The subspace iteration method is a method which combines the simultaneous inverse iteration method and the Rayleigh Ritz procedure. Since the Rayleigh Ritz procedure is usually time consuming, the solution time used i...The subspace iteration method is a method which combines the simultaneous inverse iteration method and the Rayleigh Ritz procedure. Since the Rayleigh Ritz procedure is usually time consuming, the solution time used in the subspace iteration method rises rapidly as the dimension of the subspace increases. An accelerated subspace iteration method for generalized eigenproblems is derived by obtaining a new subspace. The new subspace is composed of a dynamic condensation matrix, which relates the deformations associated with the master and slave degrees of freedom of a full model, and an identity matrix. Since the new subspace has nothing to do with the eigenpairs of the reduced model, there is no need to adopt the Rayleigh Ritz procedure in every iteration. This makes the proposed method computationally much more efficient and easier to be accelerated. The accelerated method converges any integer times as fast as the basic subspace iteration method. An eigenvalue shifting technique is also applied to make the stiffness matrix non singular, to accelerate the convergence and to calculate the eigenpairs in any given frequency range. Numerical examples demonstrate that the proposed method is feasible.展开更多
The parallel multisection method for solving algebraic eigenproblem has been presented in recent years with the development of the parallel computers, but all the research work is limited in standard eigenproblems of ...The parallel multisection method for solving algebraic eigenproblem has been presented in recent years with the development of the parallel computers, but all the research work is limited in standard eigenproblems of symmetric tridiagonal matrix. The multisection method for solving the generalized eigenproblem applied significantly in many science and engineering domains has not been studied. The parallel region preserving multisection method (PRM for short) for solving generalized eigenproblems of large sparse and real symmetric matrix is presented in this paper. This method not only retains the advantages of the conventional determinant search method (DS for short), but also overcomes its disadvantages such as leaking roots and disconvergence. We have tested the method on the YH 1 vector computer, and compared it with the parallel region preserving determinant search method the parallel region preserving bisection method (PRB for short). The numerical results show that PRM has a higher speed up, for instance, it attains the speed up of 7.7 when the scale of the problem is 2 114 and the eigenpair found is 3, and PRM is superior to PRB when the scale of the problem is large.展开更多
The interior Radon transform arises from a limited data problem in computerized tomography. The corresponding operator R is investigated as a mapping between wightedL 2-spaces. Our result is the explicit construction ...The interior Radon transform arises from a limited data problem in computerized tomography. The corresponding operator R is investigated as a mapping between wightedL 2-spaces. Our result is the explicit construction of a singular value decomposition for R. This immediately leads to an inversion formula by series expansion and range characterizations.展开更多
Let ∑, Г be two n-by-n diagonal matrices with σi,γi as their diagonals. For the inverse eigenvalue problem: look for y∈Rn such that Г + yyT is similar to ∑, we prove thatu also the sufficient condition for the ...Let ∑, Г be two n-by-n diagonal matrices with σi,γi as their diagonals. For the inverse eigenvalue problem: look for y∈Rn such that Г + yyT is similar to ∑, we prove thatu also the sufficient condition for the solvability of this inverse problem. Its solution (set) is given explicitly. In some case, the problem is unstable. But we prove that the sums of the square of some contigious components keep stable, i.e., small sum keeps small, large sum has a small relative perturbation, see Theorem 3.展开更多
-This paper reviews the current methodology for dynamic reanalysis. Rayleigh-Ritz approach and receptance approach are discussed in detail. Based on a general finite element structural analysis program SAPS, an eigenp...-This paper reviews the current methodology for dynamic reanalysis. Rayleigh-Ritz approach and receptance approach are discussed in detail. Based on a general finite element structural analysis program SAPS, an eigenproblem re-analysis prorgram ERP was compiled. With a very small change the program can be implemented readily with any general FEM program. Finally, some numerical examples show that the new algorithm is of high precision and efficiency. In the case of local modification in the offshore platform, the efficiency is raised by 20- 50 times when compared with the re-calculation of the whole model.展开更多
The parallel multisection method for solving algebraic eigenproblem has been presented in recent years with the developing of the parallel computers, but all the research work is limited in standard eigenproblem of sy...The parallel multisection method for solving algebraic eigenproblem has been presented in recent years with the developing of the parallel computers, but all the research work is limited in standard eigenproblem of symmetric tridiagonal matrix. The multisection method for solving generalized eigenproblem applied significantly in many secience and engineering domains has not been studied. The parallel region--preserving multisection method (PRM for shotr) for solving generalized eigenproblem of large sparse real symmetric matrix is presented in this paper. This method not only retains the advantages of the conventional determinant search method (DS for short), but also overcomes its disadvantages such as leaking roots and disconvergence. We tested the method on the YH--1 vector computer,and compared with the parallel region-preserving determinant search method (parallel region--preserving bisection method)(PRB for short). The numerical results show that PRM has a higher speed-up, for instance it attains the speed-up of 7.7 when the scale of the problem is 2114 and the eigenpair found is 3; and PRM is superior to PRB when scale of the problem is large.展开更多
Accelerating the eigensolver on GPUs is getting more and more attention due to its ubiquitous usage in scientific and engineering fields.However,it is very challenging to achieve high performance on eigensolvers becau...Accelerating the eigensolver on GPUs is getting more and more attention due to its ubiquitous usage in scientific and engineering fields.However,it is very challenging to achieve high performance on eigensolvers because of the intricate computational patterns which cause inefficient memory access and workload imbalance on GPUs.In this work,we propose a series of optimizations for generalized dense symmetric eigenvalue problems from both the system and operator perspectives on AMD GPUs.Firstly,we adjust the workload assignments between CPUs and GPUs and find the computational performance balance between different levels of computation.Besides,we propose a multi-level pre-aggregation strategy for symmetric matrix-vector multiplication(SYMV)and general matrix-vector multiplication(GEMV)operators to tackle the performance issue caused by lacking hardware support for atomic operation.Furthermore,we optimize Cholesky decomposition and SYR2K by adopting a better overlapping method and utilizing symmetry to reduce computation.Experiments on AMD MI60 GPUs show that our optimized eigensolver outperforms the previous state-of-the-art with roughly 1.8x–3.8x speedups.展开更多
This paper concerns the reconstruction of an hermitian Toeplitz matrix with prescribed eigenpairs. Based on the fact that every centrohermitian matrix can be reduced to a real matrix by a simple similarity transformat...This paper concerns the reconstruction of an hermitian Toeplitz matrix with prescribed eigenpairs. Based on the fact that every centrohermitian matrix can be reduced to a real matrix by a simple similarity transformation, the authors first consider the eigenstructure of hermitian Toeplitz matrices and then discuss a related reconstruction problem. The authors show that the dimension of the subspace of hermitian Toeplitz matrices with two given eigenvectors is at least two and independent of the size of the matrix, and the solution of the reconstruction problem of an hermitian Toeplitz matrix with two given eigenpairs is unique.展开更多
In this article, we study the solvability of nonlinear problem for p-Laplacian with nonlinear boundary conditions. We give some characterization of the first eigenvalue of an intermediary eigenvalne problem as simplic...In this article, we study the solvability of nonlinear problem for p-Laplacian with nonlinear boundary conditions. We give some characterization of the first eigenvalue of an intermediary eigenvalne problem as simplicity, isolation and its strict monotonicity. Afterward, we character also the second eigenvalue and its strictly partial monotony. On the other hand, in some sense, we establish the non-resonance below the first and furthermore between the first and second eigenvalues of nonlinear Steklov-Robin.展开更多
文摘The subspace iteration method is a method which combines the simultaneous inverse iteration method and the Rayleigh Ritz procedure. Since the Rayleigh Ritz procedure is usually time consuming, the solution time used in the subspace iteration method rises rapidly as the dimension of the subspace increases. An accelerated subspace iteration method for generalized eigenproblems is derived by obtaining a new subspace. The new subspace is composed of a dynamic condensation matrix, which relates the deformations associated with the master and slave degrees of freedom of a full model, and an identity matrix. Since the new subspace has nothing to do with the eigenpairs of the reduced model, there is no need to adopt the Rayleigh Ritz procedure in every iteration. This makes the proposed method computationally much more efficient and easier to be accelerated. The accelerated method converges any integer times as fast as the basic subspace iteration method. An eigenvalue shifting technique is also applied to make the stiffness matrix non singular, to accelerate the convergence and to calculate the eigenpairs in any given frequency range. Numerical examples demonstrate that the proposed method is feasible.
文摘The parallel multisection method for solving algebraic eigenproblem has been presented in recent years with the development of the parallel computers, but all the research work is limited in standard eigenproblems of symmetric tridiagonal matrix. The multisection method for solving the generalized eigenproblem applied significantly in many science and engineering domains has not been studied. The parallel region preserving multisection method (PRM for short) for solving generalized eigenproblems of large sparse and real symmetric matrix is presented in this paper. This method not only retains the advantages of the conventional determinant search method (DS for short), but also overcomes its disadvantages such as leaking roots and disconvergence. We have tested the method on the YH 1 vector computer, and compared it with the parallel region preserving determinant search method the parallel region preserving bisection method (PRB for short). The numerical results show that PRM has a higher speed up, for instance, it attains the speed up of 7.7 when the scale of the problem is 2 114 and the eigenpair found is 3, and PRM is superior to PRB when the scale of the problem is large.
基金Supported by the Foundation of the Ministry of Education of China and the Science Foundation of Wuhan University
文摘The interior Radon transform arises from a limited data problem in computerized tomography. The corresponding operator R is investigated as a mapping between wightedL 2-spaces. Our result is the explicit construction of a singular value decomposition for R. This immediately leads to an inversion formula by series expansion and range characterizations.
文摘Let ∑, Г be two n-by-n diagonal matrices with σi,γi as their diagonals. For the inverse eigenvalue problem: look for y∈Rn such that Г + yyT is similar to ∑, we prove thatu also the sufficient condition for the solvability of this inverse problem. Its solution (set) is given explicitly. In some case, the problem is unstable. But we prove that the sums of the square of some contigious components keep stable, i.e., small sum keeps small, large sum has a small relative perturbation, see Theorem 3.
文摘-This paper reviews the current methodology for dynamic reanalysis. Rayleigh-Ritz approach and receptance approach are discussed in detail. Based on a general finite element structural analysis program SAPS, an eigenproblem re-analysis prorgram ERP was compiled. With a very small change the program can be implemented readily with any general FEM program. Finally, some numerical examples show that the new algorithm is of high precision and efficiency. In the case of local modification in the offshore platform, the efficiency is raised by 20- 50 times when compared with the re-calculation of the whole model.
文摘The parallel multisection method for solving algebraic eigenproblem has been presented in recent years with the developing of the parallel computers, but all the research work is limited in standard eigenproblem of symmetric tridiagonal matrix. The multisection method for solving generalized eigenproblem applied significantly in many secience and engineering domains has not been studied. The parallel region--preserving multisection method (PRM for shotr) for solving generalized eigenproblem of large sparse real symmetric matrix is presented in this paper. This method not only retains the advantages of the conventional determinant search method (DS for short), but also overcomes its disadvantages such as leaking roots and disconvergence. We tested the method on the YH--1 vector computer,and compared with the parallel region-preserving determinant search method (parallel region--preserving bisection method)(PRB for short). The numerical results show that PRM has a higher speed-up, for instance it attains the speed-up of 7.7 when the scale of the problem is 2114 and the eigenpair found is 3; and PRM is superior to PRB when scale of the problem is large.
基金supported by the National Key Research and Development Program of China under Grant No.2021YFB0300203the National Natural Science Foundation of China under Grant No.12131005.
文摘Accelerating the eigensolver on GPUs is getting more and more attention due to its ubiquitous usage in scientific and engineering fields.However,it is very challenging to achieve high performance on eigensolvers because of the intricate computational patterns which cause inefficient memory access and workload imbalance on GPUs.In this work,we propose a series of optimizations for generalized dense symmetric eigenvalue problems from both the system and operator perspectives on AMD GPUs.Firstly,we adjust the workload assignments between CPUs and GPUs and find the computational performance balance between different levels of computation.Besides,we propose a multi-level pre-aggregation strategy for symmetric matrix-vector multiplication(SYMV)and general matrix-vector multiplication(GEMV)operators to tackle the performance issue caused by lacking hardware support for atomic operation.Furthermore,we optimize Cholesky decomposition and SYR2K by adopting a better overlapping method and utilizing symmetry to reduce computation.Experiments on AMD MI60 GPUs show that our optimized eigensolver outperforms the previous state-of-the-art with roughly 1.8x–3.8x speedups.
基金This work is supported by the National Natural Science Foundation of China under Grant Nos. 10771022 and 10571012, Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China under Grant No. 890 [2008], and Major Foundation of Educational Committee of Hunan Province under Grant No. 09A002 [2009] Portuguese Foundation for Science and Technology （FCT） through the Research Programme POCTI, respectively.
文摘This paper concerns the reconstruction of an hermitian Toeplitz matrix with prescribed eigenpairs. Based on the fact that every centrohermitian matrix can be reduced to a real matrix by a simple similarity transformation, the authors first consider the eigenstructure of hermitian Toeplitz matrices and then discuss a related reconstruction problem. The authors show that the dimension of the subspace of hermitian Toeplitz matrices with two given eigenvectors is at least two and independent of the size of the matrix, and the solution of the reconstruction problem of an hermitian Toeplitz matrix with two given eigenpairs is unique.
文摘In this article, we study the solvability of nonlinear problem for p-Laplacian with nonlinear boundary conditions. We give some characterization of the first eigenvalue of an intermediary eigenvalne problem as simplicity, isolation and its strict monotonicity. Afterward, we character also the second eigenvalue and its strictly partial monotony. On the other hand, in some sense, we establish the non-resonance below the first and furthermore between the first and second eigenvalues of nonlinear Steklov-Robin.
