Convergence of nonstationary cascade algorithms

Summary. A nonstationary multiresolution of is generated by a sequence of scaling functions We consider that is the solution of the nonstationary refinement equations where is finitely supported for each k and M is a dilation matrix. We study various forms of convergence in of the corresponding nonstationary cascade algorithm as k or n tends to It is assumed that there is a stationary refinement equation at with filter sequence h and that The results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h. Received September 19, 1997 / Revised version received May 22, 1998 / Published online August 19, 1999     

Biorthogonal M -Channel Compactly Supported Wavelets

We study a class of M -channel subband coding schemes with perfect reconstruction. Along the lines of [8] and [10], we construct compactly supported biorthogonal wavelet bases of L ² (R) , with dilation factor M , associated to these schemes. In particular, we study the case of splines, and obtain explicitly simple expressions for all the relevant filters. The resulting wavelets have arbitrarily large regularity and we also obtain asymptotic estimates for the regularity exponent. September 17, 1998. Date revised: June 14, 1999. Date accepted: June 25, 1999.     

Multigrid algorithms for a vertex–centered covolume method for elliptic problems

Summary. We analyze V–cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of the multigrid algorithm of the original problem. As an application, we study the convergence of multigrid algorithms for a covolume method or a vertex–centered finite volume element method for variable coefficient elliptic problems on polygonal domains. As in standard finite element methods, the V–cycle algorithm with one pre-smoothing converges with a rate independent of the number of levels. Various types of smoothers including point or line Jacobi, and Gauss-Seidel relaxation are considered. Received August 19, 1999 / Revised version received July 10, 2000 / Published online June 7, 2001     

Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces

Under the appropriate definition of sampling density D_ϕ, a function f that belongs to a shift invariant space can be reconstructed in a stable way from its non-uniform samples only if D_ϕ≥1. This result is similar to Landau's result for the Paley-Wiener space B _1/2 . If the shift invariant space consists of polynomial splines, then we show that D_ϕ<1 is sufficient for the stable reconstruction of a function f from its samples, a result similar to Beurling's special case B _1/2 .     

Characterizations of orthonormal scale functions: A probabilistic approach

The construction of a multiresolution analysis starts with the specification of a scale function. The Fourier transform of this function is defined by an infinite product. The convergence of this product is usually discussed in the context of L ²(R).Here, we treat the convergence problem by viewing the partial products as probabilities, converging weakly to a probability defined on an appropriate sequence space. We obtain a sufficient condition for this convergence, which is also necessary in the case where the scale function is continuous. These results extend and clarify those of Cohen [2] and Hernández et al. [4]. The method also applies to more general dilation schemes that commute with translations by Z ^d .     

Frequency optimization approach for designing <Emphasis Type="Italic">M</Emphasis>-band biorthogonal symmetric wavelets

In this paper, a new method is presented for designing M-band biorthogonal symmetric wavelets. The design problem of biorthogonal linear-phase scaling filters and wavelet filters as a quadratic programming problem with the linear constraints is formulated. The closed-form solution is given and a design example is presented.     

On complete noncompact submanifolds with constant mean curvature and finite total curvature in Euclidean spaces

In this paper, we consider a complete noncompact n-submanifold M with parallel mean curvature vector h in an Euclidean space. If M has finite total curvature, we prove that M must be minimal, so that M is an affine n-plane if it is strongly stable. This is a generalization of the result on strongly stable complete hypersurfaces with constant mean curvature in Received: 30 June 2005     

Fast verification of solutions of matrix equations

Summary. In this paper, we are concerned with a matrix equation where A is an real matrix and x and b are n-vectors. Assume that an approximate solution is given together with an approximate LU decomposition. We will present fast algorithms for proving nonsingularity of A and for calculating rigorous error bounds for . The emphasis is on rigour of the bounds. The purpose of this paper is to propose different algorithms, the fastest with flops computational cost for the verification step, the same as for the LU decomposition. The presented algorithms exclusively use library routines for LU decomposition and for all other matrix and vector operations. Received June 16, 1999 / Revised version received January 25, 2001 / Published online June 20, 2001     

On convergence rates of inexact Newton regularizations

Summary. REGINN is an algorithm of inexact Newton type for the regularization of nonlinear ill-posed problems [Inverse Problems 15 (1999), pp. 309–327]. In the present article convergence is shown under weak smoothness assumptions (source conditions). Moreover, convergence rates are established. Some computational illustrations support the theoretical results. Received March 12, 1999 / Published online October 16, 2000     

Orthonormal rational function vectors

Summary In this paper, we develop a matrix framework to solve the problem of finding orthonormal rational function vectors with prescribed poles with respect to a certain discrete inner product that is defined by a set of data points and corresponding weight vectors w_i,j. Our algorithm for solving the problem is recursive, and it is of complexity If all data points are real or lie on the unit circle, then the complexity is reduced by an order of magnitude.     

A note on best conditioned preconditioners

We discuss the solution of Hermitian positive definite systemsAx=b by the preconditioned conjugate gradient method with a preconditionerM. In general, the smaller the condition number(M ^–1/2 AM ^–1/2 ) is, the faster the convergence rate will be. For a given unitary matrixQ, letM _Q = {Q* _N Q | _n is ann-by-n complex diagonal matrix} andM _Q ⁺ ={Q* _n Q | _n is ann-by-n positive definite diagonal matrix}. The preconditionerM _b that minimizes(M ^–1/2 AM ^–1/2 ) overM _Q ⁺ is called the best conditioned preconditioner for the matrixA overM _Q ⁺ . We prove that ifQAQ* has Young's Property A, thenM _b is nothing new but the minimizer of M –A _F overM _Q . Here · _F denotes the Frobenius norm. Some applications are also given here.     

Convergence of subdivision schemes in L p spaces

Abstract. In this paper it is proved that Lp solutions of a refinement equation exist if and only ifthe corresponding subdivision scheme with suitable initial function converges in Lp without anyassumption on the stability of the solutions of the refinement equation. A characterization forconvergence of subdivision scheme is also given in terms of the refinement mask. Thus a com-plete answer to the relation between the existence of Lp solutions of the refinement equation andthe convergence of the corresponding subdivision schemes is given.     

A look-ahead algorithm for the solution of general Hankel systems

Summary The solution of systems of linear equations with Hankel coefficient matrices can be computed with onlyO(n ²) arithmetic operations, as compared toO(n ³) operations for the general cases. However, the classical Hankel solvers require the nonsingularity of all leading principal submatrices of the Hankel matrix. The known extensions of these algorithms to general Hankel systems can handle only exactly singular submatrices, but not ill-conditioned ones, and hence they are numerically unstable. In this paper, a stable procedure for solving general nonsingular Hankel systems is presented, using a look-ahead technique to skip over singular or ill-conditioned submatrices. The proposed approach is based on a look-ahead variant of the nonsymmetric Lanczos process that was recently developed by Freund, Gutknecht, and Nachtigal. We first derive a somewhat more general formulation of this look-ahead Lanczos algorithm in terms of formally orthogonal polynomials, which then yields the look-ahead Hankel solver as a special case. We prove some general properties of the resulting look-ahead algorithm for formally orthogonal polynomials. These results are then utilized in the implementation of the Hankel solver. We report some numerical experiments for Hankel systems with ill-conditioned submatrices.The research of the first author was supported by DARPA via Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA).The research of the second author was supported in part by NSF grant DRC-8412314 and Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA).     

Symmetric orthonormal scaling functions and wavelets with dilation factor 4

It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor d=4. Several examples of such orthonormal scaling functions are provided in this paper. In particular, two examples of C ¹ orthonormal scaling functions, which are symmetric about 0 and 1/6, respectively, are presented. We will then discuss how to construct symmetric wavelets from these scaling functions. We explicitly construct the corresponding orthonormal symmetric wavelets for all the examples given in this paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

On multiresolution analysis of multiplicityd

In this paper we study some basic properties of multiresolution analysis of multiplicityd in several variables and discuss some examples related to the spaces of cardinal splines with respect to the unidiagonal or the crisscross partition of the plane. Furthermore, in analogy with [8], we show that if the scaling functions are compactly supported, then it is possible to find compactly supported mother wavelets _l ,l=1,...,2 ⁿ d–d, in such a way that the family {2 ^jn/2 _l (2 ^j x–v)} is a semiorthogonal basis ofL ² ( ⁿ ).     

The effect of non-optimal bases on the convergence of Krylov subspace methods

Summary There are many examples where non-orthogonality of a basis for Krylov subspace methods arises naturally. These methods usually require less storage or computational effort per iteration than methods using an orthonormal basis (optimal methods), but the convergence may be delayed. Truncated Krylov subspace methods and other examples of non-optimal methods have been shown to converge in many situations, often with small delay, but not in others. We explore the question of what is the effect of having a non-optimal basis. We prove certain identities for the relative residual gap, i.e., the relative difference between the residuals of the optimal and non-optimal methods. These identities and related bounds provide insight into when the delay is small and convergence is achieved. Further understanding is gained by using a general theory of superlinear convergence recently developed. Our analysis confirms the observed fact that in exact arithmetic the orthogonality of the basis is not important, only the need to maintain linear independence is. Numerical examples illustrate our theoretical results.This revised version was published online in June 2005 due to a typesetting mistake in the footnote on page 7.     

Nonlinear Methods of Approximation

Abstract. Our main interest in this paper is nonlinear approximation. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a fixed linear space but are allowed to depend on the function being approximated. While the scope of this paper is mostly theoretical, we should note that this form of approximation appears in many numerical applications such as adaptive PDE solvers, compression of images and signals, statistical classification, and so on. The standard problem in this regard is the problem of m -term approximation where one fixes a basis and looks to approximate a target function by a linear combination of m terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is the starting point for compression algorithms. We are interested in the quantitative aspects of this type of approximation. Namely, we want to understand the properties (usually smoothness) of the function which govern its rate of approximation in some given norm (or metric). We are also interested in stable algorithms for finding good or near best approximations using m terms. Some of our earlier work has introduced and analyzed such algorithms. More recently, there has emerged another more complicated form of nonlinear approximation which we call highly nonlinear approximation. It takes many forms but has the basic ingredient that a basis is replaced by a larger system of functions that is usually redundant. Some types of approximation that fall into this general category are mathematical frames, adaptive pursuit (or greedy algorithms), and adaptive basis selection. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. With this motivation, our recent work and the current activity focuses on nonlinear approximation both in the classical form of m -term approximation (where several important problems remain unsolved) and in the form of highly nonlinear approximation where a theory is only now emerging.     

Orthogonal polynomials and semi-iterative methods for the Drazin-inverse solution of singular linear systems

Summary. In this work we present a novel class of semi-iterative methods for the Drazin-inverse solution of singular linear systems, whether consistent or inconsistent. The matrices of these systems are allowed to have arbitrary index and arbitrary spectra in the complex plane. The methods we develop are based on orthogonal polynomials and can all be implemented by 4-term recursion relations independently of the index. We give all the computational details of the associated algorithms. We also give a complete convergence analysis for all methods. Received June 28, 2000 / Revised version received May 23, 2001 / Published online January 30, 2002     

Scaled total least squares fundamentals

Summary. The standard approaches to solving overdetermined linear systems construct minimal corrections to the vector c and/or the matrix B such that the corrected system is compatible. In ordinary least squares (LS) the correction is restricted to c, while in data least squares (DLS) it is restricted to B. In scaled total least squares (STLS) [22], corrections to both c and B are allowed, and their relative sizes depend on a real positive parameter . STLS unifies several formulations since it becomes total least squares (TLS) when , and in the limit corresponds to LS when , and DLS when . This paper analyzes a particularly useful formulation of the STLS problem. The analysis is based on a new assumption that guarantees existence and uniqueness of meaningful STLS solutions for all parameters . It makes the whole STLS theory consistent. Our theory reveals the necessary and sufficient condition for preserving the smallest singular value of a matrix while appending (or deleting) a column. This condition represents a basic matrix theory result for updating the singular value decomposition, as well as the rank-one modification of the Hermitian eigenproblem. The paper allows complex data, and the equivalences in the limit of STLS with DLS and LS are proven for such data. It is shown how any linear system can be reduced to a minimally dimensioned core system satisfying our assumption. Consequently, our theory and algorithms can be applied to fully general systems. The basics of practical algorithms for both the STLS and DLS problems are indicated for either dense or large sparse systems. Our assumption and its consequences are compared with earlier approaches. Received June 2, 1999 / Revised version received July 3, 2000 / Published online July 25, 2001