This paper describes a method of calculating the Schur complement of a sparse positive definite matrix A. The main idea of this approach is to represent matrix A in the form of an elimination tree using a reordering a...This paper describes a method of calculating the Schur complement of a sparse positive definite matrix A. The main idea of this approach is to represent matrix A in the form of an elimination tree using a reordering algorithm like METIS and putting columns/rows for which the Schur complement is needed into the top node of the elimination tree. Any problem with a degenerate part of the initial matrix can be resolved with the help of iterative refinement. The proposed approach is close to the “multifrontal” one which was implemented by Ian Duff and others in 1980s. Schur complement computations described in this paper are available in Intel®Math Kernel Library (Intel®MKL). In this paper we present the algorithm for Schur complement computations, experiments that demonstrate a negligible increase in the number of elements in the factored matrix, and comparison with existing alternatives.展开更多
A parallel hybrid linear solver based on the Schur complement method has the potential to balance the robustness of direct solvers with the efficiency of preconditioned iterative solvers.However,when solving large-sca...A parallel hybrid linear solver based on the Schur complement method has the potential to balance the robustness of direct solvers with the efficiency of preconditioned iterative solvers.However,when solving large-scale highly-indefinite linear systems,this hybrid solver often suffers from either slow convergence or large memory requirements to solve the Schur complement systems.To overcome this challenge,we in this paper discuss techniques to preprocess the Schur complement systems in parallel. Numerical results of solving large-scale highly-indefinite linear systems from various applications demonstrate that these techniques improve the reliability and performance of the hybrid solver and enable efficient solutions of these linear systems on hundreds of processors,which was previously infeasible using existing state-of-the-art solvers.展开更多
The theory of Schur complement plays an important role in many fields such as matrix theory, control theory and computational mathematics. In this paper, some new estimates of diagonally, α-diagonally and product α-...The theory of Schur complement plays an important role in many fields such as matrix theory, control theory and computational mathematics. In this paper, some new estimates of diagonally, α-diagonally and product α-diagonally dominant degree on the Schur complement of matrices are obtained, which improve some relative results. As an application, we present several new eigenvalue inclusion regions for the Schur complement of matrices. Finally, we give a numerical example to illustrate the advantages of our derived results.展开更多
Necessary and sufficient conditions for a schur complement of a con-s-k-EP matrix to be con-s-k-EP are determined. Further it is shown that in a con-s-k-EPr matrix, every secondary sub matrix of rank “r” is con-s-k-...Necessary and sufficient conditions for a schur complement of a con-s-k-EP matrix to be con-s-k-EP are determined. Further it is shown that in a con-s-k-EPr matrix, every secondary sub matrix of rank “r” is con-s-k-EPr. We have also discussed the way of expressing a matrix of rank r as a product of con-s-k-EPr matrices. Necessary and sufficient conditions for products of con-s-k-EPr partitioned matrices to be con-s-k-EPr are given.展开更多
In this paper,we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems.One type of the preconditioners are based on the nested(or recursive)Schur complement...In this paper,we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems.One type of the preconditioners are based on the nested(or recursive)Schur complement,the other is based on an additive type Schur complement after permuting the original saddle point systems.We analyze different preconditioners incorporating the exact Schur complements.We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements.These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly.Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions.展开更多
The distribution for eigenvalues of Schur complement of matrices plays an important role in many mathematical problems.In this paper,we firstly present some criteria for H-matrix.Then as application,for two class matr...The distribution for eigenvalues of Schur complement of matrices plays an important role in many mathematical problems.In this paper,we firstly present some criteria for H-matrix.Then as application,for two class matrices whose sub-matrices areγ-diagonally dominant and productγ-diagonally dominant,we show that the eigenvalues of the Schur complement are located in the Gersgorin discs and the Ostrowski discs of the original matrices under certain conditions.展开更多
Let E be a Hilbert C＊-module,and Y be an orthogonally complemented closed submodule of E.The authors generalize the definitions of Y-complementability and Y-compatibility for general（adjointable） operators from Hil...Let E be a Hilbert C＊-module,and Y be an orthogonally complemented closed submodule of E.The authors generalize the definitions of Y-complementability and Y-compatibility for general（adjointable） operators from Hilbert space to Hilbert C＊-module,and discuss the relationship between each other.Several equivalent statements about Y-complementability and Y-compatibility,and several representations of Schur complements of Y-complementable operators（especially,of Y-compatible operators and of positive Y-compatible operators） on a Hilbert C＊-module are obtained.In addition,the quotient property for Schur complements of matrices is generalized to the quotient property for Schur complements of Y-complementable operators and Y＊-complementable operators on a Hilbert C＊-module.展开更多
The definitions of θ-ray pattern proposed to establish some new results matrix and θ-ray matrix are firstly on nonsingularity/singularity and convergence of general H-matrices. Then some conditions on the matrix A ...The definitions of θ-ray pattern proposed to establish some new results matrix and θ-ray matrix are firstly on nonsingularity/singularity and convergence of general H-matrices. Then some conditions on the matrix A ∈ C^n×n and nonempty α （n） = {1,2,... ,n} are proposed such that A is an invertible H-matrix if A（α） and A/α are both invertible H-matrices. Furthermore, the important results on Schur complement for general H-matrices are presented to give the different necessary and sufficient conditions for the matrix A E HM and the subset α C （n） such that the Schur complement matrix A/α∈ HI^n-｜α｜ or A/α ∈ Hn-｜α｜^M or A/α ∈ H^n-｜ α｜^S.展开更多
关于复分块矩阵P=A BC D∈Cm×n的广义Schur补S=A-BD+C,T=D-CA+B,S=A-BDgC,T=D-CAgB,这里,D+,A+分别代表D,A的M oore-Penrose逆,Dg,Ag分别代表D,A的群逆。这篇文章我们主要给出当P是幂等矩阵时,在一定条件下,P的某些性质成立时,S,T...关于复分块矩阵P=A BC D∈Cm×n的广义Schur补S=A-BD+C,T=D-CA+B,S=A-BDgC,T=D-CAgB,这里,D+,A+分别代表D,A的M oore-Penrose逆,Dg,Ag分别代表D,A的群逆。这篇文章我们主要给出当P是幂等矩阵时,在一定条件下,P的某些性质成立时,S,T具有同样的性质。展开更多
文摘This paper describes a method of calculating the Schur complement of a sparse positive definite matrix A. The main idea of this approach is to represent matrix A in the form of an elimination tree using a reordering algorithm like METIS and putting columns/rows for which the Schur complement is needed into the top node of the elimination tree. Any problem with a degenerate part of the initial matrix can be resolved with the help of iterative refinement. The proposed approach is close to the “multifrontal” one which was implemented by Ian Duff and others in 1980s. Schur complement computations described in this paper are available in Intel®Math Kernel Library (Intel®MKL). In this paper we present the algorithm for Schur complement computations, experiments that demonstrate a negligible increase in the number of elements in the factored matrix, and comparison with existing alternatives.
基金supported in part by the Director,Office of Science,Office of Advanced Scientific Computing Research,of the U.S.Department of Energy under Contract No.DE-AC02-05CH11231.
文摘A parallel hybrid linear solver based on the Schur complement method has the potential to balance the robustness of direct solvers with the efficiency of preconditioned iterative solvers.However,when solving large-scale highly-indefinite linear systems,this hybrid solver often suffers from either slow convergence or large memory requirements to solve the Schur complement systems.To overcome this challenge,we in this paper discuss techniques to preprocess the Schur complement systems in parallel. Numerical results of solving large-scale highly-indefinite linear systems from various applications demonstrate that these techniques improve the reliability and performance of the hybrid solver and enable efficient solutions of these linear systems on hundreds of processors,which was previously infeasible using existing state-of-the-art solvers.
文摘The theory of Schur complement plays an important role in many fields such as matrix theory, control theory and computational mathematics. In this paper, some new estimates of diagonally, α-diagonally and product α-diagonally dominant degree on the Schur complement of matrices are obtained, which improve some relative results. As an application, we present several new eigenvalue inclusion regions for the Schur complement of matrices. Finally, we give a numerical example to illustrate the advantages of our derived results.
文摘Necessary and sufficient conditions for a schur complement of a con-s-k-EP matrix to be con-s-k-EP are determined. Further it is shown that in a con-s-k-EPr matrix, every secondary sub matrix of rank “r” is con-s-k-EPr. We have also discussed the way of expressing a matrix of rank r as a product of con-s-k-EPr matrices. Necessary and sufficient conditions for products of con-s-k-EPr partitioned matrices to be con-s-k-EPr are given.
基金the NIH-RCMI(Grant No.347U54MD013376)the affliated project award from the Center for Equitable Artificial Intelligence and Machine Learning Systems at Morgan State University(Project ID 02232301)+3 种基金the National Science Foundation awards(Grant No.1831950).The work of G.Ju is supported in part by the National Key R&D Program of China(Grant No.2017YFB1001604)the National Natural Science Foundation of China(Grant No.11971221)the Shenzhen Sci-Tech Fund(Grant Nos.RCJC20200714114556020,JCYJ20170818153840322,JCYJ20190809150413261)the Guangdong Provincial Key Laboratory of Computational Science and Material Design(Grant No.2019B030301001).
文摘In this paper,we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems.One type of the preconditioners are based on the nested(or recursive)Schur complement,the other is based on an additive type Schur complement after permuting the original saddle point systems.We analyze different preconditioners incorporating the exact Schur complements.We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements.These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly.Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions.
基金supported by National Natural Science Foundation of China(11571292,11471279)National Natural Science Foundation for Youths of China(11401505)+1 种基金the Key Project of National Natural Science Foundation of China(91430213),the first class General Financial Grant from the China Postdoctoral Science Foundation(2015M582819)the General Project of Hunan Provincial Natural Science Foundation of China(2015JJ2134).
文摘The distribution for eigenvalues of Schur complement of matrices plays an important role in many mathematical problems.In this paper,we firstly present some criteria for H-matrix.Then as application,for two class matrices whose sub-matrices areγ-diagonally dominant and productγ-diagonally dominant,we show that the eigenvalues of the Schur complement are located in the Gersgorin discs and the Ostrowski discs of the original matrices under certain conditions.
基金Project supported by the National Natural Science Foundation of China (Nos.10771161,11071188)
文摘Let E be a Hilbert C＊-module,and Y be an orthogonally complemented closed submodule of E.The authors generalize the definitions of Y-complementability and Y-compatibility for general（adjointable） operators from Hilbert space to Hilbert C＊-module,and discuss the relationship between each other.Several equivalent statements about Y-complementability and Y-compatibility,and several representations of Schur complements of Y-complementable operators（especially,of Y-compatible operators and of positive Y-compatible operators） on a Hilbert C＊-module are obtained.In addition,the quotient property for Schur complements of matrices is generalized to the quotient property for Schur complements of Y-complementable operators and Y＊-complementable operators on a Hilbert C＊-module.
基金Acknowledgements This work was supported in part by the Science Foundation of the Education Department of Shaanxi Province of China （No. 2013JK0593）, the Scientific Research Foundation of Xi＇an Polytechnic University （No. BS1014）, the China Postdoctoral Science Foundation （No. 20110491668）, and the National Natural Science Foundations of China （Grant Nos. 11201362, 11271297, 11101325, 11171270）.
文摘The definitions of θ-ray pattern proposed to establish some new results matrix and θ-ray matrix are firstly on nonsingularity/singularity and convergence of general H-matrices. Then some conditions on the matrix A ∈ C^n×n and nonempty α （n） = {1,2,... ,n} are proposed such that A is an invertible H-matrix if A（α） and A/α are both invertible H-matrices. Furthermore, the important results on Schur complement for general H-matrices are presented to give the different necessary and sufficient conditions for the matrix A E HM and the subset α C （n） such that the Schur complement matrix A/α∈ HI^n-｜α｜ or A/α ∈ Hn-｜α｜^M or A/α ∈ H^n-｜ α｜^S.
文摘关于复分块矩阵P=A BC D∈Cm×n的广义Schur补S=A-BD+C,T=D-CA+B,S=A-BDgC,T=D-CAgB,这里,D+,A+分别代表D,A的M oore-Penrose逆,Dg,Ag分别代表D,A的群逆。这篇文章我们主要给出当P是幂等矩阵时,在一定条件下,P的某些性质成立时,S,T具有同样的性质。
