Optimal filter and mollifier for piecewise smooth spectral data

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the function is globally smooth, while the presence of jump discontinuities is responsible for spurious Gibbs' oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order. Classical filters and mollifiers are constructed to have compact support in the Fourier (frequency) and physical (time) spaces respectively, and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region. Here we construct a noncompactly supported filter and mollifier with optimal joint time-frequency localization for a given number of vanishing moments, resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains. Not giving preference to either space allows for a more balanced error decomposition, which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters, yet obtain true exponential accuracy.

     

Edge detection from truncated Fourier data using spectral mollifiers

Edge detection from a finite number of Fourier coefficients is challenging as it requires extracting local information from global data. The problem is exacerbated when the input data is noisy since accurate high frequency information is critical for detecting edges. The noise furthermore increases oscillations in the Fourier reconstruction of piecewise smooth functions, especially near the discontinuities. The edge detection method in Gelb and Tadmor (Appl Comput Harmon Anal 7:101–135, 1999, SIAM J Numer Anal 38(4):1389–1408, 2000) introduced the idea of “concentration kernels” as a way of converging to the singular support of a piecewise smooth function. The kernels used there, however, and subsequent modifications to reduce the impact of noise, were generally oscillatory, and as a result oscillations were always prevalent in the neighborhoods of the jump discontinuities. This paper revisits concentration kernels, but insists on uniform convergence to the “sharp peaks” of the function, that is, the edge detection method converges to zero away from the jumps without introducing new oscillations near them. We show that this is achievable via an admissible class of spectral mollifiers. Our method furthermore suppresses the oscillations caused by added noise.     

Eigenfunction expansions and the pinsky phenomenon on compact manifolds

We extend results on pointwise convergence of eigenfunction expansions established for functions on flat tori in [24] and [26] to the setting of compact Riemannian manifolds, subject to a mild restriction on the order of caustics that can arise in the fundamental solution of the wave equation. This gives analyses of some endpoint cases of results treated in [3]. In particular, we are able to treat the Pinsky phenomenon for eigenfunction expansions of piecewise smooth functions with jump across the boundary of a ball on such manifolds, in dimension three. Acknowledgements and Notes. Partially supported by NSF grant DMS 9877077.     

On constrained Newton linearization and multigrid for variational inequalities

Summary. We consider the fast solution of a class of large, piecewise smooth minimization problems. For lack of smoothness, usual Newton multigrid methods cannot be applied. We propose a new approach based on a combination of convex minization with constrained Newton linearization. No regularization is involved. We show global convergence of the resulting monotone multigrid methods and give polylogarithmic upper bounds for the asymptotic convergence rates. Efficiency is illustrated by numerical experiments. Received March 22, 1999 / Revised version received February 24, 2001 / Published online October 17, 2001     

Gibbs' phenomenon and arclength

The arclengths of the graphs Γ(s_N(f)) of the partial sums s_N(f) of the Fourier series of a piecewise smooth function f with a jump discontinuity grow at the rate O(logN). This problem does not arise if f is continuous, and can be removed by using the standard summability methods.     

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     

Convergence of adaptive FEM for elliptic obstacle problems

We treat the convergence of adaptive lowest-order FEM for some elliptic obstacle problem with affine obstacle. For error estimation, we use a residual error estimator which is an extended version of the estimator from [2] and additionally controls the data oscillations. The main result states that an appropriately weighted sum of energy error, edge residuals, and data oscillations satisfies a contraction property that leads to convergence. In addition, we discuss the generalization to the case of inhomogeneous Dirichlet data and non-affine obstacles χ ∈ H²(Ω) for which similar results are obtained. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Preconditioned conjugate gradients (PCG) are widely and successfully used methods for solving a Toeplitz linear system [59,9,20,5,34,62,6,10,28,45,44,46,49]. Frobenius-optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the Frobenius distance from . The convergence features of these PCG have been naturally studied by means of the Weierstrass–Jackson Theorem [17,36,45], owing to the profound relationship between the spectral features of the matrices , generated by the Fourier coefficients of a continuous function f, and the analytical properties of the symbol f itself. In this paper, we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f. On the other hand, by modifying the Korovkin Theorem to study the Frobenius-optimal preconditioning problem, we provide a new and unifying tool for analyzing all Frobenius-optimal preconditioners in any generic matrix algebra related to trigonometric transforms. Finally, the multilevel case is sketched and discussed by showing that a Korovkin-type Theory also holds in a multivariate sense. Received October 1, 1996 / Revised version received May 7, 1998     

The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs

Approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements. Estimates for the rate of convergence in Sobolev spaces , are given. The results are applied to estimate the rate of convergence when the p-version finite element method is used to approximate the -Laplacian. It is shown that the rate of convergence of the p-version is always at least that of the h-version (measured in terms of number of degrees of freedom used). If the solution is very smooth then the p-version attains an exponential rate of convergence. If the solution has certain types of singularity, the rate of convergence of the p-version is twice that of the h-version. The analysis generalises the work of Babuska and others to the case . In addition, the approximation theoretic results find immediate application for some types of spectral and spectral element methods. Received August 2, 1995 / Revised version received January 26, 1998     

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.     

Pointwise Behavior of Double Fourier Series of Functions of Bounded Variation

We extend some recent results of S. A. Telyakovskii on the uniform boundedness of the partial sums of Fourier series of functions of bounded variation to periodic functions of two variables, which are of bounded variation in the sense of Hardy. As corollaries, we obtain the classical Parseval formula, the convergence theorem of the series involving the sine Fourier coefficients, and a lower estimate of the best approximation by trigonometric polynomials in the metric of L in a sharpened version. This research was supported by the Hungarian National Foundation for Scientific Research under Grants TS 044 782 and T 046 192.     

An F. and M. Riesz theorem for planar vector fields

We prove that solutions of the homogeneous equation Lu=0, where L is a locally integrable vector field with smooth coefficients in two variables possess the F. and M. Riesz property. That is, if is an open subset of the plane with smooth boundary, satisfiesLu=0 on , has tempered growth at the boundary, and its weak boundary value is a measure , then is absolutely continuous with respect to Lebesgue measure on the noncharacteristic portion of . Received March 10, 2000 / Published online April 12, 2001     

Convergence of adaptive FEM for some elliptic obstacle problem

In this work, we treat the convergence of adaptive lowest-order FEM for some elliptic obstacle problem with affine obstacle. For error estimation, we use a residual error estimator from [D. Braess, C. Carstensen, and R. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math. 107 (2007), pp. 455–471]. We extend recent ideas from [J. Cascon, C. Kreuzer, R. Nochetto, and K. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), pp. 2524–2550] for the unrestricted variational problem to overcome the lack of Galerkin orthogonality. The main result states that an appropriately weighted sum of energy error, edge residuals and data oscillations satisfies a contraction property within each step of the adaptive feedback loop. This result is superior to a prior result from Braess et al. (2007) in two ways: first, it is unnecessary to control the decay of the data oscillations explicitly; second, our analysis avoids the use of some discrete local efficiency estimate so that the local mesh-refinement is fairly arbitrary.     

On the Upper Bound of Second Eigenvalues for Uniformly Elliptic Operators of any Orders

Abstract Let Ω R~m(m≥1)be a bounded domain with piecewise smooth boundary Ω.Let t and r bepositive integers with t>r+1. We consider the eigenvalue problems(1.1)and(12),and obtain Theorem 1and Theorem 2, which generalize the results in[1,2.5].     

Coupling of FEM and BEM for a nonlinear interface problem: The h–p version

This article presents some numerical examples for coupling the finite element method (FEM) and the boundary element method (BEM) as analyzed in [11]. This coupling procedure combines the advantages of boundary elements (problems in unbounded regions) and of finite elements (nonlinear problems with inhomogeneous data). In [28], experimental rates of convergence for the h version are presented, where the accuracy of the Galerkin approximation is achieved by refining the mesh. In this article we treat the h–p version, combining an increase of the degree of the piecewise polynomials with a certain mesh refinement. In our model examples, we obtain theoretically and numerically exponential convergence, which indicates a great efficiency in particular if singularities appear. © 1995 John Wiley & Sons, Inc.     

The Iterated Aluthge Transform of an Operator

The Aluthge transform (defined below) of an operator T on Hilbert space has been studied extensively, most often in connection with p-hyponormal operators. In [6] the present authors initiated a study of various relations between an arbitrary operator T and its associated , and this study was continued in [7], in which relations between the spectral pictures of T and were obtained. This article is a continuation of [6] and [7]. Here we pursue the study of the sequence of Aluthge iterates { ⁽ⁿ⁾} associated with an arbitrary operator T. In particular, we verify that in certain cases the sequence { ⁽ⁿ⁾} converges to a normal operator, which partially answers Conjecture 1.11 in [6] and its modified version below (Conjecture 5.6). Submitted: December 5, 2000? Revised: August 30, 2001.     

Bounds for the least squares distance using scaled total least squares

Summary. The standard approaches to solving overdetermined linear systems construct minimal corrections to the data to make the corrected system compatible. In ordinary least squares (LS) the correction is restricted to the right hand side c, while in scaled total least squares (STLS) [14,12] corrections to both c and B are allowed, and their relative sizes are determined by a real positive parameter . As , the STLS solution approaches the LS solution. Our paper [12] analyzed fundamentals of the STLS problem. This paper presents a theoretical analysis of the relationship between the sizes of the LS and STLS corrections (called the LS and STLS distances) in terms of . We give new upper and lower bounds on the LS distance in terms of the STLS distance, compare these to existing bounds, and examine the tightness of the new bounds. This work can be applied to the analysis of iterative methods which minimize the residual norm, and the generalized minimum residual method (GMRES) [15] is used here to illustrate our theory. Received July 20, 2000 / Revised version received February 28, 2001 / Published online July 25, 2001     

Convergent adaptive finite elements for the nonlinear Laplacian

Summary. The numerical solution of the homogeneous Dirichlet problem for the p-Laplacian, , is considered. We propose an adaptive algorithm with continuous piecewise affine finite elements and prove that the approximate solutions converge to the exact one. While the algorithm is a rather straight-forward generalization of those for the linear case p=2, the proof of its convergence is different. In particular, it does not rely on a strict error reduction. Received December 29, 2000 / Revised version received August 30, 2001 / Published online December 18, 2001 RID="*" ID="*" Current address: Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy; e-mail: veeser@mat.unimi.it     

Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets

Summary. The cascade algorithm with mask a and dilation M generates a sequence by the iterative process from a starting function where M is a dilation matrix. A complete characterization is given for the strong convergence of cascade algorithms in Sobolev spaces for the case in which M is isotropic. The results on the convergence of cascade algorithms are used to deduce simple conditions for the computation of integrals of products of derivatives of refinable functions and wavelets. Received May 5, 1999 / Revised version received June 24, 1999 / Published online June 20, 2001