Hat Average Multiresolution with Error Control in 2-D

Multiresolution representations of data are a powerful tool in data compression. For a proper adaptation to the singularities, it is crucial to develop nonlinear methods which are not based on tensor product. The hat average framework permets develop adapted schemes for all types of singularities. In contrast with the wavelet framework these representations cannot be considered as a change of basis, and the stability theory requires different considerations. In this paper, non separable two-dimensional hat average multiresolution processing algorithms that ensure stability are introduced. Explicit error bounds are presented.     

对流占优扩散问题的经济型流线扩散有限元法

In this paper, the economical finite difference-streamline diffusion (EFDSD) schemes based on the linear F.E. space for time-dependent linear and non-linear convection-dominated diffusion problems are constructed. The stability and error estimation with quasi-optimal order approximation are established in the norm stronger than L^2 - norm for the schemes considered. It is indicated by the results obtained that,for linear F.E. space, the EFDSD schemes have the same specific properties of stability and convergence as the traditional FDSD schemes for the problems discussed.     

非线性抛物方程的时空有限元方法的 估计

The space time adaptive finite element method,continuous in space but discontinuous in time for nonlinear parabolic problems is discussed.The approach is based on a combination of finite element and finite difference techniques using the properties of Lagrange interpolating polynomials on the Radau points.We ignored the restrictions of the space-time meshes which is needed in other conventional methods.Basic error estimates in L^∞(L^2) norm are obtained.     

INERTIAL ALGORITHMS FOR THE STATIONARY NAVIER-STOKES EQUATIONS

Several kind of new numerical schemes for the stationary Navier-Stokes equations based on the virtue of Inertial Manifold and Approximate Inertial Manifold, which we call them inertial algorithms in this paper, together with their error estimations are presented. All these algorithms are constructed under an uniform frame, that is to construct some kind of new projections for the Sobolev space in which the true solution is sought. It is shown that the proposed inertial algorithms can greatly improve the convergence rate of the standard Galerkin approximate solution with lower computing effort. And some numerical examples are also given to verify results of this paper.     

CLOSED SMOOTH SURFACE DEFINED FROM CUBIC TRIANGULAR SPLINES

In order to construct closed surfaces with continuous unit normal, we introduce a news pline space on an arbitrary closed mesh of three-sided faces. Our approach generalizes an idea of Goodman and is based on the concept of ‘Geometric continuity‘ for piecewise polynomial parametrizations. The functions in the spline space restricted to the faces are cubic triangular polynomials. A basis of the spline space is constructed of positive functions which sum to 1. It is also shown that the space is suitable for interpolating data at the midpoints of the faces。     

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.     

THE REDUCED BASIS TECHNIQUE AS A COARSE SOLVER FOR PARAREAL IN TIME SIMULATIONS

In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial differential equation. The fine solver is based on the finite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. Of[line-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space （typically small）. Parareal in time algorithms based on a multi-grids finite element method and a multi-degrees finite element method are also presented. Some numerical results are reported.     

Finite difference method and its convergent error analyses for thermistor problem

The thermistor problem is an initial-boundary value problem of coupled nonlinear differential equations. The nonlinear PDEs consist of a heat equation with the Joule heating as asource and a current conservation equation with temperature-dopendent electrical conductivity.This problem has important opplicatioJls in industry. In this paper, A new finite differencescheme is proposed on nonuniform rectangular partition for the thermistor problem. In the theo-retical analyses,the second-order error estimates are obtained for electrical potential in discrete L^2 and H^1 norms,and for the temperature in L^2 norm. In order to get these second-order errorestimates,the Joule heating source is used in a changed equivalent form.     

Economical cascadic multigrid method (ECMG)

In this paper,an economical cascadic multigrid method is proposed.Compared with the usual cascadic multigrid method developed by Bornemann and Deuflhard,the new one requires less iterations on each level,especially on the coarser grids.Many operations can be saved in the new cascadic multigrid algorithms.The main ingredient is the control of the iteration numbers on the each level to preserve the accuracy without over iterations.The theoretical justification is based on the observations that the error reduction rate of an iteration scheme in terms of the smoothing property is no longer accurate while the iteration number is big enough.A new formulae of the error reduction rate is employed in our new algorithm.Numerical experiments are reported to support our theory.     

EDGE-ORIENTED HEXAGONAL ELEMENTS

In this paper, two new nonconforming hexagonal elements are presented, which are based on the trilinear function space Q1^（3） and are edge-oriented, analogical to the case of the rotated Q1 quadrilateral element. A priori error estimates are given to show that the new elements achieve first-order accuracy in the energy norm and second-order accuracy in the L^2 norm. This theoretical result is confirmed by the numerical tests.     

Painless Reconstruction from Magnitudes of Frame Coefficients

The goal of this paper is to develop fast algorithms for signal reconstruction from magnitudes of frame coefficients. This problem is important to several areas of research in signal processing, especially speech recognition technology, as well as state tomography in quantum theory. We present linear reconstruction algorithms for tight frames associated with projective 2-designs in finite-dimensional real or complex Hilbert spaces. Examples of such frames are two-uniform frames and mutually unbiased bases, which include discrete chirps. The number of operations required for reconstruction with these frames grows at most as the cubic power of the dimension of the Hilbert space. Moreover, we present a very efficient algorithm which gives reconstruction on the order of d operations for a d-dimensional Hilbert space.     

QMC Rules of Arbitrary High Order: Reproducing Kernel Hilbert Space Approach

In this paper we consider numerical integration of smooth functions lying in a particular reproducing kernel Hilbert space. We show that the worst-case error of numerical integration in this space converges at the optimal rate, up to some power of a log?N factor. A similar result is shown for the mean square worst-case error, where the bound for the latter is always better than the bound for the square worst-case error. Finally, bounds for integration errors of functions lying in the reproducing kernel Hilbert space are given. The paper concludes by illustrating the theory with numerical results.     

Approximation of lebesgue integrals by Riemann sums and lattice points in domains with fractal boundary

Sets thrown at random in space contain, on average, a number of integer points equal to the measure of these sets. We determine the mean square error in the estimate of this number when the sets are homothetic to a domain with fractal boundary. This is related to the problem of approximating Lebesgue integrals by random Riemann sums.     

Measuring the efficiency of trigonometric series estimates of a density

Some types of density estimators, particularly those based on trigonometric series, converge reasonably quickly to their limit except in the neighbourhood of one or two singularities. In this situation the mean integrated square error, the traditional measure of the efficiency of a density estimator, is an unsatisfactory measure. The notion of partial mean integrated square error is introduced and used to compare the performance of trigonometric series estimators. The results lead to consideration of some new estimators which have excellent properties from the points of view of both efficiency and ease of computation.     

多元向量值正交小波包的刻画

In this paper,the notion of orthogonal vector-valued wavelet packets of space L2(Rs,Cn)is introduced.A procedure for constructing the orthogonal vector-valued wavelet packets is presented.Their properties are characterized by virtue of time-frequency analysis method,matrix theory and finite group theory,and three orthogonality formulas are obtained.Finally,new orthonormal bases of space L2(Rs,Cn)are extracted from these wavelet packets.     

Best two‐parameter rational approximation algorithm of given order: a variation of adaptive Fourier decomposition

The concept of mono‐component is widely used in nonstationary signal processing and time‐frequency analysis. In this paper, we construct several classes of complete rational function systems in the Hardy space, whose boundary values are mono‐components. Then, we propose a best approximation algorithm (BAA) based on optimal selections of two parameters in the orthonormal bases according to the approximation error. Effectiveness of BAA is evaluated by comparison experiments with the classical Fourier decomposition algorithm. It is also shown that BAA has promising results for filtering out noises and dealing with real‐world signals. Copyright © 2014 John Wiley & Sons, Ltd.     

Spectral methods for orthogonal rational functions

We present an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two alternatives are discussed, leading to representations which are linear fractional transformations with matrix coefficients acting on infinite Hessenberg or five-diagonal unitary matrices. This approach permits us to recover the orthogonality measure throughout the spectral analysis of an infinite matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg or five-diagonal matrices. As an application we use operator perturbation theory results to obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation for the orthogonal rational functions.     

On the regular representation of a nonunimodular locally compact group

We study the discrete part of the regular representation of a locally compact group and also its Type I part if the group is separable. Our results extend to nonunimodular groups' known results for unimodular groups about formal degrees of square integrable representations, and the Plancherel formula. We establish orthogonality relations for matrix coefficients of square integrable representations and we show that the formal degree in general is not a positive number, but a positive self-adjoint unbounded operator, semi-invariant under the representation. Integrable representations are also studied in this context. Finally we show that when the group is nonunimodular, “Plancherel measure” is not a true measure, but a measure multiplied by a section of a certain real oriented line bundle on the dual space of the group.     

Optimal Recovery of Functions on the Sphere on a Sobolev Spaces with a Gaussian Measure in the Average Case Setting

In this paper, we study optimal recovery(reconstruction) of functions on the sphere in the average case setting. We obtain the asymptotic orders of average sampling numbers of a Sobolev space on the sphere with a Gaussian measure in the Ld-1q(S) metric for 1 ≤ q ≤∞, and show that some worst-case asymptotically optimal algorithms are also asymptotically optimal in the average case setting in the Ldq(S-1)metric for 1 ≤ q ≤∞.     

On the hellinger square integral with respect to an operator valued measure and stationary processes

A construction of the Hellinger square integral with respect to a semispectral measure in a Banach space B is given. It is proved that the space of values of a B-valued stationary stochastic process is unitarily isomorphic to the space of all B¹-valued measures that are Hellinger square integrable with respect to the spectral measure of the process. Some applications of the above theorem in the prediction theory (especially to interpolation problem) are also considered.