Block Toeplitz Research Articles

Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol

This note is devoted to preconditioning strategies for non-Hermitian multilevel block Toeplitz linear systems associated with a multivariate Lebesgue integrable matrix-valued symbol. In particular, we consider special preconditioned matrices, where the preconditioner has a band multilevel block Toeplitz structure, and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices. In this respect, our main result is as follows. Let Ik≔(-π,π)k, let Ms be the linear space of complex s×s matrices, and let f,g:Ik→Ms be functions whose components fij,gij:Ik→C,i,j=1,…,s, belong to L∞. Consider the matrices Tn-1(g)Tn(f), where n≔(n1,…,nk) varies in Nk and Tn(f),Tn(g) are the multilevel block Toeplitz matrices of size n1⋯nks generated by f,g. Then {Tn-1(g)Tn(f)}n∈Nk∼λg-1f, i.e. the family of matrices {Tn-1(g)Tn(f)}n∈Nk has a global (asymptotic) spectral distribution described by the function g-1f, provided g possesses certain properties (which ensure in particular the invertibility of Tn(g) for all n) and the following topological conditions are met: the essential range of g-1f, defined as the union of the essential ranges of the eigenvalue functions λj(g-1f),j=1,…,s, does not disconnect the complex plane and has empty interior. This result generalizes the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work, concerning the non-preconditioned case g=1. The last part of this note is devoted to numerical experiments, which confirm the theoretical analysis and suggest the choice of optimal GMRES preconditioning techniques to be used for the considered linear systems.

A parallel linear solver for multilevel Toeplitz systems with possibly several right-hand sides

Abstract A Toeplitz matrix has constant diagonals; a multilevel Toeplitz matrix is defined recursively with respect to the levels by replacing the matrix elements with Toeplitz blocks. Multilevel Toeplitz linear systems appear in a wide range of applications in science and engineering. This paper discusses an MPI implementation for solving such a linear system by using the conjugate gradient algorithm. The implementation techniques can be generalized to other iterative Krylov methods besides conjugate gradient. These techniques include the use of an arbitrary dimensional process grid for handling the multilevel Toeplitz structure, a communication-hiding approach for performing matrix–vector multiplications, the incorporation of multilevel circulant preconditioning for accelerating convergence, an efficient orthogonalization manager for detecting linear dependence in block iterations, and an algorithmic rearrangement to eliminate all-reduce synchronizations. The combined use of these techniques leads to a scalable solver for large multilevel Toeplitz systems, possibly with several right-hand sides. We show experimental results on matrices of size up to the order of one billion with nearly perfect scaling by using up to 1024 MPI processes. We also demonstrate an application of the solver in parameter estimation for analyzing large-scale climate data.

Multigrid methods for cubic spline solution of two point (and 2D) boundary value problems

In this paper we propose a scheme based on cubic splines for the solution of the second order two point boundary value problems. The solution of the algebraic system is computed by using optimized multigrid methods. In particular the transformation of the stiffness matrix essentially in a block Toeplitz matrix and its spectral analysis allow to choose smoothers able to reduce error components related to the various frequencies and to obtain an optimal method. The main advantages of our strategy can be listed as follows: (i) a fourth order of accuracy combined with a quadratic conditioning matrix, (ii) a resulting matrix structure whose eigenvalues can be compactly described by a symbol (this information is the key for designing an optimal multigrid method). Finally, some numerics that confirm the predicted behavior of the method are presented and a discussion on the two dimensional case is given, together with few 2D numerical experiments.

Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models

AbstractThe stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

THE HYPONORMAL TOEPLITZ OPERATORS ON THE VECTOR VALUED BERGMAN SPACE

In this paper, we give a necessary and sufficient condition for the hyponormality of the block Toeplitz operators <TEX>$T_{\Phi}$</TEX>, where <TEX>${\Phi}$</TEX> = <TEX>$F+G^*$</TEX>, F(z), G(z) are some matrix valued polynomials on the vector valued Bergman space <TEX>$L^2_a(\mathbb{D},\mathbb{C}^n</TEX><TEX>)$</TEX>. We also show some necessary conditions for the hyponormality of <TEX>$T_{F+G^*}$</TEX> with <TEX>$F+G^*{\in}h^{\infty}{\otimes}M_{n{\times}n}$</TEX> on <TEX>$L^2_a(\mathbb{D},\mathbb{C}^n)$</TEX>.

The palindromic cyclic reduction and related algorithms

The cyclic reduction algorithm is specialized to palindromic matrix polynomials and a complete analysis of applicability and convergence is provided. The resulting iteration is then related to other algorithms as the evaluation/interpolation at the roots of unity of a certain Laurent matrix polynomial, the trapezoidal rule for a certain integral and an algorithm based on the finite sections of a tridiagonal block Toeplitz matrix.

Covariance Estimation in High Dimensions Via Kronecker Product Expansions

This paper presents a new method for estimating high dimensional covariance matrices. The method, permuted rank-penalized least-squares (PRLS), is based on a Kronecker product series expansion of the true covariance matrix. Assuming an i.i.d. Gaussian random sample, we establish high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity. For covariance matrices of low separation rank, our results establish that PRLS has significantly faster convergence than the standard sample covariance matrix (SCM) estimator. The convergence rate captures a fundamental tradeoff between estimation error and approximation error, thus providing a scalable covariance estimation framework in terms of separation rank, similar to low rank approximation of covariance matrices. The MSE convergence rates generalize the high dimensional rates recently obtained for the ML Flip-flop algorithm for Kronecker product covariance estimation. We show that a class of block Toeplitz covariance matrices is approximatable by low separation rank and give bounds on the minimal separation rank $r$ that ensures a given level of bias. Simulations are presented to validate the theoretical bounds. As a real world application, we illustrate the utility of the proposed Kronecker covariance estimator for spatio-temporal linear least squares prediction of multivariate wind speed measurements.

Algorithm for the Characterization of the Cross-Correlation Structure in Multivariate Time Series

In this paper, it is shown that the block circulant matrix decomposition technique makes the multivariate singular spectrum analysis (M-SSA) a well suited tool for detecting of changes of the correlation structure in non-stationary multivariate time series in the presence of high observational noise levels. The major drawback of M-SSA, that it operates on a large covariance matrix and becomes computationally expensive, can be avoided by reordering the Toeplitz-block covariance matrix into a block Toeplitz matrix, embedding this into a block circulant matrix and efficiently block-diagonalizing this by the means of the Fast Fourier Transform (FFT) using the well known algorithm. The overall degree of synchronization among multiple-channel signals is defined by the synchronization index (the S-estimator) of the rearranged and truncated eigenvalue spectrum. Throughout the experiment, the high capability of the proposed algorithm to detect the lag-synchronized state under the influence of strong noise is validated with simulated data—a network of time series generated by autoregressive models (AR) and a network of coupled chaotic Roessler oscillators.

Trace formulas for a class of truncated block Toeplitz operators

The strong Szegő limit theorem may be formulated in terms of finite-dimensional operators of the form(PNGPN)n−PNGnPNfor n=1,2,…, where G denotes the operator of multiplication by a suitably restricted d×d mvf (matrix-valued function) acting on the space of d×1 vvfʼs (vector-valued functions) f that meet the constraint ∫02πf(eiθ)⁎Δ(eiθ)f(eiθ)dθ<∞, where Δ(eiθ)=Id and PN denotes the orthogonal projection onto the space of trigonometric vector polynomials of degree at most N that are subject to the same summability constraint. In this paper, we study these operators for a class of mvfʼs Δ which admit factorizations Δ(eiθ)=Q(eiθ)⁎Q(eiθ)=R(eiθ)R(eiθ)⁎, where Q±1, R±1 belong to the Wiener plus algebra of d×d mvfʼs on the unit circle. We show thatκn(G)=deflimN↑∞trace{(PNGPN)n−PNGnPN} exists and is independent of Δ when the commutativity conditions GQ=QG and R⁎G=R⁎G are in force. The space of trigonometric vector polynomials of degree at most N is identified as a de Branges reproducing kernel Hilbert space of vector polynomials of degree at most N and weighted analogs of the strong Szegő limit theorem are established. If Q−1 and R−1 are matrix polynomials, then the inverse of the block Toeplitz matrix corresponding to Δ is of the band type. Explicit formulas for trace{PNGnPN} are obtained in this case.

Symbol approach in a signal-restoration problem involving block Toeplitz matrices

We consider a special type of signal restoration problem where some of the sampling data are not available. The formulation related to samples of the function and its derivative leads to a possibly large linear system associated to a nonsymmetric block Toeplitz matrix which can be equipped with a 2×2 matrix-valued symbol. The aim of the paper is to study the eigenvalues of the matrix. We first identify in detail the symbol and its analytical features. Then, by using recent results on the eigenvalue distribution of block Toeplitz matrix-sequences, we formally describe the cluster sets and the asymptotic spectral distribution of the matrix-sequences related to our problem. The localization areas, the extremal behavior, and the conditioning are only observed numerically, but their behavior is strongly related to the analytical properties of the symbol, even though a rigorous proof is still missing in the block case.

Analytically solvable model of an electronic Mach-Zehnder interferometer

We consider a class of models of non-equilibrium electronic Mach-Zehnder interferometers built on integer quantum Hall edges states. The models are characterized by the electron-electron interaction being restricted to the inner part of the interferometer and transmission coefficients of the quantum quantum point contacts, defining the interferometer, which may take arbitrary values from zero to one. We establish an exact solution of these models in terms of single-particle quantities --- determinants and resolvents of Fredholm integral operators. In the general situation, the results can be obtained numerically. In the case of strong charging interaction, the operators acquire the block Toeplitz form. Analyzing the corresponding Riemann-Hilbert problem, we reduce the result to certain singular single-channel determinants (which are a generalization of Toeplitz determinants with Fisher-Hartwig singularities), and obtain an analytic result for the interference current (and, in particular, for the visibility of Aharonov-Bohm oscillations). Our results, which are in good agreement with experimental observations, show an intimate connection between the observed "lobe" structure in the visibility of Aharonov-Bohm oscillations and multiple branches in the asymptotics of singular integral determinants.

A HYPONORMAL TOEPLITZ COMPLETION PROBLEM

AbstractIn this paper we consider the following ‘Toeplitz completion’ problem: Complete the unspecified analytic Toeplitz entries of the partial block Toeplitz matrix $ \begin{linenomath} A:=\begin{bmatrix} T_{\overline\psi_1}&amp; ?\\[4pt] \T_{\overline\psi_2} \end{bmatrix} \end{linenomath} $ to make A hyponormal, where ψi∈H∞ is a non-constant rational function for i=1,2.

A family of Nikishin systems with periodic recurrence coefficients

Suppose we have a Nikishin system of measures with the th generating measure of the Nikishin system supported on an interval with for all . It is well known that the corresponding staircase sequence of multiple orthogonal polynomials satisfies a -term recurrence relation whose recurrence coefficients, under appropriate assumptions on the generating measures, have periodic limits of period . (The limit values depend only on the positions of the intervals .) Taking these periodic limit values as the coefficients of a new -term recurrence relation, we construct a canonical sequence of monic polynomials , the so-called Chebyshev-Nikishin polynomials. We show that the polynomials themselves form a sequence of multiple orthogonal polynomials with respect to some Nikishin system of measures, with the th generating measure being absolutely continuous on . In this way we generalize a result of the third author and Rocha [22] for the case . The proof uses the connection with block Toeplitz matrices, and with a certain Riemann surface of genus zero. We also obtain strong asymptotics and an exact Widom-type formula for functions of the second kind of the Nikishin system for .Bibliography: 27 titles.

A fast reduced-rank interpolation method for prestack seismic volumes that depend on four spatial dimensions

Rank reduction strategies can be employed to attenuate noise and for prestack seismic data regularization. We present a fast version of Cadzow reduced-rank reconstruction method. Cadzow reconstruction is implemented by embedding 4D spatial data into a level-four block Toeplitz matrix. Rank reduction of this matrix via the Lanczos bidiagonalization algorithm is used to recover missing observations and to attenuate random noise. The computational cost of the Lanczos bidiagonalization is dominated by the cost of multiplying a level-four block Toeplitz matrix by a vector. This is efficiently implemented via the 4D fast Fourier transform. The proposed algorithm significantly decreases the computational cost of rank-reduction methods for multidimensional seismic data denoising and reconstruction. Synthetic and field prestack data examples are used to examine the effectiveness of the proposed method.

Semicommutators and Zero Product of Block Toeplitz Operators with Harmonic Symbols

We completely characterize the finite rank semicommutators, commutators, and zero product of block Toeplitz operatorsTFandTGwithF,G∈h∞⊗Mn×non the vector valued Bergman spaceLa2(𝔻,ℂn).

The Inverses of Block Toeplitz Matrices

We study the inverses of block Toeplitz matrices based on the analysis of the block cyclic displacement. New formulas for the inverses of block Toeplitz matrices are proposed. We show that the inverses of block Toeplitz matrices can be decomposed as a sum of products of block circulant matrices. In the scalar case, the inverse formulas are proved to be numerically forward stable, if the Toeplitz matrix is nonsingular and well conditioned.

An operator theoretic approach to the brickwork Ising model with second-neighbor interactions

Using results on block Toeplitz matrices, in particular the Szegő–Widom limit theorem and Gohberg and Feldman’s convergence theorem for the finite section method, we compute the spontaneous magnetization of an Ising model. In this way, the transition from the ferromagnetic phase to the paramagnetic phase may be understood as the transition from invertibility to non-invertibility of a certain block Toeplitz operator associated with the model.

Some block Toeplitz composition operators

The composition operators on H2(D) with minimal polynomial zn−1 are shown to be block Toeplitz with Toeplitz symbol equal to an n×n matrix-valued polynomial of degree 1. This result is used to prove that the numerical range of a composition operator on H2(D) with minimal polynomial z3−1 cannot be a circular disk.

Real scene capturing using spherical single-element lens camera and improved restoration algorithm for radially variant blur

A camera module employing spherical single-element lens imaging system (SSLIS) is introduced in this study. This type of imaging system can be used in compact digital cameras or mobile phone cameras, and it provides the advantages of simple design, reduced device bulkiness, and reduced manufacturing costs. When compared with conventional camera modules, our system produces radially variant blurred images, which can be satisfactorily restored by means of a polar domain deconvolution algorithm proposed in our previous study. In this study, we demonstrate an improved version of this algorithm that enables full-field-of-view (FOV) image restoration instead of the partial FOV restoration obtained via our previous algorithm. This improvement is realized by interpolating the upper and arc-shaped boundaries of the panoramic polar image such that the ringing artifacts around the center and four boundaries of the restored Cartesian image are greatly suppressed. The effectiveness of the improved algorithm is verified by image restoration of both computer simulated images and real-world scenes captured by the spherical single lens camera module. The quality of the restored image depends on the overall sparsity of all the point spread function (PSF) block Toeplitz with circulant blocks (BTCB) matrices used to restore a radially blurred image.

Development of the block-cycling inversion in computer tomography

This paper describes an improved block-cyclic inversion (BCI) method [1] for the three-dimensional (3D) inverse Radon problem: a major problem in computer tomography. A spiral-fan scheme of scanning (SFSS) and the cylindrical domain of inspection are taken. The 3D problem is reduced by decomposition to the series P of 2D problems (with the same Radon matrix). The account taken for a priori information about the circular FSS invariance allows the direct block-cyclic inversion of the Radon matrix by use of the Greville-1 block algorithm instead of the classic block Toeplitz inversion (BTI) [2,3], based on the concept of the Teoplitz range. The performance of the BCI versus BTI algorithm is better by a factor N both at the stage of preliminary inversion and in the flow due to vectorization. The required volume of memory is a sixth, but the main advantage of BCI is its simplicity of implementation, because of the absence of the degeneration problem, which is inherent in the classic method, as well as a higher stability. This has allowed a space resolution of up to 201 × 201. In 101 × 101 resolution, the counting time of a 2 s order in the variant of the PC PENTIUM-4 model (language Visual Fortran 90), and the time of the Radon matrix inversion itself is 20 s with the stability coefficient of ∼10 (in L2 metrics), 75 (in C metrics), i.e., by a factor of 3–10 better than in [23]. This is achieved due to the filtration of noise in the right-hand part, smoothing the solution itself and some other improvements of the algorithm. The problem mentioned in [1] was also solved. The obtained results can be applied to the software support of 4th generation tomographs.