On the spectrum of semi-classical Witten-Laplacians and Schrödinger operators in large dimension

We investigate the low-lying spectrum of Witten–Laplacians on forms of arbitrary degree in the semi-classical limit and uniformly in the space dimension. We show that under suitable assumptions implying that the phase function has a unique local minimum one obtains a number of clusters of discrete eigenvalues at the bottom of the spectrum. Moreover, we are able to count the number of eigenvalues in each cluster. We apply our results to certain sequences of Schrödinger operators having strictly convex potentials and show that some well-known results of semi-classical analysis hold also uniformly in the dimension.     

The correction operator for the canonical interpolation operator of the Adini element and the lower bounds of eigenvalues

In this paper,we develop a correction operator for the canonical interpolation operator of the Adini element.We use this new correction operator to analyze the discrete eigenvalues of the Adini element method for the fourth order elliptic eigenvalue problem in the three dimensions.We prove that the discrete eigenvalues are smaller than the exact ones.     

Non-consistent approximations of self-adjoint eigenproblems: application to the supercell method

In this article, we introduce a general theoretical framework to analyze non-consistent approximations of the discrete eigenmodes of a self-adjoint operator. We focus in particular on the discrete eigenvalues laying in spectral gaps. We first provide a priori error estimates on the eigenvalues and eigenvectors in the absence of spectral pollution. We then show that the supercell method for perturbed periodic Schrödinger operators falls into the scope of our study. We prove that this method is spectral pollution free, and we derive optimal convergence rates for the planewave discretization method, taking numerical integration errors into account. Some numerical illustrations are provided.     

Quasi-interpolation operators based on a cubic spline and applications in SAMR simulations

In this paper, we consider the properties of monotonicity-preserving and global conservation-preserving for interpolation operators. These two properties play important role when interpolation operators used in many real numerical simulations. In order to attain these two aspects, we propose a one-dimensional (1D) new cubic spline, and extend it to two-dimensional (2D) using tensor-product operation. Based on discrete convolution, 1D and 2D quasi-interpolation operators are presented using these functions. Both analysis and numerical results show that the interpolation operators constructed in this paper are monotonic and conservative. In particular, we consider the numerical simulations of 2D Euler equations based on the technique of structured adaptive mesh refinement (SAMR). In SAMR simulations, effective interpolators are needed for information transportation between the coarser/finer meshes. We applied the 2D quasi-interpolation operator to this environment, and the simulation result show the efficiency and correctness of our interpolator.     

Asymptotics of the Distribution of Eigenvalues of a Selfadjoint Second Order Hyperbolic Differential Operator on the Two-Dimensional Torus

We study the asymptotic properties of the discrete spectrum for general selfadjoint second order hyperbolic operators on the two-dimensional torus. For a broad class of operators with sufficiently smooth coefficients and the principal part coinciding with the wave operator in the light cone coordinates we prove the discreteness of the spectrum and obtain an asymptotic formula for the distribution of eigenvalues. In some cases we can indicate the first two asymptotic terms. We discuss the relations of these questions to analytic number theory and mathematical physics.     

Hermitian quadratic eigenvalue problems of restricted rank

We consider a quadratic eigenvalue problem such that the second order term is a Hermitian matrix of rank r, the linear term is the identity matrix, and the constant term is an arbitrary Hermitian matrix . Of the n+r solutions that this problem admits, we show at least n-r to be real and located in specific intervals defined by the eigenvalues of A, whence at most 2r are nonreal occuring in possibly repeated conjugate pairs.     

Eigenvalue asymptotics for an elliptic boundary problem involving an indefinite weight

We are concerned here with the eigenvalue asymptotics for a non-selfadjoint elliptic boundary problem involving an indefinite weight function which vanishes on a set of positive measure. The asymptotic behaviour of the eigenvalues is well known for the case of second order operators. However for higher order operators, results have only been established under the restriction that the order of the operator exceeds the dimension of the underlying Euclidean space in which the problem is set. In this paper we establish the eigenvalue asymptotics for the case of higher order operators without any such restriction.Supported in part by the John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand.     

Adaptive finite element methods for computing band gaps in photonic crystals

In this paper we propose and analyse adaptive finite element methods for computing the band structure of 2D periodic photonic crystals. The problem can be reduced to the computation of the discrete spectra of each member of a family of periodic Hermitian eigenvalue problems on a unit cell, parametrised by a two-dimensional parameter - the quasimomentum. These eigenvalue problems involve non-coercive elliptic operators with generally discontinuous coefficients and are solved by adaptive finite elements. We propose an error estimator of residual type and show it is reliable and efficient for each eigenvalue problem in the family. In particular we prove that if the error estimator converges to zero then the distance of the computed eigenfunction from the true eigenspace also converges to zero and the computed eigenvalue converges to a true eigenvalue with double the rate. We also prove that if the distance of a computed sequence of approximate eigenfunctions from the true eigenspace approaches zero, then so must the error estimator. The results hold for eigenvalues of any multiplicity. We illustrate the benefits of the resulting adaptive method in practice, both for fully periodic structures and also for the computation of eigenvalues in the band gap of structures with defect, using the supercell method.     

On smoothers for multigrid of the second kind

We study smoothers for the multigrid method of the second kind arising from Fredholm integral equations. Our model problems use nonlocal governing operators that enforce local boundary conditions. For discretization, we utilize the Nyström method with the trapezoidal rule. We find the eigenvalues of matrices associated to periodic, antiperiodic, and Dirichlet problems in terms of the nonlocality parameter and mesh size. Knowing explicitly the spectrum of the matrices enables us to analyze the behavior of smoothers. Although spectral analyses exist for finding effective smoothers for 1D elliptic model problems, to the best of our knowledge, a guiding spectral analysis is not available for smoothers of a multigrid of the second kind. We fill this gap in the literature. The Picard iteration has been the default smoother for a multigrid of the second kind. Jacobi‐like methods have not been considered as viable options. We propose two strategies. The first one focuses on the most oscillatory mode and aims to damp it effectively. For this choice, we show that weighted‐Jacobi relaxation is equivalent to the Picard iteration. The second strategy focuses on the set of oscillatory modes and aims to damp them as quickly as possible, simultaneously. Although the Picard iteration is an effective smoother for model nonlocal problems under consideration, we show that it is possible to find better than ones using the second strategy. We also shed some light on internal mechanism of the Picard iteration and provide an example where the Picard iteration cannot be used as a smoother.     

Statistical convergence of a non-positive approximation process

Starting from a general sequence of linear and positive operators of discrete type, we associate its r-th order generalization. This construction involves high order derivatives of a signal and it looses the positivity property. Considering that the initial approximation process is A-statistically uniform convergent, we prove that the property is inherited by the new sequence. Also, our result includes information about the uniform convergence. Two applications in q-Calculus are presented. We study q-analogues both of Meyer-König and Zeller operators and Stancu operators.     

Causality properties of refinable functions and sequences

We show that the scale-space operators defined by a class of refinable kernels satisfy a version of the causality property, and a sequence of such operators converges to the corresponding operator with the Gaussian kernel, if the sequence of refinable kernels converges to the Gaussian function. In addition, we consider discrete analogs of these operators and show that a class of refinable sequences satisfies a discrete version of the causality property. The solutions of the corresponding discrete refinement equations are also investigated in detail.     

Consistent estimation of the dimensionality in sliced inverse regression

Several methods have been proposed in the literature in order to estimate the dimensionality in sliced inverse regression. Most of these methods are based on sequential tests for the nullity of the last eigenvalues of suitable operators. We first establish non consistency for estimators resulting from these methods. Then, we propose an estimator obtained by minimizing a suitable penalization of a statistic based on eigenvalues. A consistency property is established for this estimator and a simulation study is undertaken to evaluate its finite sample performance.     

On the far-field operator in elastic obstacle scattering

We investigate the far-field operator for the scattering oftime-harmonic elastic plane waves by either a rigid body, acavity, or an absorbing obstacle. Extending results of Colton& Kress for acoustic obstacle scattering, for the spectrumof the far-field operator we show that there exist an infinitenumber of eigenvalues and determine disks in the complex planewhere these eigenvalues lie. In addition, as counterpart ofan identity in acoustic scattering due to Kress & Päivärinta,we will establish a factorization for the difference of thefar-field operators for two different scatterers. Finally, extendinga sampling method for the approximate solution of the acousticinverse obstacle scattering problem suggested by Kirsch to elasticity,this factorization is used for a characterization of a rigidscatterer in terms of the eigenvalues and eigenelements of thefar-field operator.     

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.     

Limiting Absorption Principle for Singularly Perturbed Operators

Given an operator H₁ for which a limiting absorption principle holds, we study operators H₂ which are produced by perturbing H₁ in the sense that the difference between some powers of the resolvents is compact.We show that (except for possibly a discrete set of eigenvalues) a limiting absorption principle holds for H₂. We apply this theory to study potential and domain perturbations of Feller operators. While our theory mostly reproduces known results in the case of potential perturbations, for domain perturbations we get results which appear to be new.     

Spectral instability for some Schr?dinger operators

Summary. We show that the condition numbers of isolated eigenvalues of typical non-self-adjoint differential operators such as the harmonic oscillator may be extremely large. We describe a stable procedure for computing the condition numbers for Schr?dinger operators in one dimension, and apply it to the complex resonances of a typical operator with a dilation analytic potential. Received October 9, 1998 / Revised version received September 13, 1999 / Published online 16 March 2000     

Iteratively reweighted algorithm for signals recovery with coherent tight frame

We consider the problem of compressed sensing with a coherent tight frame and design an iteratively reweighted least squares algorithm to solve it. To analyze the problem, we propose a sufficient null space property under a tight frame (sufficient D‐NSP). We show that, if a measurement matrix A satisfies the sufficient D‐NSP of order s, then an s‐sparse signal under the tight frame can be exactly recovered. Furthermore, if A satisfies the restricted isometric property with tight frame D of order 2bs, then it also satisfies the sufficient D‐NSP of order as with a < b and b sufficiently large. We prove the convergence of the algorithm based on the sufficient D‐NSP and give the upper error bounds. In numerical experiments, we use the discrete cosine transform, discrete Fourier transform, and Haar wavelets to verify the effectiveness of this algorithm. With increasing measurement number, the signal‐to‐noise ratio increases monotonically.     

On the spectrum of Schrödinger operators with a finitely supported potential on the -regular tree

We study the spectrum of Schrödinger operators with a uniform potential on the lth shell of the d-regular tree. As a result, we show the relationship between the spectral structure and the intensities of the potential. Furthermore we completely determine the discrete eigenvalues with their multiplicities. In addition we give some examples.     

A box‐shaped cyclically reduced operator

We propose a new procedure of partial cyclic reduction, where we apply a 2^d‐color ordering (with d=2, 3 the dimension of the problem), and use different operators for different gridpoints according to their color. These operators are chosen so that the gridpoints can be readily decoupled, and we then eliminate all colors but one. This yields a smaller cartesian mesh and box‐shaped 9‐point (in 2D) or 27‐point (in 3D) operators that are easy to analyze and implement. Multi‐line and multi‐plane orderings are considered, and we perform convergence analysis and numerical experiments that demonstrate the merits of our approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Weyl composition of symbols in large dimension

This paper is concerned with the Weyl composition of symbols in large dimension. We specify a class of symbols in order to estimate the Weyl symbol of the product of two Weyl h-pseudodifferential operators, with constants independent of the dimension. The proof includes regularized and hybrid compositions, together with a decomposition formula. We also analyze, in this context, the remainder term of the semiclassical expansion of the Weyl composition. The class of symbols contains symbols of Schrödinger semigroups in large dimension, typically for nearest neighbors or mean field interaction potentials. The Weyl composition is applied with Kac operators.