The spectral properties of the preconditioned matrix are analyzed. Numerical results demonstrate that the preconditioning strategy for solving the large sparse systems discretized from the time-dependent problems is more effective for a wide range of mesh sizes and the value of the regularization parameter.
文摘The integration of surface normals for the purpose of computing the shape of a surface in 3D space is a classic problem in computer vision. However,even nowadays it is still a challenging task to devise a method that is flexible enough to work on non-trivial computational domains with high accuracy, robustness,and computational efficiency. By uniting a classic approach for surface normal integration with modern computational techniques, we construct a solver that fulfils these requirements. Building upon the Poisson integration model, we use an iterative Krylov subspace solver as a core step in tackling the task. While such a method can be very efficient, it may only show its full potential when combined with suitable numerical preconditioning and problem-specific initialisation. We perform a thorough numerical study in order to identify an appropriate preconditioner for this purpose.To provide suitable initialisation, we compute this initial state using a recently developed fast marching integrator. Detailed numerical experiments illustrate the benefits of this novel combination. In addition, we show on real-world photometric stereo datasets that the developed numerical framework is flexible enough to tackle modern computer vision applications.
文摘This paper deals with fast and reliable numerical solution methods for the incompress- ible non-Newtonian Navier-Stokes equations. To handle the nonlinearity of the governing equations, the Picard and Newton methods are used to linearize these coupled partial dif- ferential equations. For space discretization we use the finite element method and utilize the two-by-two block structure of the matrices in the arising algebraic systems of equa- tions. The Krylov subspace iterative methods are chosen to solve the linearized discrete systems and the development of computationally and numerically efficient preconditioners for the two-by-two block matrices is the main concern in this paper. In non-Newtonian flows, the viscosity is not constant and its variation is an important factor that effects the performance of some already known preconditioning techniques. In this paper we examine the performance of several preconditioners for variable viscosity applications, and improve them further to be robust with respect to variations in viscosity.
基金Research of R.Li is supported in part by NSF grants DMS-1115834 and DMS-1317330and a Research Gift Grant from Intel Corporation.Research of Q.Ye is supported in part by NSF Grants DMS–1318633 and DMS-1317424.
文摘We present simultaneous reduction algorithms for two(nonsymmetric)matrices A and B to upper Hessenberg and lower Hessenberg forms,respectively.One is through the simultaneous similarity reduction and the other is through a Lanczos–Arnoldi-type iteration.The algorithm that uses the Lanczos–Arnoldi-type iteration can be considered as a generalization of both the nonsymmetric Lanczos algorithm and the standard Arnoldi algorithm.We shall also apply our reduction to construct a model reduction for certain kind second-order single-input single-output system.It is proved that the model reduction has the desirable moment matching property.
基金Supported by Youth Foundation of Southwest University of Science and Technology (No.11zx3126)
文摘We propose a new algorithm for the total variation based on image denoising problem. The split Bregman method is used to convert an unconstrained minimization denoising problem to a linear system in the outer iteration. An algebraic multi-grid method is applied to solve the linear system in the inner iteration. Furthermore, Krylov subspace acceleration is adopted to improve convergence in the outer iteration. Numerical experiments demonstrate that this algorithm is efficient even for images with large signal-to-noise ratio.
基金Supported by the Natural Science Foundation of Sichuan Provincial Department of Education(No.15ZB0110)
文摘In this paper, we propose an efficient combination model of the second-order ROF model and a simple fourth-order partial differential equation (PDE) for image denoising. The split Bregman method is used to convert the nonlinear combination model into a linear system in the outer iteration, and an algebraic multigrid method is applied to solve the linear system in the inner iteration. Furthermore, Krylov subspace acceleration is adopted to improve convergence in the outer iteration. At the same time, we prove that the model is strictly convex and exists a unique global minimizer. We have also conducted a variety of numerical experiments to analyze the parameter selection criteria and discuss the performance of ~he fourth-order PDE in the combination model. The results show that our model can reduce blocky effects and our algorithm is efficient and robust to solve the proposed model.