求解加权线性最小二乘问题的预处理迭代方法

给出了求解一类加权线性最小二乘问题的预处理迭代方法,也就是预处理的广义加速超松弛方法（GAOR）,得到了一些收敛和比较结果．比较结果表明当原来的迭代方法收敛时,预处理迭代方法会比原来的方法具有更好的收敛率．而且,通过数值算例也验证了新预处理迭代方法的有效性．     

预处理CG算法解油藏模拟问题的有效性比较

1引言 在大型科学和工程计算问题的实际应用中，经常会遇到求解除椭圆型或抛物线型偏微分方程问题。经差分法或有限元方法离散化后得到一个大型稀疏线性方程组。本文比较了几     

A splitting preconditioner for the iterative solution of implicit Runge-Kutta and boundary value methods

We study preconditioned iterative methods for the linear systems arising in the numerical integration of ODEs and time-dependent PDEs by implicit Runge-Kutta and boundary value methods. A preconditioning strategy based on a Kronecker product splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matrix are established. Numerical examples are presented to illustrate the effectiveness of this approach.     

Preconditioned sor methods for generalized least-squares problems

1.IntroductionThegeneralizedleastsquaresproblem,definedasmin(Ax--b)"W--'(Ax--b),(1.1)xacwhereAERm",m>n,bERm,andWERm'misasymmetricandpositivedefinitematrix,isfrequentlyfoundwhensolvingproblemsinstatistics,engineeringandeconomics.Forexample,wegetgeneralizedleastsquaresproblemswhensolvingnonlinearregressionanalysisbyquasi-likelihoodestimation,imagereconstructionproblemsandeconomicmodelsobtainedbythemaximumlikelihoodmethod(of.[1,21).Paige[3,4]investigatestheproblemexplicitly.Hechangestheorig…     

求解加权线性最小二乘问题的一类预处理GAOR方法

为了快速求解一类来自加权线性最小二乘问题的2×2块线性系统,本文提出一类新的预处理子用以加速GAOR方法,也就是新的预处理GAOR方法.得到了一些比较结果,这些结果表明当GAOR方法收敛时,新方法比原GAOR方法和之前的一些预处理GAOR方法有更好的收敛性.而且,数值算例也验证了新预处理子的有效性．     

Preconditioned GAOR methods for solving weighted linear least squares problems

In this paper, we present the preconditioned generalized accelerated overrelaxation (GAOR) method for solving linear systems based on a class of weighted linear least square problems. Two kinds of preconditioning are proposed, and each one contains three preconditioners. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the convergence rate of the preconditioned GAOR methods is indeed better than the rate of the original method, whenever the original method is convergent. Finally, a numerical example is presented in order to confirm these theoretical results.     

非Hermite线性方程组的若干预处理迭代算法

非Hermite线性方程组在科学和工程计算中有着重要的理论研究意义和使用价值,因此如何高效求解该类线性方程组,一直是研究者所探索的方向.通过提出一种预处理方法,对非Hermite线性方程组和具有多个右端项的复线性方程组求解的若干迭代算法进行预处理,旨在提高原算法的收敛速度.最后通过数值试验表明,所提出的若干预处理迭代算法与原算法相比较,预处理算法迭代次数大大降低,且收敛速度明显优于原算法.除此之外,广义共轭A-正交残量平方法(GCORS2)的预处理算法与其他算法相比,具有良好的收敛性行为和较好的稳定性.     

Preconditioned Block-Iterative Methods on GPUs

An implementation of the incomplete-LU/Cholesky preconditioned block-iterative methods on the Graphics Processing Units (GPUs) using the CUDA parallel programming model is presented. In particular, we focus on the tradeoffs associated with the sparse matrix-vector multiplication with multiple vectors, sparse triangular solve with multiple right-hand-sides (rhs) as well as incomplete factorization with 0 fill-in. We use these building blocks to implement the block-CG and BiCGStab iterative methods for the symmetric positive definite (s.p.d.) and nonsymmetric linear systems, respectively. Also, in our numerical experiments we show that the implementation of the preconditioned block-iterative methods using the CUSPARSE library on the GPU achieves an average of 3× speedup over their MKL implementation on the CPU. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a type of matrix splitting preconditioners for a class of block two-by-two linear systems

Recently, Bai et al. (2013) proposed an effective and efficient matrix splitting iterative method, called preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method, for two-by-two block linear systems of equations. The eigenvalue distribution of the iterative matrix suggests that the splitting matrix could be advantageously used as a preconditioner. In this study, the CGNR method is utilized for solving the PMHSS preconditioned linear systems, and the performance of the method is considered by estimating the condition number of the normal equations. Furthermore, the proposed method is compared with other PMHSS preconditioned Krylov subspace methods by solving linear systems arising in complex partial differential equations and a distributed control problem. The numerical results demonstrate the difference in the performance of the methods under consideration.     

Multilevel additive and multiplicative methods for orthogonal spline collocation problems

Summary. Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet boundary value problem for a self-adjoint elliptic partial differential equation on a rectangle. Additive and multiplicative preconditioners are defined respectively as sums and products of independent operators on a sequence of nested subspaces of the fine partition approximation space. A general theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent to the collocation discretization of the Laplacian with the spectral constants independent of the fine partition stepsize and the number of levels. The preconditioned conjugate gradient and preconditioned Orthomin methods are considered for the solution of collocation problems. An implementation of the methods is discussed and the results of numerical experiments are presented. Received March 1, 1994 / Revised version received January 23, 1996     

Iterative solution of matrix games using grid variational inequalities

A new approach to the approximate solution of matrix games is proposed. It is based on the reduction of the original problem to a variational inequality of a special form. In particular, this makes it possible to design preconditioned iterative methods, which proved to be effective as a tool for the numerical solution of large and ill-conditioned systems of linear algebraic equations.     

An implicit preconditioning strategy for large-scale generalized Sylvester equations

Large-scale generalized Sylvester equations appear in several important applications. Although the involved operator is linear, solving them requires specialized techniques. Different numerical methods have been designed to solve them, including direct factorization methods suitable for small size problems, and Krylov-type iterative methods for large-scale problems. For these iterative schemes, preconditioning is always a difficult task that deserves to be addressed. We present and analyze an implicit preconditioning strategy specially designed for solving generalized Sylvester equations that uses a preconditioned residual direction at every iteration. The advantage is that the preconditioned direction is built implicitly, avoiding the explicit knowledge of the given matrices. Only the effect of the matrix-vector product with the given matrices is required. We present encouraging numerical experiments for a set of different problems coming from several applications.     

关于具优势对称部分的不定线性代数方程组的分裂极小残量算法

１．引 言 考虑大型稀疏线性代数方程组 为利用系数矩阵的稀疏结构以尽可能减少存储空间和计算开销，Ｋｒｙｌｏｖ子空间迭代算法［１，１６，２３］及其预处理变型［６，８，１３，１８，１９］通常是求解（１）的有效而实用的方法．当系数矩阵对称正定时，共轭梯度法（ＣＧ（     

Numerical Integration of Reaction–Diffusion Systems

Approximating numerically the solutions of a reaction–diffusion system in an efficient manner requires the application of implicit methods, since the Courant–Friedrichs–Lewy condition on explicit methods imposes a time step of the order of the square of the space step. In this article, we review two types of strategies which are expected to yield reasonably precise solutions within a reasonable computing time. The first examines methods for solving the linear step necessary in any resolution procedure; estimates of CPU time in terms of the error are given in the non preconditioned and in the preconditioned case – provided that it is possible to define an efficient preconditioner. The second strategy is based on splitting, with or without extrapolation. The respective faults and qualities of both strategies are examined; they lead to a list of difficult analytical and numerical problems with possible hints as to their solution.     

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.     

A generalized preconditioned HSS method for non-Hermitian positive definite linear systems

Based on the HSS (Hermitian and skew-Hermitian splitting) and preconditioned HSS methods, we will present a generalized preconditioned HSS method for the large sparse non-Hermitian positive definite linear system. Our method is essentially a two-parameter iteration which can extend the possibility to optimize the iterative process. The iterative sequence produced by our generalized preconditioned HSS method can be proven to be convergent to the unique solution of the linear system. An exact parameter region of convergence for the method is strictly proved. A minimum value for the upper bound of the iterative spectrum is derived, which is relevant to the eigensystem of the products formed by inverse preconditioner and splitting. An efficient preconditioner based on incremental unknowns is presented for the actual implementation of the new method. The optimality and efficiency are effectively testified by some comparisons with numerical results.     

一个求解无约束优化问题的条件预优共轭梯度法

1 引言 考虑无约束最优化问题minf(x)(1.1)     

A balancing domain decomposition by constraints preconditioner for a weakly over‐penalized symmetric interior penalty method

We develop a balancing domain decomposition by constraints preconditioner for a weakly over‐penalized symmetric interior penalty method for second‐order elliptic problems. We show that the condition number of the preconditioned system satisfies similar estimates as those for conforming finite element methods. Corroborating numerical results are also presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Preconditioned Krylov subspace method for the solution of least-squares problems

We consider preconditioned Krylov subspace iteration methods, e.g., CG, LSQR and GMRES, for the solution of large sparse least-squares problems min ∥Ax – b ∥₂, with A ∈ R ^m×n, based on the Krylov subspaces K_k (BA, Br) and K_k (AB, r), respectively, where B ∈ R ^n×m is the preconditioning matrix. More concretely, we propose and implement a class of incomplete QR factorization preconditioners based on the Givens rotations and analyze in detail the efficiency and robustness of the correspondingly preconditioned Krylov subspace iteration methods. A number of numerical experiments are used to further examine their numerical behaviour. It is shown that for both overdetermined and underdetermined least-squares problems, the preconditioned GMRES methods are the best for large, sparse and ill-conditioned matrices in terms of both CPU time and iteration step. Also, comparisons with the diagonal scaling and the RIF preconditioners are given to show the superiority of the newly-proposed GMRES-type methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sine transform based preconditioning techniques for space fractional diffusion equations

We study the preconditioned iterative methods for the linear systems arising from the numerical solution of the multi-dimensional space fractional diffusion equations. A sine transform based preconditioning technique is developed according to the symmetric and skew-symmetric splitting of the Toeplitz factor in the resulting coefficient matrix. Theoretical analyses show that the upper bound of relative residual norm of the GMRES method when applied to the preconditioned linear system is mesh-independent which implies the linear convergence. Numerical experiments are carried out to illustrate the correctness of the theoretical results and the effectiveness of the proposed preconditioning technique.