In this paper we consider a family of tetradiagonal (= four non-zero diagonals) Toeplitz matrices with a limiting set consisting in one analytic arc only and obtain individual asymptotic expansions ...for all the eigenvalues, as the matrix size goes to infinity. Additionally, we provide specific expansions for the extreme eigenvalues which are the eigenvalues approaching the extreme points of the limiting set. In contrast to previous related works, we study non-Hermitian Toeplitz matrices having non-canonical distribution and a real limiting set. The considered family does not belong to the so-called simple-loop class, nevertheless we manage to extend the theory to this case. The achieved formulas reveal the fine details of the eigenvalue structure and allow us to directly calculate high accuracy eigenvalues, even for matrices of relatively small size.
The main aim of this paper is to apply the Galerkin finite element method to numerically solve the nonlinear fractional Schrödinger equation with wave operator. We first construct a fully discrete ...scheme combining the Crank–Nicolson method with the Galerkin finite element method. Two conserved quantities of the discrete system are shown. Meanwhile, the prior bound of the discrete solutions are proved. Then, we prove that the discrete scheme is unconditionally convergent in the senses of L2−norm and Hα/2−norm. Moreover, by the proposed iterative algorithm, some numerical examples are given to verify the theoretical results and show the effectiveness of the numerical scheme. Finally, a fast Krylov subspace solver with suitable circulant preconditioner is designed to solve above Toeplitz-like linear system. In each iterative step, this method can effectively reduce the memory requirement of the proposed iterative finite element scheme from O(M2) to O(M), and the computational complexity from O(M3) to O(MlogM), where M is the number of grid nodes. Several numerical tests are carried out to show that this fast algorithm is more practical than the traditional backslash and LU factorization methods, in terms of memory requirement and computational cost.
The Vandermonde decomposition of Toeplitz matrices, discovered by Carathéodory and Fejér in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1-D ...frequency estimation. Many related numerical tools have also been developed for multidimensional (MD), especially 2-D, frequency estimation; however, a fundamental question has remained unresolved as to whether an analog of the Vandermonde decomposition holds for multilevel Toeplitz matrices in the MD case. In this paper, an affirmative answer to this question and a constructive method for finding the decomposition are provided when the matrix rank is lower than the dimension of each Toeplitz block. A numerical method for searching for a decomposition is also proposed when the matrix rank is higher. The new results are applied to study the MD frequency estimation within the recent super-resolution framework. A precise formulation of the atomic \ell _{0} norm is derived using the Vandermonde decomposition. Practical algorithms for frequency estimation are proposed based on the relaxation techniques. Extensive numerical simulations are provided to demonstrate the effectiveness of these algorithms compared with the existing atomic norm and subspace methods.
Fractional partial order diffusion equations are a generalization of classical partial differential equations, used to model anomalous diffusion phenomena. When using the implicit Euler formula and ...the shifted Grünwald formula, it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a Toeplitz-like structure. In this paper we focus our attention on the case of variable diffusion coefficients. Under appropriate conditions, we show that the sequence of the coefficient matrices belongs to the Generalized Locally Toeplitz class and we compute the symbol describing its asymptotic eigenvalue/singular value distribution, as the matrix size diverges. We employ the spectral information for analyzing known methods of preconditioned Krylov and multigrid type, with both positive and negative results and with a look forward to the multidimensional setting. We also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system, with Krylov methods such as CGNR and GMRES. A number of numerical examples show that our proposal is more effective than recently used circulant preconditioners.
The paper presents higher-order asymptotic formulas for the eigenvalues of large Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The ...formulas are established not only for the extreme eigenvalues, but also for the inner eigenvalues. The results extend and make more precise existing results, which so far pertain to banded matrices or to matrices with infinitely differentiable symbols. Also given is a fixed-point equation for the eigenvalues which may be solved numerically by an iteration method.
A combined coprime and nested array geometry is designed and a corresponding direction of arrival (DOA) method is proposed. The proposed array has three subarrays, where two are prototype coprime ...subarrays, and both of them are nested to the third one. Then the vectorisations of two covariance matrices yield two virtual coprime subarrays, which show a much larger aperture compared to the physical ones. Thereafter, through Toeplitz matrix reconstruction and root multiple signal classification, two coprime estimations are obtained and they are combined to obtain the final unique DOA estimation. Multiple analysis and simulations verify that the proposed method achieves larger aperture, better DOA estimation accuracy and higher angular resolution compared to other coprime array-based methods.
We present a procedure which enables the computation and the description of structures of isotropy subgroups of the group of complex orthogonal matrices with respect to the action of *congruence on ...Hermitian matrices. A key ingredient in our proof is an algorithm giving solutions of a certain rectangular block (complex-alternating) upper triangular Toeplitz matrix equation.
ESPRIT-like algorithm and its variants are widely used in preprocessing schemes that provide high performance of high-resolution parameter estimation algorithms for coherent signals. However, in the ...presence of certain phase differences between coherent signals and signal angles, this method suffers serious performance degradation and even fails. In this letter, a modified scheme based on Forward and Backward Partial Toeplitz Matrices Reconstruction named as FB-PTMR is proposed that exploits half rows of the sample covariance matrix (SCM) to reconstruct the data matrix to overcome this weakness. The new approach provides better performance on estimation and probability of resolution compared to the ESPRIT-like method by exploiting a small part of information of the sample covariance matrix. The simulation results verify the effectiveness of our proposed method.
Bessel orbits of normal operators Philipp, Friedrich
Journal of mathematical analysis and applications,
04/2017, Volume:
448, Issue:
2
Journal Article
Peer reviewed
Open access
Given a bounded normal operator A in a Hilbert space and a fixed vector x, we elaborate on the problem of finding necessary and sufficient conditions under which (Akx)k∈N constitutes a Bessel ...sequence. We provide a characterization in terms of the measure ‖E(⋅)x‖2, where E is the spectral measure of the operator A. In the separately treated special cases where A is unitary or selfadjoint we obtain more explicit characterizations. Finally, we apply our results to a sequence (Akx)k∈N, where A arises from the heat equation. The problem is motivated by and related to the new field of Dynamical Sampling which was recently initiated by Aldroubi et al. in 3.
The dependence between pairs of time series is commonly quantified by Pearson's correlation. However, if the time series are themselves dependent (i.e. exhibit temporal autocorrelation), the ...effective degrees of freedom (EDF) are reduced, the standard error of the sample correlation coefficient is biased, and Fisher's transformation fails to stabilise the variance. Since fMRI time series are notoriously autocorrelated, the issue of biased standard errors – before or after Fisher's transformation – becomes vital in individual-level analysis of resting-state functional connectivity (rsFC) and must be addressed anytime a standardised Z-score is computed. We find that the severity of autocorrelation is highly dependent on spatial characteristics of brain regions, such as the size of regions of interest and the spatial location of those regions. We further show that the available EDF estimators make restrictive assumptions that are not supported by the data, resulting in biased rsFC inferences that lead to distorted topological descriptions of the connectome on the individual level. We propose a practical “xDF” method that accounts not only for distinct autocorrelation in each time series, but instantaneous and lagged cross-correlation. We find the xDF correction varies substantially over node pairs, indicating the limitations of global EDF corrections used previously. In addition to extensive synthetic and real data validations, we investigate the impact of this correction on rsFC measures in data from the Young Adult Human Connectome Project, showing that accounting for autocorrelation dramatically changes fundamental graph theoretical measures relative to no correction.
•Autocorrelation is a problem for sample correlation, breaking the variance-stabilising property of Fisher's transformation.•We show that fMRI autocorrelation varies systematically with region of interest size, and is heterogeneous over subjects.•Existing adjustment methods are themselves biased when true correlation is non-zero due to a confounding effect.•Our “xDF” method provides accurate Z-scores based on either of Pearson's or Fisher's transformed correlations.•Resting state fMRI autocorrelation considerably alters the graph theoretical description of human connectome.
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