A preconditioning strategy for banded circulant and toeplitz systems

We study an iterative, locally quadratically convergent algorithm for solving Toeplitz systems of equations from [R. P. Brent, F. G. Gustavson and D. Y. Y. Yun. “Fast solution of Toeplitz systems of equations and computation of Padé approximations”,J. Algorithms, 1:259–295, 1980]. We introduce a new iterative algorithm that is locally quadratically convergent when used to solve symmetric positive definite Toeplitz systems. We present a set of numerical experiments on randomly generated symmetric positive definite Toeplitz matrices. In these experiments, our algorithm performed significantly better than the previously proposed algorithm.

For solving the Poisson equation it is usually possible to discretize it into solving the corresponding linear system $Ax=b$. Variational quantum algorithms (VQAs) for \textcolor{red}{the discreted} Poisson equation have been studied before. We give a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix $A$. In detail, we decompose the matrix $A$ and $A^2$ into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements. For the one-dimensional Poisson equation with different boundary conditions and the $d$-dimensional Poisson equation with Dirichlet boundary conditions, \textcolor{red}{the number of decomposition terms is less than the work in [Phys. Rev. A108, 032418 (2023)]}. Based on the decomposition of the matrix, we design quantum circuits that \textcolor{red}{evaluate efficiently} the cost function. \textcolor{red}{Additionally, numerical simulation verifies the feasibility of the proposed algorithm. In the end,} the VQAs for linear systems of equations and matrix–vector multiplications with $K$-banded Teoplitz matrix $T_n^K$ \textcolor{red}{are} given, where $T_n^K\in R^{n\times n}$ and $K\in O(ploy\log n)$.

An algorithm for computing solutions to ill-conditioned banded Toeplitz least squares problems by a rank revealing URV factorization is considered. The factorization is computed in O((beta) nlogn + (beta) n<SUP>2</SUP>), where (beta) is the bandwidth of the coefficient matrix. An approximate solution to ill-conditioned banded Toeplitz systems, in the presence of noise, is then obtained by truncating the factorization. Numerical results are provided that illustrate truncated URV can compute solutions comparable to the more expensive truncated singular value decomposition.

A fast algorithm for solving banded Toeplitz systems

In this paper we present new preconditioning techniques for the solution by the preconditioned conjugate gradient (PCG) method of Hermitian Toeplitz systems with real and non- definite generating functions: actually we extend some results of Chan [IMA J. Numer. Anal., 11 (1991), pp. 333–345] and Di Benedetto, Fiorentino, and Serra [Comput. Math. Appl., 25 (1993), pp. 33–45] proved for positive definite Toeplitz systems. Moreover we demonstrate some density properties of the spectra of the preconditioned matrices. Finally, we show that the convergence speed of this PCG method is independent of the dimension of the involved matrices.

A fast fourth-order ADI scheme is proposed to solve 2D Riesz space-fractional diffusion equations (RSFDEs). This scheme involves a sequence of symmetric positive definite (SPD) Toeplitz linear systems, thus allowing the fast sine transform to be employed for computational efficiency. Furthermore, those discretized SPD systems are solved using the preconditioned conjugate gradient (PCG) method. Our theoretical analysis demonstrates that the preconditioned matrix can be expressed as the sum of an identity matrix, a small-norm matrix, and a low-rank matrix. Numerical results verify the method’s efficacy.

Various preconditioners for symmetric positive-definite (SPD) Toeplitz matrices in circulant matrix form have recently been proposed. The spectral properties of the preconditioned SPD Toeplitz matrices have also been studied. In this research, Strang's preconditioner SN and our preconditioner KN are applied to an N N nonsymmetric (or nonhermitian) Toeplitz system TNX b. For a large class of Toeplitz matrices, it is proved that the singular values of SITN and KvlTN are clustered around unity except for a fixed number independent ofN. IfTN is additionally generated by a rational function, the eigenvalues of SITN and KITN can be characterized directly. Let the eigenvalues ofSITN andKITN be classified into the outliers and the clustered eigenvalues dependingon whether they converge to I asymptotically. Then, the number ofoutliers depends on the order of the rational generating function, and the clustering radius is proportional to the magnitude of the last elements in the generating sequence used to construct the preconditioner. Numerical experiments are provided to illustrate our theoretical study.

Various Toeplitz preconditioners $P_N $ have recently been proposed so that an $N \times N$ symmetric positive definite Toeplitz system $T_N {\bf x} = {\bf b}$ can be solved effectively by the preconditioned conjugate gradient (PCG) method. It has been proven that if $T_N $ is generated by a positive function in the Wiener class, the eigenvalues of the preconditioned matrices $P_N^{ - 1} T_N $ are clustered between $( 1 - \epsilon ,1 + \epsilon )$ except for a fixed number independent of N. In this research, the spectra of $P_N^{ - 1} T_N $ are characterized more precisely for rational Toeplitz matrices $T_N $ with preconditioners proposed by Strang [Stud. Appl. Math., 74 (1986), pp. 171–176] and Ku and Kuo [IEEE Trans. Signal Process., 40 (1992), pp. 129–141]. The eigenvalues of $P_N^{ - 1} T_N $ are classified into two classes, i.e., the outliers and the clustered eigenvalues, depending on whether they converge to 1 asymptotically. It is proved that the number of outliers depends on the order of the rat...

A direct algorithm is presented for the solution of linear systems having banded Toeplitz coefficient matrix with unbalanced bandwidths. It is derived from the cyclic reduction algorithm, it makes use of techniques based on the displacement rank and it relies on the Morrison–Sherman–Woodbury formula. The algorithm always equals and sometimes outperforms the already known direct ones in terms of asymptotic computational cost. The case where the coefficient matrix is a block banded block Toeplitz matrix in block Hessenberg form is analyzed as well. The algorithm is numerically stable if applied to M‐matrices that are point diagonally dominant by columns. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Convergence analysis of superoptimal PCG algorithm for Toeplitz systems with a Fisher–Hartwig singularity

We describe an implementation of the generalized Schur algorithm for the superfast solution of real positive definite Toeplitz systems of order $n + 1$, where $n = 2^\nu$. Our implementation uses the split-radix Fast Fourier Transform algorithms for real data of Duhamel. We are able to obtain the nth Szegö polynomial using less than $8n\log _2^2 n$ real arithmetic operations without explicit use of the bit-reversal permutation. Since Levinson’s algorithm requires slightly more than $2n^2 $ operations to obtain this polynomial, we achieve crossover with Levinson’s algorithm at $n = 256$.

The first-order system least-squares (FOSLS) finite element method for solving partial differential equations has many advantages, including the construction of symmetric positive definite algebraic linear systems that can be solved efficiently with multilevel iterative solvers. However, one drawback of the method is the potential lack of conservation of certain properties. One such property is conservation of mass. This paper describes a strategy for achieving mass conservation for a FOSLS system by changing the minimization process to that of a constrained minimization problem. If the space of corresponding Lagrange multipliers contains the piecewise constants, then local mass conservation is achieved similarly to the standard mixed finite-element method. To make the strategy more robust and not add too much computational overhead to solving the resulting saddle-point system, an overlapping Schwarz process is used.KeywordsConservationFirst-order system least-squaresFinite elementsDomain decompositionTwo-level

The parallelization of the least-squares spectral element formulation of the Stokes problem has recently been discussed for incompressible flow problems on structured grids. In the present work, the extension to unstructured grids is discussed. It will be shown that, to obtain an efficient and scalable method, two different kinds of distribution of data are required involving a rather complicated parallel conversion between the data. Once the data conversion has been performed, a large symmetric positive definite algebraic system has to be solved iteratively. It is well known that the Conjugate Gradient method is a good choice to solve such systems. To improve the convergence rate of the Conjugate Gradient process, both Jacobi and Additive Schwarz preconditioners are applied. The Additive Schwarz preconditioner is based on domain decomposition and can be implemented such that a preconditioning step corresponds to a parallel matrix-by-vector product. The new results reveal that the Additive Schwarz preconditioner is very suitable for the p-refinement version of the least-squares spectral element method. To obtain good portable programs which may run on distributed-memory multiprocessors, networks of workstations as well as shared-memory machines we use MPI (Message Passing Interface). Numerical simulations have been performed to validate the scalability of the different parts of the proposed method. The experiments entailed simulating several large scale incompressible flows on a Cray T3E and on an SGI Origin 3800 with the number of processors varying from one to more than one hundred. The results indicate that the present method has very good parallel scaling properties making it a powerful method for numerical simulations of incompressible flows.

Towards essential improvement for the Parareal-TR and Parareal-Gauss4 algorithms

This paper presents linear algebra techniques used in the implementation of an interior point method for solving linear programs and convex quadratic programs with linear constraints. New regularization techniques for Newton systems applicable to both symmetric positive definite and symmetric indefinite systems are described. They transform the latter to quasidef-inite systems known to be strongly factorizable to a form of Cholesky-like factorization.Two different regularization techniques,primal; and dual, are very well suited to the (infeasible) primal-dual interior point algorithm. This particular algorithm, with an extension of multiple centrality correctors, is implemented in our solver HOPDM. Computational results are given to illustrate the potential advantages of the approach when applied to the solution of very large linear and convex quadratic programs.