Block Toeplitz Systems Research Articles

Explicit formulas for the inverses of Toeplitz matrices, with applications

We derive novel explicit formulas for the inverses of truncated block Toeplitz matrices that correspond to a multivariate minimal stationary process. The main ingredients of the formulas are the Fourier coefficients of the phase function attached to the spectral density of the process. The derivation of the formulas is based on a recently developed finite prediction theory applied to the dual process of the stationary process. We illustrate the usefulness of the formulas by two applications. The first one is a strong convergence result for solutions of general block Toeplitz systems for a multivariate short-memory process. The second application is closed-form formulas for the inverses of truncated block Toeplitz matrices corresponding to a multivariate ARMA process. The significance of the latter is that they provide us with a linear-time algorithm to compute the solutions of corresponding block Toeplitz systems.

Optimal block circulant preconditioners for block Toeplitz systems with application to evolutionary PDEs

In this work, we propose a preconditioned minimal residual (MINRES) method for a class of non-Hermitian block Toeplitz systems. Namely, considering an mn-by-mn non-Hermitian block Toeplitz matrix T(n,m) with m-by-m commuting Hermitian blocks, we first premultiply it by a simple permutation matrix to obtain a Hermitian matrix and then construct a Hermitian positive definite block circulant preconditioner for the modified matrix. Under certain conditions, we show that the eigenvalues of the preconditioned matrix are clustered around ±1 when n is sufficiently large. Due to the Hermitian nature of the modified matrix, MINRES with our proposed preconditioner can achieve theoretically guaranteed superlinear convergence under suitable conditions. In addition, we provide several useful properties of block circulant matrices with commuting Hermitian blocks, including diagonalizability and symmetrization. A generalization of our result to the multilevel block case is also provided. We in particular indicate that our work can be applied to the all-at-once systems arising from solving evolutionary partial differential equations. Numerical examples are given to illustrate the effectiveness of our preconditioning strategy.

Efficient solution of block Toeplitz systems with multiple right-hand sides arising from a periodic boundary element formulation

Block Toeplitz matrices are a special class of matrices that exhibit reduced memory requirements and a reduced complexity of matrix-vector multiplications. We herein present an efficient computational approach to solve a sequence of block Toeplitz systems arising from a block Toeplitz system with multiple right-hand sides. Two different numerical schemes are implemented for the solution of the sequence of block Toeplitz systems based on global and block variants of the generalized minimal residual (GMRES) method. The performance of the schemes is assessed in terms of the wall clock time of the iterative solution process, the number of multiplications with the block Toeplitz system matrix and the peak memory usage. To demonstrate the method, two numerical examples are presented. In the first case study, aeroacoustic prediction of an airfoil in turbulent flow is examined, which requires multiple solutions of the wall pressure field beneath the turbulent boundary layer. The fluctuating pressure on the surface of the airfoil is synthesized in terms of uncorrelated wall plane waves, whereby each realization of the wall pressure field is an input to the acoustic solver based on the boundary element method (BEM). The total acoustic response from the airfoil in turbulent flow is then obtained from an ensemble average for the number of realizations considered. The number of realizations to yield a converged solution for the wall pressure field leads to a sequence of block Toeplitz systems. The second case study examines the nonlinear eigenvalue analysis of a sonic crystal barrier composed of locally resonant C-shaped sound-hard scatterers. The periodicity of the sound barrier leads to a block Toeplitz system matrix whereas the nonlinear eigenvalue problem requires the solution of sequences of linear systems. The combined technique to solve the sequences of block Toeplitz systems using the proposed variants of the GMRES is shown to yield a computationally efficient approach for flow noise prediction and nonlinear eigenvalue analysis.

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.

On the decay of the off-diagonal singular values in cyclic reduction

It was recently observed in [10] that the singular values of the off-diagonal blocks of the matrix sequences generated by the Cyclic Reduction algorithm decay exponentially. This property was used to solve, with a higher efficiency, certain quadratic matrix equations encountered in the analysis of queuing models. In this paper, we provide a theoretical bound to the basis of this exponential decay together with a tool for its estimation based on a rational interpolation problem. Numerical experiments show that the bound is often accurate in practice. Applications to solving n×n block tridiagonal block Toeplitz systems with n×n quasiseparable blocks and certain generalized Sylvester equations in O(n2log⁡n) arithmetic operations are shown.

Improved Schur complement preconditioners for block-Toeplitz systems with small size blocks

In this paper, we employ the preconditioned conjugate gradient method with the Improved Schur complement preconditioners for Hermitian positive definite block-Toeplitz systems with small size blocks. Schur complement preconditioners have been proved to be an effective method for such block-Toeplitz systems (Ching et al. 2007). The modification is based on Taylor expansion approximation. We prove that the matrices preconditioned by improved Schur preconditioners have more clustered spectra compared to that of the Schur complement preconditioners. Hence, preconditioned conjugate gradient type methods will converge faster. Numerical examples are given to demonstrate the efficiency of the proposed method.

Efficient cyclic reduction for Quasi-Birth–Death problems with rank structured blocks

We provide effective algorithms for solving block tridiagonal block Toeplitz systems with m×m quasiseparable blocks, as well as quadratic matrix equations with m×m quasiseparable coefficients, based on cyclic reduction and on the technology of rank-structured matrices. The algorithms rely on the exponential decay of the singular values of the off-diagonal submatrices generated by cyclic reduction. We provide a formal proof of this decay in the Markovian framework. The results of the numerical experiments that we report confirm a significant speed up over the general algorithms, already starting with the moderately small size m≈102.

Negative refraction of acoustic waves in phononic crystals using recursive algorithms for block toeplitz matrices

Recent improvements in design and manufacturing have significantly improved our ability to manage phononic crystals (PC). Researchers have been studied the negative refraction in PC experimentally and theoretically using multiple scattering (MS) theory. This problem requires solving a large complex valued linear system that has a special multilevel block Toeplitz (BT) structure. We study negative refraction of acoustic waves in 2D PC by means of MS theory, by taking advantage of the PC structure and using specific recursive algorithms for BT matrices. We present new efficient and accurate algorithms for solving acoustic MS problem by the cluster of closely spaced cylinders to design acoustic negative refraction imaging in a PC. The unit cell of the PC consists, in general, of a solid cylinder in an acoustic medium. The dispersion curves of PC have a negative refraction dispersion branch producing the focusing effect. Particular attention is given to the dynamic behavior in the vicinity of dispersion branches with negative slope, and to the band gaps. We explore the effect of focal point on structural parameters, and employ a parallelization technique that allows efficient application of the proposed recursive algorithms for solving BT systems on high performance computer clusters. Numerical comparisons of CPU time and total elapsed time taken to solve the linear system using the direct LAPACK and TOEPLITZ libraries on Intel FORTRAN show the advantage of high performance recursive algorithms over the Gaussian elimination.

Acoustic multiple scattering using recursive algorithms

Acoustic multiple scattering by a cluster of cylinders in an acoustic medium is considered. A fast recursive technique is described which takes advantage of the multilevel Block Toeplitz structure of the linear system. A parallelization technique is described that enables efficient application of the proposed recursive algorithm for solving multilevel Block Toeplitz systems on high performance computer clusters. Numerical comparisons of CPU time and total elapsed time taken to solve the linear system using the direct LAPACK and TOEPLITZ libraries on Intel FORTRAN, show the advantage of the TOEPLITZ solver. Computations are optimized by multi-threading which displays improved efficiency of the TOEPLITZ solver with the increase of the number of scatterers and frequency.

Using the Sherman–Morrison–Woodbury inversion formula for a fast solution of tridiagonal block Toeplitz systems

A fast numerical algorithm for solving systems of linear equations with tridiagonal block Toeplitz matrices is presented. The algorithm is based on a preliminary factorization of the generating quadratic matrix polynomial associated with the Toeplitz matrix, followed by the Sherman–Morrison–Woodbury inversion formula and solution of two bidiagonal and one diagonal block Toeplitz systems. Tight estimates of the condition numbers are provided for the matrix system and the main matrix systems generated during the preliminary factorization. The emphasis is put on rigorous stability analysis to rounding errors of the Sherman–Morrison–Woodbury inversion. Numerical experiments are provided to illustrate the theory.

Partitioned least-squares operator for large-scale geophysical inversion

Least-squares (LS) problems are encountered in many geophysical estimation and data analysis problems where a large number of observations (data) are combined to determine a model (some aspect of the earth structure). Examples of least squares in seismic exploration include several data processing algorithms, theoretically accurate LS migration, inversion for reservoir parameters, and background velocity estimation. A frequently encountered problem is that the volume of data in 3D is so large that the matrices required for the LS solution cannot be stored within the memory of a single computer. A new technique is described for parallel computation of the LS operator that is based on a partitioned-matrix algorithm. The classical LS method for solution of block-Toeplitz systems of normal equation (NE) to the general case of block-Hermitian and non-Toeplitz systems of NE. is generalized. Specifically, a solution of a block-Hermitian system of NE is shown that may be obtained recursively by linearly combining the solutions of lesser order that are related to the forward and backward subsystems of equations. This results in an efficient parallel algorithm in which each partitioned system can be evaluated independently. The application of the algorithm to the problem of 3D plane wave transformation is demonstrated.

A direct method to solve block banded block Toeplitz systems with non-banded Toeplitz blocks

A fast solution algorithm is proposed for solving block banded block Toeplitz systems with non-banded Toeplitz blocks. The algorithm constructs the circulant transformation of a given Toeplitz system and then by means of the Sherman–Morrison–Woodbury formula transforms its inverse to an inverse of the original matrix. The block circulant matrix with Toeplitz blocks is converted to a block diagonal matrix with Toeplitz blocks, and the resulting Toeplitz systems are solved by means of a fast Toeplitz solver. The computational complexity in the case one uses fast Toeplitz solvers is equal to ξ ( m , n , k ) = O ( m n 3 ) + O ( k 3 n 3 ) flops, there are m block rows and m block columns in the matrix, n is the order of blocks, 2 k + 1 is the bandwidth. The validity of the approach is illustrated by numerical experiments.

Block Diagonal and Schur Complement Preconditioners for Block-Toeplitz Systems with Small Size Blocks

In this paper we consider the solution of Hermitian positive definite block-Toeplitz systems with small size blocks. We propose and study block diagonal and Schur complement preconditioners for such block-Toeplitz matrices. We show that for some block-Toeplitz matrices, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers where this fixed number depends only on the size of the block. Hence, conjugate gradient type methods, when applied to solving these preconditioned block-Toeplitz systems with small size blocks, converge very fast. Recursive computation of such block diagonal and Schur complement preconditioners is considered by using the nice matrix representation of the inverse of a block-Toeplitz matrix. Applications to block-Toeplitz systems arising from least squares filtering problems and queueing networks are presented. Numerical examples are given to demonstrate the effectiveness of the proposed method.

Multigrid Methods for Block Toeplitz Matrices with Small Size Blocks

In this paper we discuss multigrid methods for ill-conditioned symmetric positive definite block Toeplitz matrices. Our block Toeplitz systems are general in the sense that the individual blocks are not necessarily Toeplitz, but we restrict our attention to blocks of small size. We investigate how transfer operators for prolongation and restriction have to be chosen such that our multigrid algorithms converge quickly. We point out why these transfer operators can be understood as block matrices as well and how they relate to the zeroes of the generating matrix function. We explain how our new algorithms can also be combined efficiently with the use of a natural coarse grid operator. We clearly identify a class of ill-conditioned block Toeplitz matrices for which our algorithmic ideas are suitable. In the final section we present an outlook to well-conditioned block Toeplitz systems and to problems of vector Laplace type. In the latter case the small size blocks can be interpreted as degrees of freedom associated with a node. A large number of numerical experiments throughout the article confirms convincingly that our multigrid solvers lead to optimal order convergence.

Some algorithms for solving special tridiagonal block Toeplitz linear systems

This paper is focused on different methods and algorithms for solving tridiagonal block Toeplitz systems of linear equations. We consider the El-Sayed method (Ph.D. Thesis, 1996) for such systems and propose several modifications that lead to different algorithms, which we discuss in detail. Our algorithms are then compared with some classical techniques as far as implementation time is concerned, number of operations and storage. Comments and conclusions for computing efficiency of the proposed new algorithms are given. Numerical experiments corroborating the theoretical results are also presented.

A note on the fast algorithm for block Toeplitz systems with tensor structure

We study the solutions of block Toeplitz systems T mn x= b by using the preconditioned conjugate gradient (PCG) method. Here T mn = T m ⊗ T n and T i, i=m,n are Toeplitz matrices. In [X. Jin, Appl. Math. Comput. 73 (1995) 115–124], Jin introduced a fast algorithm for these systems by applying the PCG method. This fast algorithm allows a tensor problem to be reduced to a one-dimensional problem. It was proved that if the mn-by- mn system is well conditioned, then the PCG method converges superlinearly and only O( mnlog mn) operations are required in solving the preconditioned system. However, only well-conditioned systems were considered in Jin, 1995. In this paper, we apply this fast algorithm with the { ω}-circulant preconditioners proposed in [D. Potts, G. Steidl, Preconditioners for Ill-Conditioned toeplitz matrices, BIT, to be appeared] to solve the ill-conditioned systems. Numerical results are included to illustrate the effectiveness of the algorithm for solving the preconditioned systems by using the PCG method. An application in image restoration is also given.

Deconvolução preditiva de reflexões múltiplas e peg-legs utilizando filtragem Wiener - Levinson multicanal

No presente trabalho utilizamos o método de deconvolução preditiva multicanal (DPM) de Wiener-Levinson (WL) para a supressão de reflexões múltiplas associadas à lâmina d'agua e peg-legs. Os filtros multicanais de WL são obtidos resolvendo-se sistemas de equações normais com estrutura bloco-Toeplitz, formados a partir dos coeficientes da função de auto-correlação e cross-correlação dos traços sísmicos. Tais filtros exploram de forma mais efetiva a redundância espacial e temporal dos sismogramas, comparado aos correspondentes filtros monocanais. A DPM foi aplicada em painéis de traços sísmicos corrigidos de NMO utilizando-se a velocidade da múltipla a ser removida. Neste domínio as múltiplas apresentam caráter periódico e o método multicanal se mostrou bastante eficaz. Exemplos numéricos utilizando dados sintéticos e dados sísmicos marítimos do Golfo do México ilustram o desempenho da filtragem WL multicanal. Os resultados obtidos demonstram que o método de DPM utilizado é robusto e eficaz na remoção de múltiplas e peg-legs.

Convergence of the Multigrid Method of Ill-conditioned Block Toeplitz Systems

We study the solutions of block Toeplitz systems Amnu = b by the multigrid method (MGM). Here the block Toeplitz matrices Amn are generated by a nonnegative function f (x,y) with zeros. Since the matrices Amn are ill-conditioned, the convergence factor of classical iterative methods will approach 1 as the size of the matrices becomes large. These classical methods, therefore, are not applicable for solving ill-conditioned systems. The MGM is then proposed in this paper. For a class of block Toeplitz matrices, we show that the convergence factor of the two-grid method (TGM) is uniformly bounded below 1 independent of mn and the full MGM has convergence factor depending only on the number of levels. The cost per iteration for the MGM is of O(mn log mn) operations. Numerical results are given to explain the convergence rate.

Preconditioners for block Toeplitz systems based on circulant preconditioners

We study the numerical solution of a block system T m,n x=b by preconditioned conjugate gradient methods where T m,n is an m×m block Toeplitz matrix with n×n Toeplitz blocks. These systems occur in a variety of applications, such as two-dimensional image processing and the discretization of two-dimensional partial differential equations. In this paper, we propose new preconditioners for block systems based on circulant preconditioners. From level-1 circulant preconditioner we construct our first preconditioner q 1(T m,n ) which is the sum of a block Toeplitz matrix with Toeplitz blocks and a sparse matrix with Toeplitz blocks. By setting selected entries of the inverse of level-2 circulant preconditioner to zero, we get our preconditioner q 2(T m,n ) which is a (band) block Toeplitz matrix with (band) Toeplitz blocks. Numerical results show that our preconditioners are more efficient than circulant preconditioners.

Preconditioners for ill-conditioned Toeplitz matrices

This paper is concerned with the solution of systems of linear equations A N x = b, where $$\{ A_N \} _{N \in \mathbb{N}}$$ denotes a sequence of positive definite Hermitian ill-conditioned Toeplitz matrices arising from a (real-valued) nonnegative generating function f ∈ C 2π with zeros. We construct positive definite Hermitian preconditioners M N such that the eigenvalues of M N −1 A N are clustered at 1 and the corresponding PCG-method requires only O(N log N) arithmetical operations to achieve a prescribed precision. We sketch how our preconditioning technique can be extended to symmetric Toeplitz systems, doubly symmetric block Toeplitz systems with Toeplitz blocks and non-Hermitian Toeplitz systems. Numerical tests confirm the theoretical expectations.