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121.
Zhong-Zhi Bai. 《Mathematics of Computation》2006,75(254):791-815
For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.
122.
Yuliang Han 《Journal of Mathematical Analysis and Applications》2007,336(2):962-974
In this paper almost periodic random sequence in probability is defined and investigated. It is also applied to random difference equations by means of exponential dichotomy. The existence of such kind of solutions of random difference equations is discussed. 相似文献
123.
Approximation of an Eigenvalue Problem Associated with the Stokes Problem by the Stream Function-Vorticity-Pressure Method 总被引:3,自引:2,他引:1
By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure
method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence
for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of
the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical
experiments are reported.
The first author was supported by China Postdoctoral Sciences Foundation. 相似文献
124.
Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation 总被引:7,自引:0,他引:7
It is well-known that on uniform meshes the piecewise linearconforming finite element solution of the Poisson equation approximatesthe interpolant to a higher order than the solution itself.In this paper, this type of superclose property is studied forthe canonical interpolant defined by the nodal functionals ofseveral non-conforming finite elements of lowest order. By givingexplicit examples we show that some non-conforming finite elementsdo not admit the superclose property. In particular, we discusstwo non-conforming finite elements which satisfy the supercloseproperty. Moreover, applying a postprocessing technique, wecan also state a superconvergence property for the discretizationerror of the postprocessed discrete solution to the solutionitself. Finally, we show that an extrapolation technique leadsto a further improvement of the accuracy of the finite elementsolution. 相似文献
125.
Strict feasibility plays an important role in the development of the theoryand algorithms of complementarity problems. In this paper, we establishsufficient conditions to ensure strict feasibility of a nonlinearcomplementarity problem. Our analysis method, based on a newly introducedconcept of -exceptional sequence, can be viewed as a unified approachfor proving the existence of a strictly feasible point. Some equivalentconditions of strict feasibility are also developed for certaincomplementarity problems. In particular, we show that aP*-complementarity problem is strictly feasible if and only ifits solution set is nonempty and bounded. 相似文献
126.
Recently, we have extended SDP by adding a quadratic term in the objective function and give a potential reduction algorithm using NT directions. This paper presents a predictor–corrector algorithm using both Dikin-type and Newton centering steps and studies properties of Dikin-type step. In this algorithm, when the condition K(XS) is less than a given number K
0, we use Dikin-type step. Otherwise, Newton centering step is taken. In both cases, step-length is determined by line search. We show that at least a constant reduction in the potential function is guaranteed. Moreover the algorithm is proved to terminate in O
log(1/)) steps. In the end of this paper, we discuss how to compute search direction (X,S) using the conjugate gradient method. 相似文献
127.
New properties of a nonlinear conjugate gradient method 总被引:6,自引:0,他引:6
Yu-Hong Dai 《Numerische Mathematik》2001,89(1):83-98
Summary. This paper provides several new properties of the nonlinear conjugate gradient method in [5]. Firstly, the method is proved
to have a certain self-adjusting property that is independent of the line search and the function convexity. Secondly, under
mild assumptions on the objective function, the method is shown to be globally convergent with a variety of line searches.
Thirdly, we find that instead of the negative gradient direction, the search direction defined by the nonlinear conjugate
gradient method in [5] can be used to restart any optimization method while guaranteeing the global convergence of the method.
Some numerical results are also presented.
Received March 12, 1999 / Revised version received April 25, 2000 / Published online February 5, 2001 相似文献
128.
Conjugate gradient methods are a class of important methods for unconstrained optimization, especially when the dimension
is large. This paper proposes a new conjugacy condition, which considers an inexact line search scheme but reduces to the
old one if the line search is exact. Based on the new conjugacy condition, two nonlinear conjugate gradient methods are constructed.
Convergence analysis for the two methods is provided. Our numerical results show that one of the methods is very efficient
for the given test problems.
Accepted 15 September 2000. Online publication 8 December 2000. 相似文献
129.
130.