共查询到20条相似文献,搜索用时 15 毫秒
1.
《Applied Mathematics Letters》2007,20(3):284-289
In the present work, we give some new results for block minimal residual methods when applied to multiple linear systems. Using the Schur complement, we develop new expressions for the approximation obtained, for the corresponding residual and for the Frobenius residual norm. These results could be used to derive new convergence properties for the block minimal residual methods. 相似文献
2.
《Journal of Computational and Applied Mathematics》1997,84(2):243-256
Norm-minimizing-type methods for solving large sparse linear systems with symmetric and indefinite coefficient matrices are considered. The Krylov subspace can be generated by either the Lanczos approach, such as the methods MINRES, GMRES and QMR, or by a conjugate-gradient approach. Here, we propose an algorithm based on the latter approach. Some relations among the search directions and the residuals, and how the search directions are related to the Krylov subspace are investigated. Numerical experiments are reported to verify the convergence properties. 相似文献
3.
Two minimal residual methods for solving linear systems of the form (αU + βI)x = b, where U is a unitary matrix, are compared numerically. The first method uses conventional Krylov subspaces, while the second involves generalized Krylov subspaces. Experiments favor the second method if |α| > |β|. Moreover, the greater the ratio |α|/|β|, the higher the superiority of the second method. 相似文献
4.
We consider two Krylov subspace methods for solving linear systems, which are the minimal residual method and the orthogonal
residual method. These two methods are studied without referring to any particular implementations. By using the Petrov–Galerkin
condition, we describe the residual norms of these two methods in terms of Krylov vectors, and the relationship between there
two norms. We define the Ritz singular values, and prove that the convergence of these two methods is governed by the convergence
of the Ritz singular values.
AMS subject classification 65F10 相似文献
5.
A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = ρU + ζI, where ρ and ζ are constants, U is a unitary matrix and I is the identity matrix. We show that for such matrices a Krylov subspace basis can be generated by recursion formulas with few terms. This leads to a minimal residual algorithm that requires little storage and makes it possible to determine each iterate with fairly little arithmetic work. This algorithm provides a model for iterative methods for non-Hermitian linear systems of equations, in a similar way to the conjugate gradient and conjugate residual algorithms. Our iterative scheme illustrates that results by Faber and Manteuffel [3,4] on the existence of conjugate gradient algorithms with short recurrence relations, and related results by Joubert and Young [13], can be extended. 相似文献
6.
Hongmin Ren 《Journal of Computational and Applied Mathematics》2010,235(1):218-228
A local convergence analysis of inexact Newton-type methods using a new type of residual control was recently presented by C. Li and W. Shen. Here, we introduce the center-Hölder condition on the operator involved, and use it in combination with the Hölder condition to provide a new local convergence analysis with the following advantages: larger radius of convergence, and tighter error bounds on the distances involved. These results are obtained under the same hypotheses and computational cost. Numerical examples further validating the theoretical results are also provided in this study. 相似文献
7.
N. S. Chernikov 《Journal of Mathematical Sciences》2009,163(6):774-784
The author establishes a number of new propositions asserting that groups with various minimal conditions are Chernikov under
some additional extremely weak finiteness conditions. These propositions include a great many known theorems on groups with
minimal conditions. 相似文献
8.
Summary In this paper, we discuss an approach to the obstacle problem for minimal boundaries via penalty techniques. After investigating some classes of penalized problems, a general method is introduced, based on the minimization of a suitable functional containing an extra term related to the mean curvature of the given obstacle. 相似文献
9.
A minimal residual method, called MINRES-N2, that is based on the use of unconventional Krylov subspaces was previously proposed by the authors for solving a system of linear equations Ax = b with a normal coefficient matrix whose spectrum belongs to an algebraic second-degree curve Γ. However, the computational scheme of this method does not cover matrices of the form A = αU + βI, where U is an arbitrary unitary matrix; for such matrices, Γ is a circle. Systems of this type are repeatedly solved when the eigenvectors of a unitary matrix are calculated by inverse iteration. In this paper, a modification of MINRES-N2 suitable for linear polynomials in unitary matrices is proposed. Numerical results are presented demonstrating the significant superiority of the modified method over GMRES as applied to systems of this class. 相似文献
10.
For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability
and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems,
namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite
versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite
preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning)
and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean
norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations
for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where
the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this
case.
This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative
Research Grant CRG 960782 相似文献
11.
Summary.
Discretisation of the classical Stokes problem gives rise
to symmetric indefinite matrices with eigenvalues which,
in a precise way, are not symmetric about the origin, but which
do depend on a mesh size parameter. Convergence
estimates for the Conjugate Residual or Minimum Residual
iterative solution of such systems are given by best
minimax polynomial approximations on an inclusion set for the
eigenvalues.
In this paper, an analytic convergence estimate for such
problems is given in terms of an asymptotically small
mesh size parameter.
Received
November 16, 1993 / Revised version received August 2,
1994 相似文献
12.
In this paper, we consider simple random sampling without replacement from a dichotomous finite population and derive a necessary and sufficient condition on the finite population parameters for a valid large sample Normal approximation to Hypergeometric probabilities. We then obtain lower and upper bounds on the difference between the Normal and the Hypergeometric distributions solely under this necessary and sufficient condition.
13.
We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution
of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type
instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our
error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions
are violated. Numerical examples are also provided. 相似文献
14.
15.
Summary. Stabilisation methods are often used to circumvent the difficulties associated with the stability of mixed finite element
methods. Stabilisation however also means an excessive amount of dissipation or the loss of nice conservation properties.
It would thus be desirable to reduce these disadvantages to a minimum. We present a general framework, not restricted to mixed
methods, that permits to introduce a minimal stabilising term and hence a minimal perturbation with respect to the original problem. To do so, we rely on the fact that
some part of the problem is stable and should not be modified. Sections 2 and 3 present the method in an abstract framework. Section 4 and 5 present
two classes of stabilisations for the inf-sup condition in mixed problems. We present many examples, most arising from the
discretisation of flow problems. Section 6 presents examples in which the stabilising terms is introduced to cure coercivity
problems.
Received August 9, 1999 / Revised version received May 19, 2000 / Published online March 20, 2001 相似文献
16.
Vera Zeidan 《Applied Mathematics and Optimization》1989,20(1):19-31
The aim of this article is to provide second-order sufficiency criteria that extend known ones in [6] and [7] to the case where the control candidate and some of the data are merely essentially bounded, and/or the control setU is any convex subset of
m
. In the classical setting, where a restriction on the velocity is imposed
, it is shown that whenU is compact the known strengthened Weierstrass condition is equivalent to the Weierstrass condition with strict inequality.The research was supported by an NSERC Grant and by GNAFA of CNR, which are gratefully acknowledged. 相似文献
17.
This paper concerns the use of conjugate residual methods for the solution of nonsymmetric linear systems arising in applications to differential equations. We focus on an application derived from a seismic inverse problem. The linear system is a small perturbation to a symmetric positive-definite system, the nonsymmetries arising from discretization errors in the solution of certain boundary-value problems. We state and prove a new error bound for a class of generalized conjugate residual methods; we show that, in some cases, the perturbed symmetric problem can be solved with an error bound similar to the one for the conjugate residual method applied to the symmetric problem. We also discuss several applications for special distributions of eigenvalues.This work was supported in part by the National Science Foundation, Grants DMS-84-03148 and DCR-81-16779, and by the Office of Naval Research, Contract N00014-85-K-0725. 相似文献
18.
Projection methods have emerged as competitive techniques for solving large scale matrix Lyapunov equations. We explore the numerical solution of this class of linear matrix equations when a Minimal Residual (MR) condition is used during the projection step. We derive both a new direct method, and a preconditioned operator-oriented iterative solver based on CGLS, for solving the projected reduced least squares problem. Numerical experiments with benchmark problems show the effectiveness of an MR approach over a Galerkin procedure using the same approximation space. 相似文献
19.
C.A. Brebbia 《Applied Mathematical Modelling》1978,2(3):160-164
A rational classification of the different approximating methods which points out their fundamental differences is presented here. The classification is based on the weighted residual formulation and the type of approximating and weighting functions required.Laplace's equation is used throughout the paper to illustrate the different techniques and to permit critical comparisons, but the conclusions are equally applicable to any weighted residual formulation. A fluid flow application is presented and solved numerically using three different approximate techniques, each of them belonging to one of the three groups into which each of the methods have been classified. 相似文献
20.
We illustrate the use of the recent approach by P. Albrecht to the derivation of order conditions for partitioned Runge-Kutta methods for ordinary differential equations. 相似文献