对称正定Toeplitz型方程组的混合预处理

本文提出一个新的预条件子,用共轭梯度法求解对称正定的Teoplitz型线性方程组.该预处理子构造简单,易于实施快速傅里叶变换.理论和数值实验显示,我们的预处理子与T.Chan预处理子收敛性相近.     

Multilevel approaches for FSAI preconditioning

Factorized sparse approximate inverse (FSAI) preconditioners are robust algorithms for symmetric positive matrices, which are particularly attractive in a parallel computational environment because of their inherent and almost perfect scalability. Their parallel degree is even redundant with respect to the actual capabilities of the current computational architectures. In this work, we present two new approaches for FSAI preconditioners with the aim of improving the algorithm effectiveness by adding some sequentiality to the native formulation. The first one, denoted as block tridiagonal FSAI, is based on a block tridiagonal factorization strategy, whereas the second one, domain decomposition FSAI, is built by reordering the matrix graph according to a multilevel k‐way partitioning method followed by a bandwidth minimization algorithm. We test these preconditioners by solving a set of symmetric positive definite problems arising from different engineering applications. The results are evaluated in terms of performance, scalability, and robustness, showing that both strategies lead to faster convergent schemes regarding the number of iterations and total computational time in comparison with the native FSAI with no significant loss in the algorithmic parallel degree.     

Some Properties of the Optimal Preconditioner and the Generalized Superoptimal Preconditioner

<正>The optimal preconditioner and the superoptimal preconditioner were proposed in 1988 and 1992 respectively.They have been studied widely since then.Recently, Chen and Jin[6]extend the superoptimal preconditioner to a more general case by using the Moore-Penrose inverse.In this paper,we further study some useful properties of the optimal and the generalized superoptimal preconditioners.Several existing results are extended and new properties are developed.     

Two-level Preconditioner with Small Coarse Grid Appropriate for Unstructured Meshes

The presented paper contains a suggestion and an analysis of a two-level preconditioner appropriate for unstructured meshes. The convergence analysis is done for H¹-equivalent forms in the two-dimensional case; ways for generalization are discussed, too. The efficiency is demonstrated by numerical experiments.     

A preconditioning strategy for boundary element Galerkin methods

The Dirichlet and Neumann problems for the Laplacian are reformulated in the usual way as boundary integral equations of the first kind with symmetric kernels. These integral equations are solved using Galerkin's method with piecewise-constant and piecewise-linear boundary elements, respectively. In both cases, the stiffness matrix is symmetric and positive-definite, and has a condition number of order N, the number of degrees of freedom. By contrast, the condition number of the product of the two stiffness matrices is bounded independently of N. Hence, we can use the Neumann stiffness matrix to precondition the Dirichlet stiffness matrix, and vice versa. © 1997 John Wiley & Sons, Inc.     

A New Class AOR Preconditioner for $L$-Matrices

Hadjidimos(1978) proposed a classical accelerated overrelaxation(AOR) iterative method to solve the system of linear equations, and discussed its convergence under the conditions that the coefficient matrices are irreducible diagonal dominant, L-matrices, and consistently orders matrices. Several preconditioned AOR methods have been proposed to solve system of linear equations Ax = b, where A ∈ R~(n×n) is an L-matrix. In this work, we introduce a new class preconditioners for solving linear systems and give a comparison result and some convergence result for this class of preconditioners. Numerical results for corresponding preconditioned GMRES methods are given to illustrate the theoretical results.     

Toeplitz approximate inverse preconditioner for banded Toeplitz matrices

In this paper we introduce a new preconditioner for banded Toeplitz matrices, whose inverse is itself a Toeplitz matrix. Given a banded Hermitian positive definite Toeplitz matrixT, we construct a Toepliz matrixM such that the spectrum ofMT is clustered around one; specifically, if the bandwidth ofT is , all but eigenvalues ofMT are exactly one. Thus the preconditioned conjugate gradient method converges in +1 steps which is about half the iterations as required by other preconditioners for Toepliz systems that have been suggested in the literature. This idea has a natural extension to non-banded and non-Hermitian Toeplitz matrices, and to block Toeplitz matrices with Toeplitz blocks which arise in many two dimensional applications in signal processing. Convergence results are given for each scheme, as well as numerical experiments illustrating the good convergence properties of the new preconditioner.Partly supported by a travel fund from the Deutsche Forschungsgemeinschaft.Research supported in part by Oak Ridge Associated Universities grant no. 009707.     

鞍点问题基于半增广的松弛分裂预条件子

对于(1,1)块为正定的鞍点问题,本文给出了半增广松弛分裂预条件子.文中分析了预条件矩阵特征值分布情况,并用数值实验验证了半增广松弛分裂预条件子的有效性.     

Multilevel Additive Schwarz Method for the Version of the Galerkin Boundary Element Method

We study a multilevel additive Schwarz method for the - version of the Galerkin boundary element method with geometrically graded meshes. Both hypersingular and weakly singular integral equations of the first kind are considered. As it is well known the - version with geometric meshes converges exponentially fast in the energy norm. However, the condition number of the Galerkin matrix in this case blows up exponentially in the number of unknowns . We prove that the condition number of the multilevel additive Schwarz operator behaves like . As a direct consequence of this we also give the results for the -level preconditioner and also for the - version with quasi-uniform meshes. Numerical results supporting our theory are presented.

     

Multilevel Preconditioners for Mixed Methods for Second Order Elliptic Problems

A new approach for constructing algebraic multilevel preconditioners for mixed finite element methods for second order elliptic problems with tensor coefficients on general geometry is proposed. The linear system arising from the mixed methods is first algebraically condensed to a symmetric, positive definite system for Lagrange multipliers, which corresponds to a linear system generated by standard nonconforming finite element methods. Algebraic multilevel preconditioners for this system are then constructed based on a triangulation of the domain into tetrahedral substructures. Explicit estimates of condition numbers and simple computational schemes are established for the constructed preconditioners. Finally, numerical results for the mixed finite element methods are presented to illustrate the present theory.     

On the eigenvalue distribution of preconditioned nonsymmetric saddle point matrices

In this paper, we derive bounds for the complex eigenvalues of a nonsymmetric saddle point matrix with a symmetric positive semidefinite (2,2) block, that extend the corresponding previous bounds obtained by Bergamaschi. For the nonsymmetric saddle point problem, we propose a block diagonal preconditioner for the conjugate gradient method in a nonstandard inner product. Numerical experiments are also included to test the performance of the presented preconditioner. Copyright © 2013 John Wiley & Sons, Ltd.     

Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Efficient multilevel preconditioners are developed and analyzed for the quadrature finite element Galerkin approximation of the biharmonic Dirichlet problem. The quadrature scheme is formulated using the Bogner–Fox–Schmit rectangular element and the product two‐point Gaussian quadrature. The proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the quadrature scheme. The preconditioners are implemented by optimal algorithms, and they are used to accelerate convergence of the preconditioned conjugate gradient method. Numerical results are presented demonstrating efficiency of the preconditioners. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Alternating oblique projections for coupled linear systems

In this work we propose the use of alternating oblique projections (AOP) for the solution of the saddle points systems resulting from the discretization of domain decomposition problems. These systems are called coupled linear systems. The AOP method is a descent method in which the descent direction is defined by using alternating oblique projections onto the search subspaces. We prove that this method is a preconditioned simple gradient (Uzawa) method with a particular preconditioner. Finally, a preconditioned conjugate gradient based version of AOP is proposed. AMS subject classification 65F10, 65N22, 65Y05     

Flexible incomplete Cholesky factorization with multi‐parameters to control the number of nonzero elements in preconditioners

An incomplete Cholesky (IC) factorization with multi‐parameters is presented. The marked virtue of the proposed IC factorization algorithm is to dynamically control the number of nonzero elements in each column of the IC factorization preconditioner L with the help of these involved parameters. Parameter setting strategies are also given. Numerical results show that the total computing time for both computation of the preconditioner L and iterative solution is evidently reduced for almost all test matrices. In general, these parameters can obviously enhance the effectiveness and performance of the IC factorization. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability of T. Chan＇s Preconditioner from Numerical Range

A matrix is said to be stable if the real parts of all the eigenvalues are negative. In this paper, for any matrix An, we discuss the stability properties of T. Chan＇s preconditioner cU（An） from the viewpoint of the numerical range. An application in numerical ODEs is also given.     

Generalization of Strang's Preconditioner with Applications to Toeplitz Least Squares Problems

In this paper, we propose a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices A_n. The th column of our circulant preconditioner Sn is equal to the th column of the given matrix A_n. Thus if A_n is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be defined as . This construction is similar to the forward-backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. We show that if the matrix A_n has decaying coefficients away from the main diagonal, then is a good preconditioner for An. Comparisons of our preconditioner with other circulant-based preconditioners are carried out for some 1-D Toeplitz least squares problems: min ∥ b - Ax∥₂. Preliminary numerical results show that our preconditioner performs quite well, in comparison to other circulant preconditioners. Promising test results are also reported for a 2-D deconvolution problem arising in ground-based atmospheric imaging.     

Analysis of a novel preconditioner for a class of p‐level lower rank extracted systems

This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, Âx=b, which we call p‐level lower rank extracted systems (p‐level LRES), by the preconditioned conjugate gradient method. The study of these systems is motivated by the numerical approximation of integral equations with convolution kernels defined on arbitrary p‐dimensional domains. This is in contrast to p‐level Toeplitz systems which only apply to rectangular domains. The coefficient matrix, Â, is a principal submatrix of a p‐level Toeplitz matrix, A, and the preconditioner for the preconditioned conjugate gradient algorithm is provided in terms of the inverse of a p‐level circulant matrix constructed from the elements of A. The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix which leads to a substantial reduction in the computational cost of solving LRE systems. Copyright © 2006 John Wiley & Sons, Ltd.     

Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs

This paper addresses the solution of parabolic evolution equations simultaneously in space and time as may be of interest in, for example, optimal control problems constrained by such equations. As a model problem, we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual Petrov–Galerkin variational formulation of the heat equation in space–time results in a generalized least squares problem. This formulation admits a unique, quasi‐optimal solution in the natural space–time Hilbert space and serves as a basis for the development of space–time compressive solution algorithms. The solution of the heat equation is obtained by applying the conjugate gradient method to the normal equations of the generalized least squares problem. Starting from stable subspace splittings in space and in time, multilevel space–time preconditioners for the normal equations are derived. In order to reduce the complexity of the full space–time problem, all computations are performed in a compressed or sparse format called the hierarchical Tucker format, supposing that the input data are available in this format. In order to maintain sparsity, compression of the iterates within the hierarchical Tucker format is performed in each conjugate gradient iteration. Its application to vectors in the hierarchical Tucker format is detailed. Finally, numerical results in up to five spatial dimensions based on the recently developed htucker toolbox for MATLAB are presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Two‐level preconditioners for serendipity finite element matrices

In this paper an approach to construct algebraic multilevel preconditioners for serendipity finite element matrices is presented. Two‐level preconditioners constructed in the paper allow to obtain multilevel preconditioners in serendipity case using multilevel preconditioners for linear finite element matrices. Copyright © 2006 John Wiley & Sons, Ltd.     

An Overlapping Domain Decomposition Preconditioner for High Order BEM with Anisotropic Elements

We study a preconditioner for the boundary element method with high order piecewise polynomials for hypersingular integral equations in three dimensions. The meshes may consist of anisotropic quadrilateral and triangular elements. Our preconditioner is based on an overlapping domain decomposition which is assumed to be locally quasi-uniform. Denoting the subdomain sizes by H _j and the overlaps by _j , we prove that the condition number of the preconditioned system is bounded essentially by max _j O(1+log H _j / _j )². The appearing constant depends linearly on the maximum ratio H _i /H _j for neighboring subdomains, but is independent of the elements' aspect ratios. Numerical results supporting our theory are reported.