Superlinear convergence for PCG using band plus algebra preconditioners for Toeplitz systems

AbstractThe paper studies fast and efficient solution algorithms for n×n symmetric ill conditioned Toeplitz systems Tn(f)x=b where the generating function f is known a priori, real valued, nonnegative, and has isolated roots of even order. The preconditioner that we propose is a product of a band Toeplitz matrix and matrices that belong to a certain trigonometric algebra. The basic idea behind the proposed scheme is to combine the advantages of all components of the product that are well known when every component is used as a stand-alone preconditioner. As a result we obtain a flexible preconditioner which can be applied to the system Tn(f)x=b infusing superlinear convergence to the PCG method. The important feature of the proposed technique is that it can be extended to cover the 2D case, i.e. ill-conditioned block Toeplitz matrices with Toeplitz blocks. We perform many numerical experiments, whose results confirm the theoretical analysis and effectiveness of the proposed strategy

Recursive-Based PCG Methods for Toeplitz Systems with Nonnegative Generating Functions

In this paper, we consider the solutions of symmetric positive definite, but ill-conditioned, Toeplitz systems An x = b. Here we propose to solve the system by the recursive-based preconditioned conjugate gradient method. The idea is to use the inverse of Am (the principal submatrix of An with the Gohberg--Semencul formula as a preconditioner for An. The inverse of Am can be generated recursively by using the formula until m is small enough. The construction of the preconditioners requires only the entries of An and does not require the explicit knowledge of the generating function f of An. We show that if f is a nonnegative, bounded, and piecewise continuous even function with a finite number of zeros of even order, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers. Hence the conjugate gradient method, when applied to solving the preconditioned system, converges very quickly. Numerical results are included to illustrate the effectiveness of our approach.published_or_final_versio

PRECONDITIONING STRATEGIES FOR 2D FINITE DIFFERENCE MATRIX SEQUENCES

In this paper we are concerned with the spectral analysis of the sequence of preconditioned matrices {P-1 n An(a, m1, m2, k)}n, where n = (n1, n2), N(n) = n1n2 and where An (a, m1, m2, k) 08 \u211dN(n) 7 N(n) is the symmetric two-level matrix coming from a high-order Finite Difference (FD) discretization of the problem (equation presented) with \u3bd denoting the unit outward normal direction and where m1 and m2 are parameters identifying the precision order of the used FD schemes. We assume that the coefficient a(x, y) is nonnegative and that the set of the possible zeros can be represented by a finite collection of curves. The proposed preconditioning matrix sequences correspond to two different choices: the Toeplitz sequence {An(1, m1, m2, k)}n and a Toeplitz based sequence that adds to the Toeplitz structure the informative content given by the suitable scaled diagonal part of An(a, m1, m2 k). The former case gives rise to optimal preconditioning sequences under the assumption of positivity and boundedness of a. With respect to the latter, the main result is the proof of the asymptotic clustering at unity of the eigenvalues of the preconditioned matrices, where the "strength" of the cluster depends on the order k, on the regularity features of a(x, y) and on the presence of zeros of a(x, y)

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

Some fast algorithms in signal and image processing.

Kwok-po Ng.Thesis (Ph.D.)--Chinese University of Hong Kong, 1995.Includes bibliographical references (leaves 138-139).AbstractsSummaryIntroduction --- p.1Summary of the papers A-F --- p.2Paper A --- p.15Paper B --- p.36Paper C --- p.63Paper D --- p.87Paper E --- p.109Paper F --- p.12

V-cycle optimal convergence for certain (multilevel) structured linear systems

In this paper we are interested in the solution by multigrid strategies of multilevel linear systems whose coefficient matrices belong to the circulant, Hartley, or \u3c4 algebras or to the Toeplitz class and are generated by (the Fourier expansion of) a nonnegative multivariate polynomial f. It is well known that these matrices are banded and have eigenvalues equally distributed as f, so they are ill-conditioned whenever f takes the zero value; they can even be singular and need a low-rank correction. We prove the V-cycle multigrid iteration to have a convergence rate independent of the dimension even in presence of ill-conditioning. If the (multilevel) coefficient matrix has partial dimension nr at level r, r = 1, . . . ,d, then the size of the algebraic system is N(n) = \u3a0r=1 d nr, O(N(n)) operations are required by our technique, and therefore the corresponding method is optimal. Some numerical experiments concerning linear systems arising in applications, such as elliptic PDEs with mixed boundary conditions and image restoration problems, are considered and discussed.cussed

Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications

In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively. Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given. All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator. In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8