Error bounds for approximations of traces of products of truncated Toeplitz operators

The paper establishes error orders for integral limit approximations to the traces of products of truncated Toeplitz operators generated by integrable real symmetric functions defined on the real line. These approximations and the corresponding error bounds are of importance in the statistical analysis of continuous-time stationary processes (asymptotic distributions and large deviations of Toeplitz type quadratic functionals, estimation of the spectrum, etc.). The results improve the rates obtained by the authors (in an earlier paper). An explicit second-order asymptotic expansion is found for the trace of a product of two truncated Toeplitz operators generated by the spectral densities of continuous-time stationary fractional Riesz-Bessel motions. The order of magnitude of the second term in this expansion is shown to depend on the long-memory parameters of the processes. The second-order term provides a substantially better approximation to the original functional, as compared with the first-order approximation. M. Ginovyan research was partially supported by National Science Foundation Grant #DMS-0706786.     

On the trace approximation problem for truncated Toeplitz operators and matrices

The paper is devoted to the problem of approximation of the traces of products of truncated Toeplitz operators and matrices generated by integrable real symmetric functions defined on the real line (resp. on the unit circle), and estimation of the corresponding errors. These approximations and the corresponding error bounds are of importance in the statistical analysis of continuous- and discrete-time stationary processes (asymptotic distributions and large deviations of Toeplitz type quadratic functionals and forms, parametric and nonparametric estimation, etc.)We review and summarize the known results concerning the trace approximation problem and prove some new results.     

Extrapolation of Mixed Finite Element Approximations for the Maxwell Eigenvalue Problem

In this paper,a general method to derive asymptotic error expansion formulas for the mixed finite element approximations of the Maxwell eigenvalue problem is established.Abstract lemmas for the error of the eigenvalue approximations are obtained.Based on the asymptotic error expansion formulas,the Richardson extrapolation method is employed to improve the accuracy of the approximations for the eigenvalues of the Maxwell system from θ(h2) to θ(h4) when applying the lowest order Nédé1ec mixed finite element and a nonconforming mixed finite element.To our best knowledge,this is the first superconvergence result of the Maxwell eigenvalue problem by the extrapolation of the mixed finite element approximation.Numerical experiments are provided to demonstrate the theoretical results.     

Optimal convergence rate for Yosida approximations of bounded holomorphic semigroups

In this paper, we obtain optimal bounds for convergence rate for Yosida approximations of bounded holomorphic semigroups. We also provide asymptotic expansions for semigroups in terms of Yosida approximations and obtain optimal error bounds for these expansions.     

Asymptotic Toeplitz and Hankel operators on the Bergman space

In this paper the concept of asymptotic Toeplitz and asymptotic Hankel operators on the Bergman space are introduced and properties of these classes of operators are studied. The importance of this notion is that it associates with a class of operators a Toeplitz operator and with a class of operators a Hankel operator where the original operators are not even Toeplitz or Hankel. Thus it is possible to assign a symbol to an operator that is not Toeplitz or Hankel and hence a symbol calculus is obtained. Further a relation between Toeplitz operators and little Hankel operators on the Bergman space is established in some asymptotic sense.     

Asymptotic expansions and Richardson extrapolation of approximate solutions for integro-differential equations by mixed finite element methods

In this paper asymptotic error expansions for mixed finite element approximations of the integro-differential equation are derived, and Richardson extrapolation is applied to improve the accuracy of the approximations by two different schemes with the help of an interpolation post-processing technique. The results of this paper provide new asymptotic expansions. As a by-product, we illustrate that all the approximations of higher accuracy can be used to form a class of a-posteriori error estimators for this mixed finite element method. Finally, a numerical example is provided to validate the theoretical results. This project was supported in part by the Special Funds for Major State Basic Research Project (2007CB8149), the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education of China (Numerical Methods for Convertible Bonds, 06JA630047), the NSERC, Tianjin Natural Science Foundation (07JCYBJC14300), Tianjin Educational Committee, Liu Hui Center for Applied Mathematics of Nankai University and Tianjin University, and Tianjin University of Finance and Economics.     

Richardson extrapolation of iterated discrete projection methods for eigenvalue approximation

In this paper, the eigenvalue approximation of a compact integral operator with a smooth kernel is discussed. We propose asymptotic error expansions of the iterated discrete Galerkin and iterated discrete collocation methods, and asymptotic error expansion of approximate eigenvalues. We then apply Richardson extrapolation to obtain higher order super-convergence of eigenvalue approximations. Numerical examples are presented to illustrate the theoretical estimate.     

Discretization error of irregular sampling approximations of stochastic integrals

This paper studies the limit distributions for discretization error of irregular sampling approximations of stochastic integral. The irregular sampling approximation was first presented in Hayashi et al.[3], which was more general than the sampling approximation in Lindberg and Rootz′en [10]. As applications, we derive the asymptotic distribution of hedging error and the Euler scheme of stochastic differential equation respectively.     

Dirichlet空间上Toeplitz算子与Berezin型变换相关的一些性质

本文研究了单位圆盘D 的Dirichlet 空间上Toeplitz 算子和小Hankel 算子. 利用Berezin 型变换讨论了Toeplitz 算子的不变子空间问题, 具有Berezin 型符号的Toeplitz 算子的渐进可乘性以及Toeplitz 算子的Riccati 方程的可解性. 应用Berezin 变换得到了Toeplitz 算子和小Hankel 算子可逆的充分条件. 此外, 还利用Hankel 算子和Berezin 变换刻画了算子2T_uv-T_uT_v-T_vT_u 的紧性, 其中函数u,v ∈ L^2,1.     

The spectral approximation of multiplication operators via asymptotic (structured) linear algebra

A multiplication operator on a Hilbert space may be approximated with finite sections by choosing an orthonormal basis of the Hilbert space. Multiplication operators with nonzero symbols, defined on L² spaces of functions, are never compact and then such approximations cannot converge in the norm topology. Instead, we consider how well the spectra of the finite sections approximate the spectrum of the multiplication operator whose expression is simply given by the essential range of the symbol (i.e. the multiplier). We discuss the case of real orthogonal polynomial bases and the relations with the classical Fourier basis whose choice leads to the well studied Toeplitz case. Indeed, the asymptotic approximation of the spectrum by the spectra of the associated Toeplitz sections is possible only under precise geometric assumptions on the range of the symbol. Conversely, the use of circulant approximations leads to constructive algorithms, with O(N log(N)) complexity (N = number of sections), working in general and generalizable to the separable multivariate and matrix-valued cases as well.     

Asymptotic Formulas for Determinants of a Sum of Finite Toeplitz and Hankel Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.     

Asymptotic solution of singularly perturbed linear-quadratic optimal control problems with discontinuous coefficients

An asymptotic solution of a singularly perturbed linear-quadratic optimal control problem with discontinuous coefficients is constructed by directly substituting an boundary-layer asymptotic expansion of the solution into the condition of the problem and considering a series of problems for finding the asymptotic terms. The error in the approximate solution is estimated. It is shown that the values of the minimized functional do not increase when the next approximations of the optimal control are used.     

On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation

The main theme of this paper is the construction of efficient, reliable and affordable error bounds for two families of quadrature methods for highly oscillatory integrals. We demonstrate, using asymptotic expansions, that the error can be bounded very precisely indeed at the cost of few extra derivative evaluations. Moreover, in place of derivatives it is possible to use finite difference approximations, with spacing inversely proportional to frequency. This renders the computation of error bounds even cheaper and, more importantly, leads to a new family of quadrature methods for highly oscillatory integrals that can attain arbitrarily high asymptotic order without computation of derivatives. AMS subject classification (2000) Primary 65D30, secondary 34E05.Received June 2004. Accepted October 2004. Communicated by Lothar Reichel.     

On the asymptotic exactness of error estimators for linear triangular finite elements

Summary This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution.One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.     

Maximal eigenvalue and norm of a product of Toeplitz matrices. Study of a particular case

In this paper we describe the asymptotic behaviour of the spectral norm of the product of two finite Toeplitz matrices as the matrix dimension goes to infinity. These Toeplitz matrices are generated by functions with Fisher–Hartwig singularities of negative order. If these functions are positives the product of the two matrices has positive eigenvalues and it is known that the spectral norm is also the largest eigenvalue of this product.     

Asymptotic Spectra of Dense Toeplitz Matrices Are Unstable

The paper is concerned with the limiting set of the eigenvalues of the truncations of an infinite Toeplitz matrix whose symbol is continuous but not rational. This limiting set is shown to be unstable with respect to small perturbations of the symbol in the uniform norm, which reveals that the numerical computation of the asymptotic spectra of dense Toeplitz matrices is a genuine mathematical challenge.     

An efficient approximation for stochastic differential equations on the partition ofsymmetricalirst

In certain applications of stochastic differential equations a numerical solution must be found corresponding to a particular sample path of the driving process. The order of convergence of approximations based on regular samples of the path is limited, and some approximations are asymptotically efficient in that they minimise the leading coefficient in the expansion of mean-square errors as power series in the sample step size. This paper considers approximations based on irregular samples taken at the passage times of the driving process through a series of thresholds. Such approximations can involve less computation than their regular sample counterparts, particularly for real-time applications. The orders of convergence of the Euler and Milshtein approximations are derived and a new approximation is defined which is asymptotically efficient with respect to the irregular samples. Its asymptotic mean-square error is less than half that of efficient approximations based on regular sample     

Additive blending of local approximations into a globally-valid approximation with application to the dilogarithm

We show how local approximations, each accurate on a subinterval, can be blended together to form a global approximation which is accurate over the entire interval. The blending functions are smoothed approximations to a step function, constructed using the error function. The local approximations may be power series, asymptotic expansion, or other more exotic species. As an example, for the dilogarithm function, we construct a one-line analytic approximation which is accurate to one part in 700. This can be generalized to higher order merely by adding more terms in the local approximations. We also show the failure of the alternative strategy of subtracting singularities.     

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q ₁^rot and EQ ₁^rot. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.