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
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
Recently numerous numerical experiments on realistic calculation have shown that the conjugate A-orthogonal residual squared （CORS） method is often competitive with other popular methods. However, the CORS method, l...Recently numerous numerical experiments on realistic calculation have shown that the conjugate A-orthogonal residual squared （CORS） method is often competitive with other popular methods. However, the CORS method, like the CGS method, shows irreg- ular convergence, especially appears large intermediate residual norm, which may lead to worse approximate solutions and slower convergence rate. In this paper, we present a new product-type method for solving complex non-Hermitian linear systems based on the bicon- jugate A-orthogonal residual （BiCOR） method, where one of the polynomials is a BiCOR polynomial, and the other is a BiCOR polynomial with the same degree corresponding to different initial residual. Numerical examples are given to illustrate the effectiveness of the proposed method.展开更多
To perform nuclear reactor simulations in a more realistic manner,the coupling scheme between neutronics and thermal-hydraulics was implemented in the HNET program for both steady-state and transient conditions.For si...To perform nuclear reactor simulations in a more realistic manner,the coupling scheme between neutronics and thermal-hydraulics was implemented in the HNET program for both steady-state and transient conditions.For simplicity,efficiency,and robustness,the matrixfree Newton/Krylov(MFNK)method was applied to the steady-state coupling calculation.In addition,the optimal perturbation size was adopted to further improve the convergence behavior of the MFNK.For the transient coupling simulation,the operator splitting method with a staggered time mesh was utilized to balance the computational cost and accuracy.Finally,VERA Problem 6 with power and boron perturbation and the NEACRP transient benchmark were simulated for analysis.The numerical results show that the MFNK method can outperform Picard iteration in terms of both efficiency and robustness for a wide range of problems.Furthermore,the reasonable agreement between the simulation results and the reference results for the NEACRP transient benchmark verifies the capability of predicting the behavior of the nuclear reactor.展开更多
The seed method is used for solving multiple linear systems A (i)x (i) =b (i) for 1≤i≤s, where the coefficient matrix A (i) and the right-hand side b (i) are different in general. It is known that the CG meth...The seed method is used for solving multiple linear systems A (i)x (i) =b (i) for 1≤i≤s, where the coefficient matrix A (i) and the right-hand side b (i) are different in general. It is known that the CG method is an effective method for symmetric coefficient matrices A (i). In this paper, the FOM method is employed to solve multiple linear sy stems when coefficient matrices are non-symmetric matrices. One of the systems is selected as the seed system which generates a Krylov subspace, then the resi duals of other systems are projected onto the generated Krylov subspace to get t he approximate solutions for the unsolved ones. The whole process is repeated u ntil all the systems are solved.展开更多
The DGMRES method for solving Drazin-inverse solution of singular linear systems is generally used with restarting. But the restarting often slows down the convergence and DGMRES often stagnates. We show that adding s...The DGMRES method for solving Drazin-inverse solution of singular linear systems is generally used with restarting. But the restarting often slows down the convergence and DGMRES often stagnates. We show that adding some eigenvectors to the subspace can improve the convergence just like the method proposed by R. Morgan in [R. Morgan, A restarted GMRES method augmented with eigenvectors, SIAM J. Matrix Anal App1., 16： 1154-1171, 1995. We derive the implementation of this method and present some numerical examples to show the advantages of this method.展开更多
In this paper, we propose another version of the full orthogonalization method (FOM) for the resolution of linear system Ax = b, based on an extended definition of Sturm sequence in the calculation of the determinant ...In this paper, we propose another version of the full orthogonalization method (FOM) for the resolution of linear system Ax = b, based on an extended definition of Sturm sequence in the calculation of the determinant of an upper hessenberg matrix in o(n2). We will also give a new version of Givens method based on using a tensor product and matrix addition. This version can be used in parallel calculation.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
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.
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The authors are grateful to the referees for their valuable comments and suggestions which helped to improve the presentation of this paper. The research is supported by the National Natural Science Foundation of China under grant No.11071118, Natural Science Foundation from Anhui Province Education Department under grant No.KJ2012B058 and AHSTU under grant No.ZRC2013388.
文摘Recently numerous numerical experiments on realistic calculation have shown that the conjugate A-orthogonal residual squared （CORS） method is often competitive with other popular methods. However, the CORS method, like the CGS method, shows irreg- ular convergence, especially appears large intermediate residual norm, which may lead to worse approximate solutions and slower convergence rate. In this paper, we present a new product-type method for solving complex non-Hermitian linear systems based on the bicon- jugate A-orthogonal residual （BiCOR） method, where one of the polynomials is a BiCOR polynomial, and the other is a BiCOR polynomial with the same degree corresponding to different initial residual. Numerical examples are given to illustrate the effectiveness of the proposed method.
基金supported by the China Postdoctoral Science Foundation(No.2021M703045)the National Natural Science Foundation of China(No.12075067)the National Key R&D Program of China(No.2018YFE0180900).
文摘To perform nuclear reactor simulations in a more realistic manner,the coupling scheme between neutronics and thermal-hydraulics was implemented in the HNET program for both steady-state and transient conditions.For simplicity,efficiency,and robustness,the matrixfree Newton/Krylov(MFNK)method was applied to the steady-state coupling calculation.In addition,the optimal perturbation size was adopted to further improve the convergence behavior of the MFNK.For the transient coupling simulation,the operator splitting method with a staggered time mesh was utilized to balance the computational cost and accuracy.Finally,VERA Problem 6 with power and boron perturbation and the NEACRP transient benchmark were simulated for analysis.The numerical results show that the MFNK method can outperform Picard iteration in terms of both efficiency and robustness for a wide range of problems.Furthermore,the reasonable agreement between the simulation results and the reference results for the NEACRP transient benchmark verifies the capability of predicting the behavior of the nuclear reactor.
基金Project supported by the National Natural Science Foundation of China (Grant No.10271075)
文摘The seed method is used for solving multiple linear systems A (i)x (i) =b (i) for 1≤i≤s, where the coefficient matrix A (i) and the right-hand side b (i) are different in general. It is known that the CG method is an effective method for symmetric coefficient matrices A (i). In this paper, the FOM method is employed to solve multiple linear sy stems when coefficient matrices are non-symmetric matrices. One of the systems is selected as the seed system which generates a Krylov subspace, then the resi duals of other systems are projected onto the generated Krylov subspace to get t he approximate solutions for the unsolved ones. The whole process is repeated u ntil all the systems are solved.
基金Supported by the National Natural Science Foundation of China(No.11171151)Natural Science Foundation of Jiangsu Province of China(No.BK2011720)
文摘The DGMRES method for solving Drazin-inverse solution of singular linear systems is generally used with restarting. But the restarting often slows down the convergence and DGMRES often stagnates. We show that adding some eigenvectors to the subspace can improve the convergence just like the method proposed by R. Morgan in [R. Morgan, A restarted GMRES method augmented with eigenvectors, SIAM J. Matrix Anal App1., 16： 1154-1171, 1995. We derive the implementation of this method and present some numerical examples to show the advantages of this method.
文摘In this paper, we propose another version of the full orthogonalization method (FOM) for the resolution of linear system Ax = b, based on an extended definition of Sturm sequence in the calculation of the determinant of an upper hessenberg matrix in o(n2). We will also give a new version of Givens method based on using a tensor product and matrix addition. This version can be used in parallel calculation.
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.
