CONVERGENCE ANALYSIS ON ITERATIVE METHODS FOR SEMIDEFINITE SYSTEMS

The convergence analysis on the general iterative methods for the symmetric and positive semidefinite problems is presented in this paper. First, formulated are refined necessary and sumcient conditions for the energy norm convergence for iterative methods. Some illustrative examples for the conditions are also provided. The sharp convergence rate identity for the Gauss-Seidel method for the semidefinite system is obtained relying only on the pure matrix manipulations which guides us to obtain the convergence rate identity for the general successive subspace correction methods. The convergence rate identity for the successive subspace correction methods is obtained under the new conditions that the local correction schemes possess the local energy norm convergence. A convergence rate estimate is then derived in terms of the exact subspace solvers and the parameters that appear in the conditions. The uniform convergence of multigrid method for a model problem is proved by the convergence rate identity. The work can be regradled as unified and simplified analysis on the convergence of iteration methods for semidefinite problems [8, 9].     

SEPARATION OF SCALES IN ELASTICITY IMAGING： A NUMERICAL STUDY

In magnetic resonance elastography, one seeks to reconstruct the shear modulus from measurements of the displacement field in the whole body. In this paper, we present an optimization approach which solves the problem by simply minimizing a discrepancy functional. In order to recover a complex anomaly in a homogenous medium, we first observe that the information contained in the wavefield should be decomposed into two parts, a ＂near-field＂ part in the region around the anomaly and a ＂far-field＂ part in the region away from the anomaly. As will be justified both theoretically and numerically, separating these scales provides a local and precise reconstruction.     

STRONG STABILITY PRESERVING PROPERTY OF THE DEFERRED CORRECTION TIME DISCRETIZATION

In this paper, we study the strong stability preserving （SSP） property of a class of deferred correction time discretization methods, for solving the method-of-lines schemes approximating hyperbolic partial differential equations.     

RICHARDSON EXTRAPOLATION AND DEFECT CORRECTION OF FINITE ELEMENT METHODS FOR OPTIMAL CONTROL PROBLEMS

Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.     

NUMERICAL RECONSTRUCTION OF SMALL PERTURBATIONS IN THE ELECTROMAGNETIC COEFFICIENTS OF A DIELECTRIC MATERIAL

The aim of this paper is to solve numerically the inverse problem of reconstructing small amplitude perturbations in the magnetic permeability of a dielectric material from partial or total dynamic boundary measurements. Our numerical algorithm is based on the resolution of the time-dependent Maxwell equations, an exact controllability method and Fourier inversion for localizing the perturbations. Two-dimensional numerical experiments illustrate the performance of the reconstruction method for different configurations even in the case of limited-view data.     

A Generalized Sampling Theorem with the Inverse of an Arbitrary Square Summable Sequence as Sampling Points

In this article a generalized sampling theorem using an arbitrary sequence of sampling points is derived. The sampling theorem is a Kramer-type sampling theorem, but unlike Kramer's theorem the sampling points are not necessarily eigenvalues of some boundary value problems. The theorem is then used to characterize a class of entire functions that can be reconstructed from their sample values at the points t_n = an + b if n = 0, 1, 2, ... and t_n = an + c if n = 0, -1, -2, ..., where a, b, c are arbitrary constants. The reconstruction formula is derived explicitly in the form of a sampling series expansion. When a = 1, b = 0 = c, the famous Whittaker-Shannon-Kotel'nikov sampling theorem is obtained as a special case.     

A Matched Asymptotic Expansions Approach to Continuity Corrections for Discretely Sampled Options. Part 2: Bermudan Options

The paper discusses the ‘continuity correction’ that should be applied to connect the prices of discretely sampled American put options (i.e. Bermudan options) and their continuously‐sampled equivalents. Using a matched asymptotic expansions approach the correction is computed and related to that discussed by Broadie, Glasserman & Kou (1997 Broadie, M. 1997. A continuity correction for discretely sampled barrier options,. Mathematical Finance, 7: 325[Crossref], [Web of Science ®] , [Google Scholar]) (Mathematical Finance, 7, p.325 for barrier options. In the Bermudan case, the continuity correction is an order of magnitude smaller than in the corresponding barrier problem. It is also shown that the optimal exercise boundary in the discrete case is slightly higher than in the continuously sampled case.     

PREDUAL SPACES FOR Q SPACES

To find the predual spaces Pα（R^n） of Qα（R^n） is an important motivation in the study of Q spaces. In this article, wavelet methods are used to solve this problem in a constructive way. First, an wavelet tent atomic characterization of Pα（Rn） is given, then its usual atomic characterization and Poisson extension characterization are given. Finally, the continuity on Pα of Calderon-Zygmund operators is studied, and the result can be also applied to give the Morrey characterization of Pα（Rn）.     

STABILIZED FEM FOR CONVECTION-DIFFUSION PROBLEMS ON LAYER-ADAPTED MESHES

The application of a standard Galerkin finite element method for convection-diffusion problems leads to oscillations in the discrete solution, therefore stabilization seems to be necessary. We discuss several recent stabilization methods, especially its combination with a Galerkin method on layer-adapted meshes. Supercloseness results obtained allow an improvement of the discrete solution using recovery techniques.     

A TWO-SCALE HIGHER-ORDER FINITE ELEMENT DISCRETIZATION FOR SCHROEDINGER EQUATION

In this paper, a two-scale higher-order finite element discretization scheme is proposed and analyzed for a Schroedinger equation on tensor product domains. With the scheme, the solution of the eigenvalue problem on a fine grid can be reduced to an eigenvalue problem on a much coarser grid together with some eigenvalue problems on partially fine grids. It is shown theoretically and numerically that the proposed two-scale higher-order scheme not only significantly reduces the number of degrees of freedom but also produces very accurate approximations.     

FINITE ELEMENTS WITH LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW PROBLEMS

In this paper we review recent developments in the analysis of finite element methods for incompressible flow problems with local projection stabilization （LPS）. These methods preserve the favourable stability and approximation properties of classical residual-based stabilization （RBS） techniques but avoid the strong coupling of velocity and pressure in the stabilization terms. LPS-methods belong to the class of symmetric stabilization techniques and may be characterized as variational multiscale methods. In this work we summarize the most important a priori estimates of this class of stabilization schemes developed in the past 6 years. We consider the Stokes equations, the Oseen linearization and the NavierStokes equations. Furthermore, we apply it to optimal control problems with linear（ized） flow problems, since the symmetry of the stabilization leads to the nice feature that the operations ＂discretize＂ and ＂optimize＂ commute.     

AN EFFICIENT METHOD FOR MULTIOBJECTIVE OPTIMAL CONTROL AND OPTIMAL CONTROL SUBJECT TO INTEGRAL CONSTRAINTS

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a ＂budget＂ remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian ＂marching＂ method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced ＂weighted sum＂ based algorithm for the same problem [25]. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.     

DIRECT MINIMIZATION FOR CALCULATING INVARIANT SUBSPACES IN DENSITY FUNCTIONAL COMPUTATIONS OF THE ELECTRONIC STRUCTURE

In this article, we analyse three related preconditioned steepest descent algorithms, which are partially popular in Hartree-Fock and Kohn-Sham theory as well as invariant subspace computations, from the viewpoint of minimization of the corresponding functionals, constrained by orthogonality conditions. We exploit the geometry of the admissible manifold, i.e., the invariance with respect to unitary transformations, to reformulate the problem on the Grassmann manifold as the admissible set. We then prove asymptotical linear convergence of the algorithms under the condition that the Hessian of the corresponding Lagrangian is elliptic on the tangent space of the Grassmann manifold at the minimizer.     

Cartesian closed categories of FƵ-domains

A subset system Z assigns to each partially ordered set P a certain collection Z（P） of subsets. In this paper, a new kind of subset systems called directable subset systems is introduced. For a directable subset system Z, the concepts of FZ-way-below relation and FZ-domain are introduced. The well-known Scott topology is naturally generalized to the Z-level and the resulting topology is called FZ-Scott topology, and the continuous functions with respect to this topology are characterized by preserving the suprema of directed Z-sets. Then, we mainly consider a generalization of the cartesian closedness of the categories DCPO of directed complete posets, BF of bifinite domains and FS of FS-domains to the Z-level. Corresponding to them, it is proved that, for a suitable subset system Z, the categories FZCPO of Z-complete posets, FSFZ of finitely separated FZ-domains and BFFZ of bifinite FZ-domains are all cartesian closed. Some examples of these categories are given.     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP --In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

FINITE ELEMENT APPROXIMATIONS FOR SCHROEDINGER EQUATIONS WITH APPLICATIONS TO ELECTRONIC STRUCTURE COMPUTATIONS

In this paper, both the standard finite element discretization and a two-scale finite element discretization for SchrSdinger equations are studied. The numerical analysis is based on the regularity that is also obtained in this paper for the Schroedinger equations. Very satisfying applications to electronic structure computations are provided, too.     

Projections,Birkhoff Orthogonality and Angles in Normed Spaces

Let X be a Minkowski plane, i.e., a real two dimensional normed linear space. We use projections to give a definition of the angle Aq（x, y） between two vectors x and y in X, such that x is Birkhoff orthogonal to y if and only if Aq（x,y）=π/2. Some other properties of this angle are also discussed.     

K-controllability and Approximate K-controllability of Nonlinear Neutral Systems in Banach Spaces

In this paper, K-controllability and approximate K-controllability of non- linear neutral differential equations in Banach spaces are studied. Sufficient conditions are established for each of these types of controllability. The results are obtained by using Leray-Schauder theory.     

More about the kernel convergence and the ideal convergence

Let I 2N be an ideal and let XI = span{χI ： I ∈ I}, and let pI be the quotient norm of l∞/XI. In this paper, we show first that for each proper ideal I 2N, the ideal convergence deduced by I is equivalent to pI-kernel convergence. In addition, let K = {x＊oχ（·） ： x＊∈ p（e）}, where p（x） = lim supn→∞1/n（∑k=1n｜x（k）｜, and let Iμ = {A N ： μ（A） = 0} for all μ = x＊oχ（·） ∈ K. Then Iμ is a proper ideal. We also show that the ideal convergence deduced by the proper ideal Iμ, the p-kernel convergence and the statistical convergence are also equivalent.     

Spectral gap and convergence rate for discrete-time Markov chains

Let P be a transition matrix which is symmetric with respect to a measure π.The spectral gap of P in L2(π)-space,denoted by gap(P),is defined as the distance between 1 and the rest of the spectrum of P.In this paper,we study the relationship between gap(P) and the convergence rate of Pn.When P is transient,the convergence rate of P n is equal to 1 gap(P).When P is ergodic,we give the explicit upper and lower bounds for the convergence rate of Pn in terms of gap(P).These results are extended to L∞(π)-space.