Block Toeplitz Research Articles

Modified Szegö–Widom Asymptotics for Block Toeplitz Matrices with Zero Modes

The Szego-Widom theorem provides an expression for the determinant of block Toeplitz matrices in the asymptotic limit of large matrix dimension n. We show that the presence of zero modes, i.e, eigenvalues that vanish as \alpha^n, |\alpha|<1, when n \to \infty, require a modification of the Szego-Widom theorem. A new asymptotic expression for the determinant of a certain class of block Toeplitz matrices with one pair of zero modes is derived. The result is inspired by 1-dimensional topological superconductors, and the relation with the latter systems is discussed.

Isogeometric analysis for 2D and 3D curl–div problems: Spectral symbols and fast iterative solvers

Alfvén-like operators are of interest in magnetohydrodynamics, which is used in plasma physics to study the macroscopic behavior of plasma. Motivated by this important and complex application, we focus on a parameter-dependent curl–div problem that can be seen as a prototype of an Alfvén-like operator, and we discretize it using isogeometric analysis based on tensor-product B-splines. The involved coefficient matrices can be very ill-conditioned, so that standard numerical solution methods perform quite poorly here. In order to overcome the difficulties caused by such ill-conditioning, a two-step strategy is proposed. First, we conduct a detailed spectral study of the coefficient matrices, highlighting the critical dependence on the different physical and approximation parameters. Second, we exploit such spectral information to design fast iterative solvers for the corresponding linear systems. For the first goal we apply the theory of (multilevel block) Toeplitz and generalized locally Toeplitz sequences, while for the second we use a combination of multigrid techniques and preconditioned Krylov solvers. Several numerical tests are provided both for the study of the spectral problem and for the solution of the corresponding linear systems.

New “Verblunsky-type” coefficients of block Toeplitz and Hankel matrices and of corresponding Dirac and canonical systems

We introduce Verblunsky-type coefficients of Toeplitz and Hankel matrices, which correspond to the discrete Dirac and canonical systems generated by Toeplitz and Hankel matrices, respectively. We prove one to one correspondences between positive-definite Toeplitz (Hankel) matrices and their Verblunsky-type coefficients as analogs of the well-known Verblunsky’s theorem. Several interconnections with the spectral theory are described as well.

Tau Functions as Widom Constants

We define a tau function for a generic Riemann-Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed as the Fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices. Its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the Riemann-Hilbert problem. In the case of one circle, the tau function coincides with Widom's determinant arising in the asymptotics of block Toeplitz matrices. Our construction gives the Jimbo-Miwa-Ueno tau function for Riemann-Hilbert problems of isomonodromic origin (Painlev\'e VI, V, III, Garnier system, etc) and the Sato-Segal-Wilson tau function for integrable hierarchies such as Gelfand-Dickey and Drinfeld-Sokolov.

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix T_n(f), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f. A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where mathbf {f} is an stimes s matrix-valued trigonometric polynomial with sge 1, and T_n(mathbf {f}) is the block Toeplitz matrix generated by mathbf {f}, whose size is N(n,s)=sn. The case s=1 corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrix-less. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the mathbb {Q}_p Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.

Entanglement entropy in the long-range Kitaev chain

In this paper we complete the study on the asymptotic behavior of the entanglement entropy for Kitaev chains with long-range pairing. We discover that when the couplings decay with the distance with a critical exponent new properties for the asymptotic growth of the entropy appear. The coefficient of the leading term is not universal any more and the connection with conformal field theories is lost. We perform a numerical and analytical approach to the problem showing a perfect agreement. In order to carry out the analytical study, a technique for computing the asymptotic behavior of block Toeplitz determinants with discontinuous symbols has been developed.

Singular boundary method for wave propagation analysis in periodic structures

A strong-form boundary collocation method, the singular boundary method (SBM), is developed in this paper for the wave propagation analysis at low and moderate wavenumbers in periodic structures. The SBM is of several advantages including mathematically simple, easy-to-program, meshless with the application of the concept of origin intensity factors in order to eliminate the singularity of the fundamental solutions and avoid the numerical evaluation of the singular integrals in the boundary element method. Due to the periodic behaviors of the structures, the SBM coefficient matrix can be represented as a block Toeplitz matrix. By employing three different fast Toeplitz-matrix solvers, the computational time and storage requirements are significantly reduced in the proposed SBM analysis. To demonstrate the effectiveness of the proposed SBM formulation for wave propagation analysis in periodic structures, several benchmark examples are presented and discussed The proposed SBM results are compared with the analytical solutions, the reference results and the COMSOL software.

Jointly hyponormal block Toeplitz pairs with rational symbols

Jointly hyponormal block Toeplitz pairs with rational symbols

Preconditioning and Iterative Solution of All-at-Once Systems for Evolutionary Partial Differential Equations

Standard Krylov subspace solvers for self-adjoint problems have rigorous convergence bounds based solely on eigenvalues. However, for non-self-adjoint problems, eigenvalues do not determine behavior even for widely used iterative methods. In this paper, we discuss time-dependent PDE problems, which are always non-self-adjoint. We propose a block circulant preconditioner for the all-at-once evolutionary PDE system which has block Toeplitz structure. Through reordering of variables to obtain a symmetric system, we are able to rigorously establish convergence bounds for MINRES which guarantee a number of iterations independent of the number of time-steps for the all-at-once system. If the spatial differential operators are simultaneously diagonalizable, we are able to quickly apply the preconditioner through use of a sine transform; and for those that are not, we are able to use an algebraic multigrid process to provide a good approximation. Results are presented for solution to both the heat and convection diffusion equations.

Matrix orthogonal polynomials associated with perturbations of block Toeplitz matrices

Let {c j } =0 be a sequence of matrix moments associated with a matrix of measures supported on the unit circle, and let {P j } =0 be its corresponding sequence of monic matrix orthogonal polynomials. In this contribution, we consider a perturbation on the moments and find an explicit relation for the perturbed orthogonal polynomials in terms of {P j } =0 . We also obtain an expression for the corresponding second kind polynomials.

Contravariant Adaptation on the Manifold of Causal, FIR, Invertible Multivariable Matrix Systems

This letter extends previous work on contravariant adaptation by providing a formula for the contravariant (natural) gradient on the manifold of multivariable, causal, invertible, finite impulse response (FIR) systems. The right action on the manifold of multivariable causal Toeplitz systems is defined. Using this right action, a bilinear form defined on the tangent space at the identity is extended throughout the entire manifold by invariance. This results in a formula analogous to the natural gradient for matrices, but which preserves the causal, FIR nature of the system. The contravariant conversion factor is block Toeplitz structured, so that implementations may employ fast Fourier transform based convolutions to produce lower complexity than would a comparably-sized generic matrix natural gradient.

Asymptotics of determinants of block Toeplitz matrices

This paper is a survey of results that show how, in some cases, to compute the constant term which occurs in the Strong Szegö–Widom Limit Theorem for block Toeplitz matrices. While this constant in the scalar case has a long history, it is really only in a few instances that it can be calculated explicitly in the case of matrix-valued symbols.

Hyponormality of block Toeplitz operators on the weighted Bergman spaces

In this paper we consider the block Toeplitz operators TPhi on the weighted Bergman space Aα2(D, Cn) and we give a necessary and sufficient condition for the hyponormality of block Toeplitz operators with symbol in the class of functions Φ=F+G* with matrix-valued polynomial functions F and G with degree 2.

Convex recovery of continuous domain piecewise constant images from nonuniform Fourier samples.

We consider the recovery of a continuous domain piecewise constant image from its non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities/edges of the image are localized to the zero level-set of a bandlimited function. This assumption induces linear dependencies between the Fourier coefficients of the image, which results in a two-fold block Toeplitz matrix constructed from the Fourier coefficients being low-rank. The proposed algorithm reformulates the recovery of the unknown Fourier coefficients as a structured low-rank matrix completion problem, where the nuclear norm of the matrix is minimized subject to structure and data constraints. We show that exact recovery is possible with high probability when the edge set of the image satisfies an incoherency property. We also show that the incoherency property is dependent on the geometry of the edge set curve, implying higher sampling burden for smaller curves. This paper generalizes recent work on the super-resolution recovery of isolated Diracs or signals with finite rate of innovation to the recovery of piecewise constant images.

Preconditioned conjugate gradient methods for the solution of Love’s integral equation with very small parameter

In this paper, we propose efficient numerical methods for the solution of the following Love’s integral equation f ( x ) + 1 π ∫ − 1 1 c ( x − y ) 2 + c 2 f ( y ) d y = 1 , x ∈ [ − 1 , 1 ] , where c > 0 is a very small parameter. We introduce a new unknown function h ( x ) = f ( x ) − 0 . 5 as in Lin et al. (2013), and then apply a composite Gauss–Legendre quadrature to the resulting integral equation as in Pastore (2011). The coefficient matrix of corresponding linear system is a nonsymmetric block matrix with Toeplitz blocks. We transform the nonsymmetric linear system into a symmetric linear system and introduce a preconditioner which is a block matrix with circulant blocks. Spectral properties of relevant matrices are analyzed and numerical results are presented to illustrate the efficiency of the proposed methods.

Performance Analysis of Toeplitz Block Sampling Matrix in Compressed Sensing

This paper analyzes the recovery performance when a Toeplitz block matrix is used for sampling in compressed sensing. Different from current work that mainly discusses the restricted isometry property and the applications of such matrix, we provide an upper bound for the mean squared reconstruction error. By comparing with the random matrix in which its entries are drawn independently from certain probability distributions, the results show that the Toeplitz block matrix is efficient for compressed sampling. Simulation results validate that the sampling performance of Toeplitz block matrix can approach that of random matrix by choosing its parameters properly.

Acoustic Scattering for Rotational and Translational Symmetric Structures in Nonuniform Potential Flow

An efficient approach is proposed to predict acoustic scattering with nonuniform potential flow effects for structures with rotational and translational symmetries. The convected wave equation is transformed to Helmholtz and Laplace equations using a time transformation. The boundary-element method is used to formulate scattering by rotationally symmetric structures as two separate block circulant matrix equations and, similarly, as two separate block Toeplitz matrix equations for structures with translational symmetry. Discrete Fourier transform is employed to solve the block circulant systems. The block Toeplitz systems are solved using the generalized minimal residual method along with the discrete Fourier transform. Solving the convected wave equation using structured matrices significantly reduces computational time and storage requirements. To demonstrate the application of the formulation, two exterior acoustic case studies are considered. The first case study examines acoustic scattering from a sphere submerged in potential flow under monopole source excitation. Directivity plots obtained using the proposed technique are compared with analytical results. The second case study examines flow-induced noise generated by a rigid cylinder immersed in low-Mach-number flow, with the effect of mean flow on the scattered acoustic field taken into account using nonuniform potential flow. The fluctuating flowfield is obtained using an incompressible computational fluid dynamics solver. Acoustic sources based on Lighthill’s analogy are extracted from the flowfield data using a high-order reconstruction scheme. Results from the hybrid computational fluid dynamics– boundary-element method technique are presented for turbulent flow past the cylinder, with Reynolds number based on cylinder diameter of and Mach number . The aeroacoustic results are compared with data from literature.

Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach

Abstract This paper introduces and analyzes the new grid-based tensor approach to approximate solutions of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree–Fock equation over a spatial L 1 × L 2 × L 3 {L_{1}\times L_{2}\times L_{3}} lattice for both periodic and non-periodic problem setting, discretized in the localized Gaussian-type orbitals basis. In the periodic case, the Galerkin system matrix obeys a three-level block-circulant structure that allows the FFT-based diagonalization, while for the finite extended systems in a box (Dirichlet boundary conditions) we arrive at the perturbed block-Toeplitz representation providing fast matrix-vector multiplication and low storage size. The proposed grid-based tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) in the periodic case the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems in a box with Dirichlet boundary conditions are treated numerically by our previous tensor solver for single molecules, which makes possible calculations on rather large L 1 × L 2 × L 3 {L_{1}\times L_{2}\times L_{3}} lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic L × 1 × 1 {L\times 1\times 1} lattice chain in a 3D rectangular “tube” with L up to several hundred confirm the theoretical complexity bounds for the block-structured eigenvalue solvers in the limit of large L.

Effect of a serrated trailing edge on sound radiation from nearby quadrupoles.

A periodic boundary element technique is implemented to study the noise reduction capability of a plate with a serrated trailing edge under quadrupole excitation. It is assumed for this purpose that the quadrupole source tensor is independent of the trailing edge configuration and that the effect of the trailing edge shape is to modify sound radiation from prescribed boundary layer sources. The flat plate is modelled as a continuous structure with a finite repetition of small spanwise segments. The matrix equation formulated by the periodic boundary element method for this 3D acoustic scattering problem is represented as a block Toeplitz matrix. The discrete Fourier transform is employed in an iterative algorithm to solve the block Toeplitz system. The noise reduction mechanism for a serrated trailing edge in the near field is investigated by comparing contour plots obtained from each component of the quadrupole for unserrated and serrated trailing edge plate models. The noise reduction due to the serrated trailing edge is also examined as a function of the source location.

Reversibility Problem of Multidimensional Finite Cellular Automata

While the reversibility of multidimensional cellular automata is undecidable and there exists a criterion for determining if a multidimensional linear cellular automaton is reversible, there are only a few results about the reversibility problem of multidimensional linear cellular automata under boundary conditions. This work proposes a criterion for testing the reversibility of a multidimensional linear cellular automaton under null boundary condition and an algorithm for the computation of its reverse, if it exists. The investigation of the dynamical behavior of a multidimensional linear cellular automaton under null boundary condition is equivalent to elucidating the properties of block Toeplitz matrix. The proposed criterion significantly reduce the computational cost whenever the number of cells or the dimension is large; the discussion can also apply to cellular automata under periodic boundary condition with a minor modification.