鞍点问题的广义位移分裂预条件子

对于大型稀疏非Hermitian正定线性方程组,Bai等人提出了一种位移分裂预条件子(J.Comput.Math.,24(2006)539-552).本文将这种思想用到鞍点问题上并提出了一种广义位移分裂(Generalized Shift Splitting,GSS)预条件子,同时证明了该预条件子所对应分裂迭代法的无条件收敛性.最后用数值算例验证了新预条件子的有效性.     

Domain decomposition preconditioners of Neumann-Neumann type for hp-approximations on boundary layer meshes in three dimensions

We develop and analyse Neumann–Neumann methods for hpfinite-element approximations of scalar elliptic problems ongeometrically refined boundary layer meshes in three dimensions.These are meshes that are highly anisotropic where the aspectratio typically grows exponentially with the polynomial degree.The condition number of our preconditioners is shown to be independentof the aspect ratio of the mesh and of potentially large jumpsof the coefficients. In addition, it only grows polylogarithmicallywith the polynomial degree, as in the case of p approximationson shape-regular meshes. This work generalizes our previousone on two-dimensional problems in Toselli & Vasseur (2003a,submitted to Numerische Mathematik, 2003c to appear in Comput.Methods Appl. Mech. Engng.) and the estimates derived here canbe employed to prove condition number bounds for certain typesof FETI methods.     

A note on skewcirculant preconditioners for elliptic problems

In a recent paper Chan and Chan study the use of circulant preconditioners for the solution of elliptic problems. They prove that circulant preconditioners can be chosen so that the condition number of the preconditioned system can be reduced fromO(n ² ) toO(n). In addition, using the Fast Fourier Transform, the computation of the preconditioner is highly parallelizable. To obtain their result, Chan and Chan introduce a shift /p/n ² for some >0. The aim of this paper is to consider skewcirculant preconditioners, and to show that in this case the condition number ofO(n) can easily be shown without using the somewhat unsatisfactory shift /p/n ². Furthermore, our estimates are more precise.     

A Note on the Problem of Multivariable Adaptive Tracking

In this note we consider the problem of adaptive tracking formultivariable relative degree one, minimum phase systems. Ourapproach is in the spirit of Helmke et al. (1990) who consideredadaptive tracking of sinusoidal reference signals for single-input,single-output systems. The case of relative degree r, single-input,single-output, minimum-phase systems has been considered inMareels (1984). The main contribution we make is in establishingthat the sufficient conditions for universal adaptive tracking,namely minimum phase, relative degree one and the internal modelprinciple, hold for the suitably precompensated plant. Thisenables us to apply the techniques of Helmke et al. (1990).     

Preconditioners Based on Fit Techniques for the Iterative Regularization in the Image Deconvolution Problem

For large-scale image deconvolution problems, the iterative regularization methods can be favorable alternatives to the direct methods. We analyze preconditioners for regularizing gradient-type iterations applied to problems with 2D band Toeplitz coefficient matrix. For problems having separable and positive definite matrices, the fit preconditioner we have introduced in a previous paper has been shown to be effective in conjunction with CG. The cost of this preconditioner is of O(n²) operations per iteration, where n² is the pixels number of the image, whereas the cost of the circulant preconditioners commonly used for this type of problems is of O(n² log n) operations per iteration. In this paper the extension of the fit preconditioner to more general cases is proposed: namely the nonseparable positive definite case and the symmetric indefinite case. The major difficulty encountered in this extension concerns the factorization phase, where a further approximation is required. Three approximate factorizations are proposed. The preconditioners thus obtained have still a cost of O(n²) operations per iteration. A numerical experimentation shows that the fit preconditioners are competitive with the regularizing Chan preconditioner, both in the regularizing efficiency and the computational cost. AMS subject classification (2000) 65F10, 65F22.Received October 2003. Accepted December 2004. Communicated by Lars Eldén.     

A note on an SOR-like method for augmented systems

Golub et al. (2001, BIT, 41, 71–85) gave a generalizedsuccessive over-relaxation method for the augmented systems.In this paper, the connection between the SOR-like method andthe preconditioned conjugate gradient (PCG) method for the augmentedsystems is investigated. It is shown that the PCG method isat least as accurate (fast) as the SOR-like method. Numericalexamples demonstrate that the PCG method is much faster thanthe SOR-like method.     

A new iterative method for solving a class of complex symmetric system of linear equations

We present a new stationary iterative method, called Scale-Splitting (SCSP) method, and investigate its convergence properties. The SCSP method naturally results in a simple matrix splitting preconditioner, called SCSP-preconditioner, for the original linear system. Some numerical comparisons are presented between the SCSP-preconditioner and several available block preconditioners, such as PGSOR (Hezari et al. Numer. Linear Algebra Appl. 22, 761–776, 2015) and rotate block triangular preconditioners (Bai Sci. China Math. 56, 2523–2538, 2013), when they are applied to expedite the convergence rate of Krylov subspace iteration methods for solving the original complex system and its block real formulation, respectively. Numerical experiments show that the SCSP-preconditioner can compete with PGSOR-preconditioner and even more effective than the rotate block triangular preconditioners.     

Local and global behaviour of steady-state solutions of the Sel'kov model

In this paper we discuss steady-state solutions of the systemof reaction-diffusion equations known as the Sel'kov model.This model has been the subject of much discussion; in particular,analytical and numerical results have been discussed by Lopez-Gomezet al. (1992, IMA J. Num. Anal. 12, 405–28). We show thata simple analysis of the bifurcation function associated withthe system can explain many of the numerical observations, suchas the formation and development of loops of nontrivial solutions,in a simpler and more complete manner than the analysis of Lopez-Gomezet al. This allows for a clearer understanding of the qualitativebehaviour of the set of nontrivial solutions and hence of thebifurcation diagram.     

Approximation theorems for total quasi-?-asymptotically nonexpansive mappings with applications

The purpose of this article is to prove some approximation theorems of common fixed points for countable families of total quasi-?-asymptotically nonexpansive mappings which contain several kinds of mappings as its special cases in Banach spaces. In order to get the approximation theorems, the hybrid algorithms are presented and are used to approximate the common fixed points. Using this result, we also discuss the problem of strong convergence concerning the maximal monotone operators in a Banach space. The results of this article extend and improve the results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 134 (2005) 257-266], Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 149 (2007) 103-115], Li, Su [H. Y. Li, Y. F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72(2) (2010) 847-855], Su, Xu and Zhang [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3960], Wang et al. [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364-2371], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. 73 (2010) 2260-2270], Chang et al. [S.S. Chang, C.K. Chan, H.W. Joseph Lee, Modified block iterative algorithm for quasi-?-asymptotically nonexpansive mappings and equilibrium problem in Banach spaces, Appl. Math. Comput. 217 (2011) 7520-7530], Ofoedu and Malonza [E.U. Ofoedu, D.M. Malonza, Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type, Appl. Math. Comput. 217 (2011) 6019-6030] and Yao et al. [Y.H. Yao, Y.C. Liou, S.M. Kang, Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings, Appl. Math. Comput. 208 (2009) 211-218].     

Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay

In this note we propose a method for the integration of y'(t) = f(t, y(t), y(rt)), 0 t t_f y(0) = y₀, where 0 < r < 1, by a superconvengent s-stage continuousRK method of discrete global order p and continuous uniformorder q < p – 1 for the approximation of the delayedterm y(rt). We prove that, although the maximum attainable orderof the method on an arbitrary mesh is q' = min{p, q + 1}, byusing a quasi-geometric mesh, introduced by Bellen et al. (1997,Appl. Numer. Math. 24, 1997, 279–293), the optimal accuracyorder p is preserved.     

Local preconditioners for two‐level non‐overlapping domain decomposition methods

We consider additive two‐level preconditioners, with a local and a global component, for the Schur complement system arising in non‐overlapping domain decomposition methods. We propose two new parallelizable local preconditioners. The first one is a computationally cheap but numerically relevant alternative to the classical block Jacobi preconditioner. The second one exploits all the information from the local Schur complement matrices and demonstrates an attractive numerical behaviour on heterogeneous and anisotropic problems. We also propose two implementations based on approximate Schur complement matrices that are cheaper alternatives to construct the given preconditioners but that preserve their good numerical behaviour. Through extensive computational experiments we study the numerical scalability and the robustness of the proposed preconditioners and compare their numerical performance with well‐known robust preconditioners such as BPS and the balancing Neumann–Neumann method. Finally, we describe a parallel implementation on distributed memory computers of some of the proposed techniques and report parallel performances. Copyright © 2001 John Wiley & Sons, Ltd.     

Matrix fractions and full system equivalence

Following the definition by Hayton et al. (1990) of full systemequivalence for linear multivariables systems, the concept offull unimodular equivalence is defined for matrix fraction descriotions(MFDs).Full unimodular equivalence is an extension of unimodular equivalence,which has been proposed by Kailath (1980) and Smith (1981) forsystems described by polynomial matrix models in either leftor right MFD form. Full unimodular equivalence has the propertyof leaving invariant both the finite and infinite structureof the system. It is shown that full unimodular equivalencefor MFDs and full system equivalence coincide.     

A generalization of the HSS-based sequential two-stage method for solving non-Hermitian saddle point problems

For non-Hermitian saddle point problems with non-Hermitian positive definite (1,1)-block, Zhu et al. studied the HSS-based sequential two-stage method (see Zhu et al. Appl. Math. Comput. 242, 907–916 19). However, this approach may not work when the (1,1)-block of the saddle point problems is weakly Hermitian or skew-Hermitian dominant. By introducing a new preconditioning matrix, a generalization of the HSS-based sequential two-stage method is proposed for solving non-Hermitian saddle-point problems with non-Hermitian positive definite and Hermitian or skew-Hermitian dominant (1,1)-block. Theoretical analysis shows that the proposed iterative method is convergent. Numerical experiments are provided to confirm the theoretical results, which demonstrate that the generalized method is effective and feasible for solving saddle point problems with non-Hermitian positive definite and Hermitian or skew-Hermitian dominant (1,1)-block.     

How to prove that a preconditioner cannot be superlinear

In the general case of multilevel Toeplitz matrices, we recently proved that any multilevel circulant preconditioner is not superlinear (a cluster it may provide cannot be proper). The proof was based on the concept of quasi-equimodular matrices, although this concept does not apply, for example, to the sine-transform matrices. In this paper, with a new concept of partially equimodular matrices, we cover all trigonometric matrix algebras widely used in the literature. We propose a technique for proving the non-superlinearity of certain frequently used preconditioners for some representative sample multilevel matrices. At the same time, we show that these preconditioners are, in a certain sense, the best among the sublinear preconditioners (with only a general cluster) for multilevel Toeplitz matrices.

     

Robust tracking for MIMO nonlinear time-dependent mismatched uncertain systems: non-adaptive versus adaptive control

This paper considers the problem of robust output-tracking controlfor multi-input multi-output (MIMO) nonlinear systems in thepresence of mismatched time-dependent uncertainties. It firstgeneralizes the stable combined-variable perturbation method(SCVPM) which was proposed by Li et al. (1996) to derive a newrobust controller and its adaptive version for MIMO systemsto track a desired trajectory. It is shown that both controllersnot only stabilize the closed-loop systems but also guaranteethat the tracking errors remain in an O() neighbourhood of theorgin, where is a small design parameter of the controller.Moreover, the adaptive robust controller is only based on thenominal system, and no a priori data on the uncertainties areneeded. Therefore, it is more feasibly implemented than thenon-adaptive controller.     

Strang-type preconditioners for systems of LMF-based ODE codes

We consider the solution of ordinary differential equations(ODEs) using boundary value methods. These methods require thesolution of one or more unsymmetric, large and sparse linearsystems. The GMRES method with the Strang-type block-circulantpreconditioner is proposed for solving these linear systems.We show that if an A_k1,k2 -stable boundary value method is usedfor an m-by-m system of ODEs, then our preconditioners are invertibleand all the eigenvalues of the preconditioned systems are 1except for at most 2m(k₁ + k₂) outliers. It follows that whenthe GMRES method is applied to solving the preconditioned systems,the method will converge in at most 2m(k₁ + k₂) + 1 iterations.Numerical results are given to illustrate the effectivenessof our methods. Received 8 October 1999. Accepted 30 May 2000.     

Block diagonally preconditioned PIU methods of saddle point problem

For large sparse saddle point problems, we firstly introduce the block diagonally preconditioned Gauss-Seidl method (PBGS) which reduces to the GSOR method [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1-38] and PIU method [Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900-2932] when the preconditioners equal to different matrices, respectively. Then we generalize the PBGS method to the PPIU method and discuss the sufficient conditions such that the spectral radius of the PPIU method is much less than one. Furthermore, some rules are considered for choices of the preconditioners including the splitting method of the (1, 1) block matrix in the PIU method and numerical examples are given to show the superiority of the new method to the PIU method.     

A posteriori L2 error estimation on anisotropic tetrahedral finite element meshes

A new a posteriori L₂ norm error estimator is proposed for thePoisson equation. The error estimator can be applied to anisotropictetrahedral or triangular finite element meshes. The estimatoris rigorously analysed for Dirichlet and Neumann boundary conditions. The lower error bound relies on specifically designed anisotropicbubble functions and the corresponding inverse inequalities.The upper error bound utilizes non-standard anisotropic interpolationestimates. Its proof requires H² regularity of the Poisson problem,and its quality depends on how good the anisotropic mesh resolvesthe anisotropy of the problem. This is measured by a so-called‘matching function’. A numerical example supports the anisotropic error analysis.     

Error expansion for a first-order upwind difference scheme applied to a model convection-diffusion problem

A singularly perturbed convection–diffusion problem isconsidered. The problem is discretized using a simple first-orderupwind difference scheme on general meshes. We derive an expansionof the error of the scheme that enables uniform error boundswith respect to the perturbation parameter in the discrete maximumnorm for both a defect correction method and the Richardsonextrapolation technique. This generalizes and simplifies resultsobtained in earlier publications by Fröhner et al.(2001,Numer. Algorithms, 26, 281–299) and by Natividad &Stynes (2003, Appl. Numer. Math., 45, 315–329). Numericalexperiments complement our theoretical results.     

A nonlocal equation arising in the theory of wet combustion

A steady-state equation describing the self-heating of a materialdue to oxidation, hydrolysis, and the evaporation and condensationof water was introduced recently by Sisson et al. (1993, IMAJ. Appl. Math. 50, 285), and some preliminary existence anduniqueness results were derived. In this paper a numerical studyof the equation is undertaken. Critical behaviour and the disappearanceof criticality is observed.