Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form

where the vector of functions is in and is a finitely supported sequence of matrices called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the subdivision scheme associated with , i.e., the convergence of the sequence in the -norm.

Our main result characterizes the convergence of a subdivision scheme associated with the mask in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations.

Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

  

A Generalized Discrepancy and Quadrature Error Bound

An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the -star discrepancy and that arises in the study of lattice rules.

  

An adaptive trust region method and its convergence

In this paper, a new trust region subproblem is proposed. The trust radius in the new subproblem adjusts itself adaptively. As a result, an adaptive trust region method is constructed based on the new trust region subproblem. The local and global convergence results of the adaptive trust region method are proved. Numerical results indicate that the new method is very efficient.  

A simple global synchronization criterion for coupled chaotic systems

Based on the Lyapunov stabilization theory and Gerschgorin theorem, a simple generic criterion is derived for global synchronization of two coupled chaotic systems with a unidirectional linear error feedback coupling. This simple criterion is applicable to a large class of chaotic systems, where only a few algebraic inequalities are involved. To demonstrate the efficiency of design, the suggested approach is applied to some typical chaotic systems with different types of nonlinearities, such as the original Chua’s circuit, the modified Chua’s circuit with a sine function, and the Rössler chaotic system. It is proved that these synchronizations are ensured by suitably designing the coupling parameters.  

Theorems of the Alternative and Optimization with Set-Valued Maps

In this paper, the concept of generalized cone subconvexlike set-valued mapsis presented and a theorem of alternative for the system of generalizedinequality–equality set-valued maps is established. By applying thetheorem of the alternative and other results, necessary and sufficientoptimality conditions for vector optimization problems with generalizedcone subconvexlike set-valued maps are obtained.  

Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities

In this paper, we study the relationship of some projection-type methods for monotone nonlinear variational inequalities and investigate some improvements. If we refer to the Goldstein–Levitin–Polyak projection method as the explicit method, then the proximal point method is the corresponding implicit method. Consequently, the Korpelevich extragradient method can be viewed as a prediction-correction method, which uses the explicit method in the prediction step and the implicit method in the correction step. Based on the analysis in this paper, we propose a modified prediction-correction method by using better prediction and correction stepsizes. Preliminary numerical experiments indicate that the improvements are significant.  

On E-Convex Sets, E-Convex Functions, and E-Convex Programming

Recently, E-convex sets and E-convex functions were introduced in Ref. 1. However, some results seem to be incorrect. In this note, some counterexamples are given.  

Integration and approximation in arbitrary dimensions

We study multivariate integration and approximation for various classes of functions of d variables with arbitrary d. We consider algorithms that use function evaluations as the information about the function. We are mainly interested in verifying when integration and approximation are tractable and strongly tractable. Tractability means that the minimal number of function evaluations needed to reduce the initial error by a factor of ɛ is bounded by C(d)ɛ^−p for some exponent p independent of d and some function C(d). Strong tractability means that C(d) can be made independent of d. The ‐exponents of tractability and strong tractability are defined as the smallest powers of ɛ{^-1} in these bounds. We prove that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure. We obtain bounds on the ‐exponents, and for some cases we find their exact values. For some weighted Korobov and Sobolev spaces, the strong ‐exponent is the same as the ‐exponent for d=1, whereas for the third space it is 2. For approximation we also consider algorithms that use general evaluations given by arbitrary continuous linear functionals as the information about the function. Our main result is that the ‐exponents are the same for general and function evaluations. This holds under the assumption that the orthonormal eigenfunctions of the covariance operator have uniformly bounded L∞ norms. This assumption holds for spaces with shift-invariant kernels. Examples of such spaces include weighted Korobov spaces. For a space with non‐shift‐invariant kernel, we construct the corresponding space with shift-invariant kernel and show that integration and approximation for the non-shift-invariant kernel are no harder than the corresponding problems with the shift-invariant kernel. If we apply this construction to a weighted Sobolev space, whose kernel is non-shift-invariant, then we obtain the corresponding Korobov space. This enables us to derive the results for weighted Sobolev spaces. This revised version was published online in June 2006 with corrections to the Cover Date.  

A new superconvergence property of Wilson nonconforming finite element

Summary. In this paper the Wilson nonconforming finite element method is considered to solve a class of two-dimensional second-order elliptic boundary value problems. A new superconvergence property at the vertices and the midpoints of four edges of rectangular meshes is obtained. Received May 5, 1995 / Revised version received November 11, 1996