New results on the algebraic theory of linear systems: the solution of the cover problems

We give the complete solution of a large class of problems in linear system theory, the so-called cover problems. These problems are formulated and solved both in the state-space and in the input-output frameworks. The key concept, which allows the effective parametrization of all solutions of the cover problems, is that of the partial realizations of a sequence of matrices. It is shown that the solutions of the state-space cover problems can be expressed as state spaces of the partial realizations of appropriately defined sequences of matrices, and the solution of the input-output cover problems can be expressed as a simple function of the functions of the partial realizations of the sequences of mentioned above.     

Banach spaces that admit support sets

It is shown that the existence of a closed convex set all of whose points are properly supported in a Banach space is equivalent to the existence of a certain type of uncountable ordered one-sided biorthogonal system. Under the continuum hypothesis, we deduce that this notion is weaker than the existence of an uncountable biorthogonal system.

     

Numerical stability of biorthogonal wavelet transforms

Biorthogonal wavelets are essential tools for numerous practical applications. It is very important that wavelet transforms work numerically stable in floating point arithmetic. This paper presents new results on the worst-case analysis of roundoff errors occurring in floating point computation of periodic biorthogonal wavelet transforms, i.e. multilevel wavelet decompositions and reconstructions. Both of these wavelet algorithms can be realized by matrix–vector products with sparse structured matrices. It is shown that under certain conditions the wavelet algorithms can be remarkably stable. Numerous tests demonstrate the performance of the results.      

Splines and biorthogonal systems

The paper constructs coordinate splines on a closed interval, provides realizations of the corresponding biorthogonal system, and constructs finite-dimensional spaces of splines (nonpolynomialin general) of the class C ¹. Bibliography: 7 titles.     

The use of discrete dyadic wavelets in image processing

In this paper, using the discreteWalsh transform, we construct orthogonal and biorthogonal wavelets for complex periodic sequences similar to those studied earlier for the Cantor group. Results of numerical experiments demonstrate the effectiveness of the use of constructed discrete wavelets in image processing.     

Biorthogonal systems for solving Volterra integral equation systems of the second kind

With the aid of biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a system of nonlinear Volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically.     

Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function φ which has sufficient regularity and which is fundamental for interpolation [that means, φ(0)=1 and φ(α)=0 for all α∈ Z ^s ∖{0}].
Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces.
A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed.     

Construction of Biorthogonal Multiwavelets Using the Lifting Scheme

The lifting scheme has been found to be a flexible method for constructing scalar wavelets with desirable properties. Here it is extended to the construction of multiwavelets. It is shown that any set of compactly supported biorthogonal multiwavelets can be obtained from the Lazy matrix filters with a finite number of lifting steps. As an illustration of the general theory, compactly supported biorthogonal multiwavelets with optimum time–frequency resolution are constructed. In addition, experimental results of applying these multiwavelets to image compression are presented.     

Biorthogonal sequences of solutions of the generalized feller equation

The generalized Feller equation is a linear, autonomous, parabolic equation of a positive space variable and a time variable. Its coefficients are power functions of the space variable, and they depend on four parameters. In general, the equation is singular at the origin and at infinity. It contains as special cases the special Feller equation, the Kepinski equation, and the standard heat equation. The main objective of the present paper is to establish series expansions of solutions of the generalized Feller equation in terms of the elements of two sequences of particular solutions. The elements of one of these sequences are particular initial condition solutions. The two sequences are biorthogonal. The main result is that a solution does have the desired expansion property if and only if it has the Huygens property in some neighborhood of the origin of the time variable.     

A method for least squares solution of systems with a cyclic rectangular coefficient matrix

An orthogonalization procedure is given for a sequence of vectors having the special feature that consecutive vectors are related by a_k+1 = Pa_k, (k = 1, 2, 3, …), where P is a unitary operator.This orthogonalization procedure is applied to the least squares solution of linear equations with a cyclic rectangular coefficient matrix. Furthermore, it is shown how the pseudoinverse of such matrices can be obtained.     

四元数Hilbert空间中Riesz基的刻画*

四元数Hilbert空间在应用物理科学特别是量子物理中占有重要地位.本文讨论四元数Hilbert空间的框架理论, 在四元数Hilbert空间中引入了Riesz基的概念, 在此基础上刻画了Riesz基,给出了它们的一些等价条件; 特别地, 得到了四元数Hilbert空间中的一个序列是Riesz基的充要条件是它是一个具有双正交序列的完备Bessel序列,且它的双正交序列也是一个完备Bessel序列; 并进一步证明了双正交序列中一个序列的完备性可以从特征刻画中去除.文中举例说明了双正交性、完备性和Bessel性质之间的关系.     

On 2-switches and isomorphism classes

A 2-switch is an edge addition/deletion operation that changes adjacencies in the graph while preserving the degree of each vertex. A well-known result states that graphs with the same degree sequence may be changed into each other via sequences of 2-switches. We show that if a 2-switch changes the isomorphism class of a graph, then it must take place in one of four configurations. We also present a sufficient condition for a 2-switch to change the isomorphism class of a graph. As consequences, we give a new characterization of matrogenic graphs and determine the largest hereditary graph family whose members are all the unique realizations (up to isomorphism) of their respective degree sequences.     

Noncommutative biorthogonal polynomials

We define and study biorthogonal sequences of polynomials over noncommutative rings, generalizing previous treatments of biorthogonal polynomials over commutative rings and of orthogonal polynomials over noncommutative rings. We extend known recurrence relations for specific cases of biorthogonal polynomials and prove a general version of Favard?s theorem.     

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.     

Bell-shaped nonstationary refinable ripplets

We study the approximation properties of the class of nonstationary refinable ripplets introduced in Gori and Pitolli (2008). These functions are the solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions, convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the C ² nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented.     

Construction of biorthogonal wavelets from pseudo-splines

Pseudo-splines constitute a new class of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. Pseudo-splines were first introduced by Daubechies, Han, Ron and Shen in [Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14(1) (2003), 1–46] and Selenick in [Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10(2) (2001) 163–181], and their properties were extensively studied by Dong and Shen in [Pseudo-splines, wavelets and framelets, 2004, preprint]. It was further shown by Dong and Shen in [Linear independence of pseudo-splines, Proc. Amer. Math. Soc., to appear] that the shifts of an arbitrarily given pseudo-spline are linearly independent. This implies the existence of biorthogonal dual refinable functions (of pseudo-splines) with an arbitrarily prescribed regularity. However, except for B-splines, there is no explicit construction of biorthogonal dual refinable functions with any given regularity. This paper focuses on an implementable scheme to derive a dual refinable function with a prescribed regularity. This automatically gives a construction of smooth biorthogonal Riesz wavelets with one of them being a pseudo-spline. As an example, an explicit formula of biorthogonal dual refinable functions of the interpolatory refinable function is given.     

Multiwavelet approximation methods for pseudodifferential equations on curves. Stability and convergence analysis

We develop a stability and convergence analysis of Galerkin–Petrov schemes based on a general setting of multiresolution generated by several refinable functions for the numerical solution of pseudodifferential equations on smooth closed curves. Particular realizations of such a multiresolution analysis are trial spaces generated by biorthogonal wavelets or by splines with multiple knots. The main result presents necessary and sufficient conditions for the stability of the numerical method in terms of the principal symbol of the pseudodifferential operator and the Fourier transforms of the generating multiscaling functions as well as of the test functionals. Moreover, optimal convergence rates for the approximate solutions in a range of Sobolev spaces are established. This revised version was published online in June 2006 with corrections to the Cover Date.     

Biorthogonal Eigenfunction System in the Triple-Deck Limit

The solutions of receptivity problems for a periodic-in-time actuator placed on the wall in a two-dimensional boundary layer and for a two-dimensional hump are discussed within the scope of the biorthogonal eigenfunction expansion technique in the limit of high Reynolds number when the triple-deck scaling is imposed. It is shown that the solutions obtained with the help of the biorthogonal eigenfunction system are equivalent to the solutions derived within the scope of the triple-deck theory.     

On IGMRES: An incomplete generalized minimal residual method for large unsymmetric linear systems

The truncated version of the generalized minimal residual method (GMRES), the incomplete generalized minimal residual method (IGMRES), is studied. It is based on an incomplete orthogonalization of the Krylov vectors in question, and gives an approximate or quasi-minimum residual solution over the Krylov subspace. A convergence analysis of this method is given, showing that in the non-restarted version IGMRES can behave like GMRES once the basis vectors of Krylov subspace generated by the incomplete orthogonalization are strongly linearly independent. Meanwhile, some relationships between the residual norms for IOM and IGMRES are established. Numerical experiments are reported to show convergence behavior of IGMRES and of its restarted version IGMRES(m). Project supported by the China State Key Basic Researches, the National Natural Science Foundation of China (Grant No. 19571014), the Doctoral Program (97014113), the Foundation of Returning Scholars of China and the Natural Science Foundation of Liaoning Province.     

Convergent and Orthogonality Preserving Schemes for Approximating the Kohn-Sham Orbitals

To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure calculations. In this paper, we propose and analyze a class of iteration schemes for the discretized Kohn-Sham Density Functional Theory model, with which the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals without any orthogonalization as long as the initial orbitals are orthogonal and the time step sizes are given properly. In addition, we present a feasible and efficient approach to get suitable time step sizes and report some numerical experiments to validate our theory.