Abstract This paper presents a restarted conjugate gradient iterative algorithm for solving ill-posed problems. The damped Morozov's discrepancy principle is used as a stopping rule. Numerical experiments are give...Abstract This paper presents a restarted conjugate gradient iterative algorithm for solving ill-posed problems. The damped Morozov's discrepancy principle is used as a stopping rule. Numerical experiments are given to illustrate the efficiency of the method.展开更多
In this paper we study the algorithms and their parallel implementation for solving large-scale generalized eigenvalue problems in modal analysis.Three predominant subspace algorithms,i.e.,Krylov-Schur method,implicit...In this paper we study the algorithms and their parallel implementation for solving large-scale generalized eigenvalue problems in modal analysis.Three predominant subspace algorithms,i.e.,Krylov-Schur method,implicitly restarted Arnoldi method and Jacobi-Davidson method,are modified with some complementary techniques to make them suitable for modal analysis.Detailed descriptions of the three algorithms are given.Based on these algorithms,a parallel solution procedure is established via the PANDA framework and its associated eigensolvers.Using the solution procedure on a machine equipped with up to 4800processors,the parallel performance of the three predominant methods is evaluated via numerical experiments with typical engineering structures,where the maximum testing scale attains twenty million degrees of freedom.The speedup curves for different cases are obtained and compared.The results show that the three methods are good for modal analysis in the scale of ten million degrees of freedom with a favorable parallel scalability.展开更多
In this paper, we present a compact version of the Heart iteration. One that requires less matrix-vector products per iteration and attains faster convergence. The Heart iteration is a new type of Restarted Krylov met...In this paper, we present a compact version of the Heart iteration. One that requires less matrix-vector products per iteration and attains faster convergence. The Heart iteration is a new type of Restarted Krylov methods for calculating peripheral eigenvalues of symmetric matrices. The new framework avoids the Lanczos tridiagonalization process and the use of implicit restarts. This simplifies the restarting mechanism and allows the introduction of several modifications. Convergence is assured by a monotonicity property that pushes the computed Ritz values toward their limits. Numerical experiments illustrate the usefulness of the proposed approach.展开更多
In this paper we present a new type of Restarted Krylov methods for calculating peripheral eigenvalues of symmetric matrices. The new framework avoids the Lanczos tridiagonalization process, and the use of polynomial ...In this paper we present a new type of Restarted Krylov methods for calculating peripheral eigenvalues of symmetric matrices. The new framework avoids the Lanczos tridiagonalization process, and the use of polynomial filtering. This simplifies the restarting mechanism and allows the introduction of several modifications. Convergence is assured by a monotonicity property that pushes the eigenvalues toward their limits. The Krylov matrices that we use lead to fast rate of convergence. Numerical experiments illustrate the usefulness of the proposed approach.展开更多
The purpose of this paper is to employ the Adomian Decomposition Method (ADM) and Restarted Adomian Decomposition Method (RADM) with new useful techniques to resolve Bratu’s boundary value problem by using a new inte...The purpose of this paper is to employ the Adomian Decomposition Method (ADM) and Restarted Adomian Decomposition Method (RADM) with new useful techniques to resolve Bratu’s boundary value problem by using a new integral operator. The solutions obtained in this way require the use of the boundary conditions directly. The obtained results indicate that the new techniques give more suitable and accurate solutions for the Bratu-type problem, compared with those for the ADM and its modification.展开更多
In this paper, we used an efficient algorithm to obtain an analytic approximation for Volterra’s model for population growth of a species within a closed system, called the Restarted Adomian decomposition method (RAD...In this paper, we used an efficient algorithm to obtain an analytic approximation for Volterra’s model for population growth of a species within a closed system, called the Restarted Adomian decomposition method (RADM) to solve the model. The numerical results illustrate that RADM has the good accuracy.展开更多
In this paper we present a new subspace iteration for calculating eigenvalues of symmetric matrices. The method is designed to compute a cluster of k exterior eigenvalues. For example, k eigenvalues with the largest a...In this paper we present a new subspace iteration for calculating eigenvalues of symmetric matrices. The method is designed to compute a cluster of k exterior eigenvalues. For example, k eigenvalues with the largest absolute values, the k algebraically largest eigenvalues, or the k algebraically smallest eigenvalues. The new iteration applies a Restarted Krylov method to collect information on the desired cluster. It is shown that the estimated eigenvalues proceed monotonically toward their limits. Another innovation regards the choice of starting points for the Krylov subspaces, which leads to fast rate of convergence. Numerical experiments illustrate the viability of the proposed ideas.展开更多
In this paper,we are concerned with the numerical solutions for the parabolic and hyperbolic partial differential equations with nonlocal boundary conditions.Thus,we presented a new iterative algorithm based on the Re...In this paper,we are concerned with the numerical solutions for the parabolic and hyperbolic partial differential equations with nonlocal boundary conditions.Thus,we presented a new iterative algorithm based on the Restarted Adomian Decomposition Method(RADM)for solving the two equations of different types involving dissimilar boundary and nonlocal conditions.The algorithm presented transforms the given nonlocal initial boundary value problem to a local Dirichlet one and then employs the RADM for the numerical treatment.Numerical comparisons were made between our proposed method and the Adomian Decomposition Method(ADM)to demonstrate the efficiency and performance of the proposed method.展开更多
We present a three dimensional preconditioned implicit free-surface capture scheme on tetrahedral grids.The current scheme improves our recently reported method[10]in several aspects.Specifically,we modified the origi...We present a three dimensional preconditioned implicit free-surface capture scheme on tetrahedral grids.The current scheme improves our recently reported method[10]in several aspects.Specifically,we modified the original eigensystem by applying a preconditioning matrix so that the new eigensystem is virtually independent of density ratio,which is typically large for practical two-phase problems.Further,we replaced the explicit multi-stage Runge-Kutta method by a fully implicit Euler integration scheme for the Navier-Stokes(NS)solver and the Volume of Fluids(VOF)equation is now solved with a second order Crank-Nicolson implicit scheme to reduce the numerical diffusion effect.The preconditioned restarted GeneralizedMinimal RESidual method(GMRES)is then employed to solve the resulting linear system.The validation studies show that with these modifications,the method has improved stability and accuracy when dealing with large density ratio two-phase problems.展开更多
In this paper,we study shifted restated full orthogonalization method with deflation for simultaneously solving a number of shifted systems of linear equations.Theoretical analysis shows that with the deflation techni...In this paper,we study shifted restated full orthogonalization method with deflation for simultaneously solving a number of shifted systems of linear equations.Theoretical analysis shows that with the deflation technique,the new residual of shifted restarted FOM is still collinear with each other.Hence,the new approach can solve the shifted systems simultaneously based on the same Krylov subspace.Numerical experiments show that the deflation technique can significantly improve the convergence performance of shifted restarted FOM.展开更多
基金Supported by CAS Hundred Talents Program and Digital Earth (KZCX2-312)also partially supported by National Natural Science Foundation of China (No.19731010).
文摘Abstract This paper presents a restarted conjugate gradient iterative algorithm for solving ill-posed problems. The damped Morozov's discrepancy principle is used as a stopping rule. Numerical experiments are given to illustrate the efficiency of the method.
基金supported by the National Defence Basic Fundamental Research Program of China(Grant No.C1520110002)the Fundamental Development Foundation of China Academy Engineering Physics(Grant No.2012A0202008)
文摘In this paper we study the algorithms and their parallel implementation for solving large-scale generalized eigenvalue problems in modal analysis.Three predominant subspace algorithms,i.e.,Krylov-Schur method,implicitly restarted Arnoldi method and Jacobi-Davidson method,are modified with some complementary techniques to make them suitable for modal analysis.Detailed descriptions of the three algorithms are given.Based on these algorithms,a parallel solution procedure is established via the PANDA framework and its associated eigensolvers.Using the solution procedure on a machine equipped with up to 4800processors,the parallel performance of the three predominant methods is evaluated via numerical experiments with typical engineering structures,where the maximum testing scale attains twenty million degrees of freedom.The speedup curves for different cases are obtained and compared.The results show that the three methods are good for modal analysis in the scale of ten million degrees of freedom with a favorable parallel scalability.
文摘In this paper, we present a compact version of the Heart iteration. One that requires less matrix-vector products per iteration and attains faster convergence. The Heart iteration is a new type of Restarted Krylov methods for calculating peripheral eigenvalues of symmetric matrices. The new framework avoids the Lanczos tridiagonalization process and the use of implicit restarts. This simplifies the restarting mechanism and allows the introduction of several modifications. Convergence is assured by a monotonicity property that pushes the computed Ritz values toward their limits. Numerical experiments illustrate the usefulness of the proposed approach.
文摘In this paper we present a new type of Restarted Krylov methods for calculating peripheral eigenvalues of symmetric matrices. The new framework avoids the Lanczos tridiagonalization process, and the use of polynomial filtering. This simplifies the restarting mechanism and allows the introduction of several modifications. Convergence is assured by a monotonicity property that pushes the eigenvalues toward their limits. The Krylov matrices that we use lead to fast rate of convergence. Numerical experiments illustrate the usefulness of the proposed approach.
文摘The purpose of this paper is to employ the Adomian Decomposition Method (ADM) and Restarted Adomian Decomposition Method (RADM) with new useful techniques to resolve Bratu’s boundary value problem by using a new integral operator. The solutions obtained in this way require the use of the boundary conditions directly. The obtained results indicate that the new techniques give more suitable and accurate solutions for the Bratu-type problem, compared with those for the ADM and its modification.
文摘In this paper, we used an efficient algorithm to obtain an analytic approximation for Volterra’s model for population growth of a species within a closed system, called the Restarted Adomian decomposition method (RADM) to solve the model. The numerical results illustrate that RADM has the good accuracy.
文摘In this paper we present a new subspace iteration for calculating eigenvalues of symmetric matrices. The method is designed to compute a cluster of k exterior eigenvalues. For example, k eigenvalues with the largest absolute values, the k algebraically largest eigenvalues, or the k algebraically smallest eigenvalues. The new iteration applies a Restarted Krylov method to collect information on the desired cluster. It is shown that the estimated eigenvalues proceed monotonically toward their limits. Another innovation regards the choice of starting points for the Krylov subspaces, which leads to fast rate of convergence. Numerical experiments illustrate the viability of the proposed ideas.
文摘In this paper,we are concerned with the numerical solutions for the parabolic and hyperbolic partial differential equations with nonlocal boundary conditions.Thus,we presented a new iterative algorithm based on the Restarted Adomian Decomposition Method(RADM)for solving the two equations of different types involving dissimilar boundary and nonlocal conditions.The algorithm presented transforms the given nonlocal initial boundary value problem to a local Dirichlet one and then employs the RADM for the numerical treatment.Numerical comparisons were made between our proposed method and the Adomian Decomposition Method(ADM)to demonstrate the efficiency and performance of the proposed method.
基金supported by the Flood Risk from Extreme Events(FREE)Programme of the UK Natural Environment Research Council(NERC)(NE/E0002129/1)coordinated and monitored by Professor Chris Collier and Paul Hardaker.We thank Dr.Zhengyi Wang and Dr.Qun Zhao for helpful discussions.The numerical calculations have been carried out on the HPC facility at the University of Plymouth.
文摘We present a three dimensional preconditioned implicit free-surface capture scheme on tetrahedral grids.The current scheme improves our recently reported method[10]in several aspects.Specifically,we modified the original eigensystem by applying a preconditioning matrix so that the new eigensystem is virtually independent of density ratio,which is typically large for practical two-phase problems.Further,we replaced the explicit multi-stage Runge-Kutta method by a fully implicit Euler integration scheme for the Navier-Stokes(NS)solver and the Volume of Fluids(VOF)equation is now solved with a second order Crank-Nicolson implicit scheme to reduce the numerical diffusion effect.The preconditioned restarted GeneralizedMinimal RESidual method(GMRES)is then employed to solve the resulting linear system.The validation studies show that with these modifications,the method has improved stability and accuracy when dealing with large density ratio two-phase problems.
文摘In this paper,we study shifted restated full orthogonalization method with deflation for simultaneously solving a number of shifted systems of linear equations.Theoretical analysis shows that with the deflation technique,the new residual of shifted restarted FOM is still collinear with each other.Hence,the new approach can solve the shifted systems simultaneously based on the same Krylov subspace.Numerical experiments show that the deflation technique can significantly improve the convergence performance of shifted restarted FOM.
