Symmetrization Techniques in Image Deblurring

This paper presents a couple of preconditioning techniques that can be used to enhance the performance of iterative regularization methods applied to image deblurring problems with a variety of point spread functions (PSFs) and boundary conditions. More precisely, we first consider the anti-identity preconditioner, which symmetrizes the coefficient matrix associated to problems with zero boundary conditions, allowing the use of MINRES as a regularization method. When considering more sophisticated boundary conditions and strongly nonsymmetric PSFs, the anti-identity preconditioner improves the performance of GMRES. We then consider both stationary and iteration-dependent regularizing circulant preconditioners that, applied in connection with the anti-identity matrix and both standard and flexible Krylov subspaces, speed up the iterations. A theoretical result about the clustering of the eigenvalues of the preconditioned matrices is proved in a special case. The results of many numerical experiments are reported to show the effectiveness of the new preconditioning techniques, including when considering the deblurring of sparse images

A preconditioned MINRES method for optimal control of wave equations and its asymptotic spectral distribution theory

In this work, we propose a novel preconditioned Krylov subspace method for solving an optimal control problem of wave equations, after explicitly identifying the asymptotic spectral distribution of the involved sequence of linear coefficient matrices from the optimal control problem. Namely, we first show that the all-at-once system stemming from the wave control problem is associated to a structured coefficient matrix-sequence possessing an eigenvalue distribution. Then, based on such a spectral distribution of which the symbol is explicitly identified, we develop an ideal preconditioner and two parallel-in-time preconditioners for the saddle point system composed of two block Toeplitz matrices. For the ideal preconditioner, we show that the eigenvalues of the preconditioned matrix-sequence all belong to the set $\left(-\frac{3}{2},-\frac{1}{2}\right)\bigcup \left(\frac{1}{2},\frac{3}{2}\right)$ well separated from zero, leading to mesh-independent convergence when the minimal residual method is employed. The proposed {parallel-in-time} preconditioners can be implemented efficiently using fast Fourier transforms or discrete sine transforms, and their effectiveness is theoretically shown in the sense that the eigenvalues of the preconditioned matrix-sequences are clustered around $\pm 1$, which leads to rapid convergence. When these parallel-in-time preconditioners are not fast diagonalizable, we further propose modified versions which can be efficiently inverted. Several numerical examples are reported to verify our derived localization and spectral distribution result and to support the effectiveness of our proposed preconditioners and the related advantages with respect to the relevant literature

A block α\alpha-circulant based preconditioned MINRES method for wave equations

In this work, we propose an absolute value block $\alpha$-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Since the original block $\alpha$-circulant preconditioner shown successful by many recently is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Motivated by the absolute value block circulant preconditioner proposed in [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput., 40(2):A1012-A1033, 2018], we propose an absolute value version of the block $\alpha$-circulant preconditioner. Our proposed preconditioner is the first Hermitian positive definite variant of the block $\alpha$-circulant preconditioner, which fills the gap between block $\alpha$-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for properly chosen $\alpha$ the MINRES solver with the proposed preconditioner has a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved

Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices

When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments

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