arXiv subjects

Stefano Serra-Capizzano

Publications and source records attributed to Stefano Serra-Capizzano.

70 records · Page 4Linked to original sources

arXiv2018

A note on the spectral distribution of symmetrized Toeplitz sequences

The singular value and spectral distribution of Toeplitz matrix sequences with Lebesgue integrable generating functions is well studied. Early results were provided in the classical Szeg{ő} theorem and the Avram-Parter theorem, in which the singular value symbol coincides with the generating function. More general versions of the theorem were later proved by Zamarashkin and Tyrtyshnikov, and Tilli. Considering (real) nonsymmetric Toeplitz matrix sequences, we first symmetrize them via a simple permutation matrix and then we show that the singular value and spectral distribution of the symmetrized matrix sequence can be obtained analytically, by using the notion of approximating class of sequences. In particular, under the assumption that the symbol is sparsely vanishing, we show that roughly half of the eigenvalues of the symmetrized Toeplitz matrix (i.e. a Hankel matrix) are negative/positive for sufficiently large dimension, i.e. the matrix sequence is symmetric (asymptotically) indefinite.

math.NA↗

arXiv2018

A Multigrid method for nonlocal problems: non-diagonally dominant Toeplitz-plus-tridiagonal systems

The nonlocal problems have been used to model very different applied scientific phenomena, which involve the fractional Laplacian when one looks at the Lévy processes and stochastic interfaces. This paper deals with the nonlocal problems on a bounded domain, where the stiffness matrices of the resulting systems are Toeplitz-plus-tridiagonal and far from being diagonally dominant, as it occurs when dealing with linear finite element approximations. By exploiting a weakly diagonally dominant Toeplitz property of the stiffness matrices, the optimal convergence of the two-grid method is well established [Fiorentino and Serra-Capizzano, {\em SIAM J. Sci. Comput.}, {17} (1996), pp. 1068--1081; Chen and Deng, {\em SIAM J. Matrix Anal. Appl.}, {38} (2017), pp. 869--890]; and there are still questions about best ways to define coarsening and interpolation operator when the stiffness matrix is far from being weakly diagonally dominant [Stüben, {\em J. Comput. Appl. Math.}, {128} (2001), pp. 281--309]. In this work, using spectral indications from our analysis of the involved matrices, the simple (traditional) restriction operator and prolongation operator are employed in order to handle general algebraic systems which are {\em neither Toeplitz nor weakly diagonally dominant} corresponding to the fractional Laplacian kernel and the constant kernel, respectively. We focus our efforts on providing the detailed proof of the convergence of the two-grid method for such situations. Moreover, the convergence of the full multigrid is also discussed with the constant kernel. The numerical experiments are performed to verify the convergence with only $\mathcal{O}(N \mbox{log} N)$ complexity by the fast Fourier transform, where $N$ is the number of the grid points.

math.NA↗

arXiv2018

Isogeometric analysis for 2D and 3D curl-div problems: Spectral symbols and fast iterative solvers

Alfvén-like operators are of interest in magnetohydrodynamics, which is used in plasma physics to study the macroscopic behavior of plasma. Motivated by this important and complex application, we focus on a parameter-dependent curl-div problem that can be seen as a prototype of an Alfvén-like operator, and we discretize it using isogeometric analysis based on tensor-product B-splines. The involved coefficient matrices can be very ill-conditioned, so that standard numerical solution methods perform quite poorly here. In order to overcome the difficulties caused by such ill-conditioning, a two-step strategy is proposed. First, we conduct a detailed spectral study of the coefficient matrices, highlighting the critical dependence on the different physical and approximation parameters. Second, we exploit such spectral information to design fast iterative solvers for the corresponding linear systems. For the first goal we apply the theory of (multilevel block) Toeplitz and generalized locally Toeplitz sequences, while for the second we use a combination of multigrid techniques and preconditioned Krylov solvers. Several numerical tests are provided both for the study of the spectral problem and for the solution of the corresponding linear systems.

math.NA↗

arXiv2017

Uniform convergence of V-cycle multigrid algorithms for two-dimensional fractional Feynman-Kac equation

In this paper we derive new uniform convergence estimates for the V-cycle MGM applied to symmetric positive definite Toeplitz block tridiagonal matrices, by also discussing few connections with previous results. More concretely, the contributions of this paper are as follows: (1) It tackles the Toeplitz systems directly for the elliptic PDEs. (2) Simple (traditional) restriction operator and prolongation operator are employed in order to handle general Toeplitz systems at each level of the recursion. Such a technique is then applied to systems of algebraic equations generated by the difference scheme of the two-dimensional fractional Feynman-Kac equation, which describes the joint probability density function of non-Brownian motion. In particular, we consider the two coarsening strategies, i.e., doubling the mesh size (geometric MGM) and Galerkin approach (algebraic MGM), which lead to the distinct coarsening stiffness matrices in the general case: however, several numerical experiments show that the two algorithms produce almost the same error behaviour.

math.NA↗

arXiv2014

Iterated fractional Tikhonov regularization

Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order. This paper provides saturation and converse results on their convergence rates. Using the same iterative refinement strategy of iterated Tikhonov regularization, new iterated fractional Tikhonov regularization methods are introduced. We show that these iterated methods are of optimal order and overcome the previous saturation results. Furthermore, nonstationary iterated fractional Tikhonov regularization methods are investigated, establishing their convergence rate under general conditions on the iteration parameters. Numerical results confirm the effectiveness of the proposed regularization iterations.

math.NA↗

arXiv2014

Optimal order conditions for filter based regularization methods, with applications to variants of the Tikhonov method

We study filter based regularization methods for linear ill-posed problems between Hilbert spaces. We derive optimal order conditions under a-priori choice rules for the regularization parameter. Such analysis is applied to the fractional Tikhonov method and the weighted Tikhonov method recently investigated in [9] and [8], respectively. These variants of the classical Tikhonov method can be viewed as classes of methods depending on a further parameter, controlling the smoothness of the computed solution. From our previous analysis, we derive sufficient conditions on such parameter in order to have filter based regularization methods that agrees with the previous results in the literature. On the other hand, our analysis shows sufficient and necessary conditions to have an optimal order fractional Tikhonov method, which are stronger than those in [9]. The same analysis is performed also for the weighted Tikhonov method.

math.NA↗

arXiv2014

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 $I_k:=(-π,π)^k$, let $\mathcal M_s$ be the linear space of complex $s\times s$ matrices, and let $f,g:I_k\to\mathcal M_s$ be functions whose components $f_{ij},\,g_{ij}:I_k\to\mathbb C,\ i,j=1,\ldots,s,$ belong to $L^\infty$. Consider the matrices $T_n^{-1}(g)T_n(f)$, where $n:=(n_1,\ldots,n_k)$ varies in $\mathbb N^k$ and $T_n(f),T_n(g)$ are the multilevel block Toeplitz matrices of size $n_1\cdots n_ks$ generated by $f,g$. Then $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}\sim_λg^{-1}f$, i.e. the family of matrices $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}$ has a global (asymptotic) spectral distribution described by the function $g^{-1}f$, provided $g$ possesses certain properties (which ensure in particular the invertibility of $T_n^{-1}(g)$ for all $n$) and the following topological conditions are met: the essential range of $g^{-1}f$, defined as the union of the essential ranges of the eigenvalue functions $λ_j(g^{-1}f),\ j=1,\ldots,s$, 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.

math.NA↗

arXiv2013

A Fast Alternating Minimization Algorithm for Total Variation Deblurring Without Boundary Artifacts

Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang [{\em SIAM J. Imaging Sci.}, 1 (2008), pp. 248--272]. The method in a nutshell consists of a discrete Fourier transform-based alternating minimization algorithm with periodic boundary conditions and in which two fast Fourier transforms (FFTs) are required per iteration. In this paper, we propose an alternating minimization algorithm for the continuous version of the total variation image deblurring problem. We establish convergence of the proposed continuous alternating minimization algorithm. The continuous setting is very useful to have a unifying representation of the algorithm, independently of the discrete approximation of the deconvolution problem, in particular concerning the strategies for dealing with boundary artifacts. Indeed, an accurate restoration of blurred and noisy images requires a proper treatment of the boundary. A discrete version of our continuous alternating minimization algorithm is obtained following two different strategies: the imposition of appropriate boundary conditions and the enlargement of the domain. The first one is computationally useful in the case of a symmetric blur, while the second one can be efficiently applied for a nonsymmetric blur. Numerical tests show that our algorithm generates higher quality images in comparable running times with respect to the Fast Total Variation deconvolution algorithm.

math.FA↗

arXiv2012

Optimal preconditioning for image deblurring with Anti-Reflective boundary conditions

Inspired by the theoretical results on optimal preconditioning stated by Ng, R.Chan, and Tang in the framework of Reflective boundary conditions (BCs), in this paper we present analogous results for Anti-Reflective BCs, where an additional technical difficulty is represented by the non orthogonal character of the Anti-Reflective transform and indeed the technique of Ng, R.Chan, and Tang can not be used. Nevertheless, in both cases, the optimal preconditioner is the blurring matrix associated to the symmetrized Point Spread Function (PSF). The geometrical idea on which our proof is based is very simple and general, so it may be useful in the future to prove theoretical results for new proposed boundary conditions. Computational results show that the preconditioning strategy is effective and it is able to give rise to a meaningful acceleration both for slightly and highly non-symmetric PSFs.

math.NA↗

arXiv2012

A Proposal of Multigrid Methods for Hermitian Positive Definite Linear Systems enjoying an order relation

Given a multigrid procedure for linear systems with coefficient matrices $A_n$, we discuss the optimality of a related multigrid procedure with the same smoother and the same projector, when applied to properly related algebraic problems with coefficient matrices $B_n$: we assume that both $A_n$ and $B_n$ are positive definite with $A_n\le \vartheta B_n$, for some positive $\vartheta$ independent of $n$. In this context we prove the Two-Grid method optimality. We apply this elementary strategy for designing a multigrid solution for modifications of multilevel structured (Toeplitz, circulants, Hartley, sine ($τ$ class) and cosine algebras) linear systems, in which the coefficient matrix is banded in a multilevel sense and Hermitian positive definite. In such a way, several linear systems arising from the approximation of integro-differential equations with various boundary conditions can be efficiently solved in linear time (with respect to the size of the algebraic problem). Some numerical experiments are presented and discussed, both with respect to Two-Grid and multigrid procedures.

math.NA↗

arXiv2012

Korovkin results and Frobenius optimal approximants for infinite dimensional bounded linear operators

The classical as well as non commutative Korovkin-type theorems deal with convergence of positive linear maps with respect to modes of convergences such as norm convergence and weak operator convergence. In this article, Korovkin-type theorems are proved for convergence of completely positive maps with respect to weak, strong and uniform clustering of sequences of matrices of growing order. Such modes of convergence were originally considered for Toeplitz matrices (see [23],[26]). As an application, we translate the Korovkin-type approach used in the finite dimensional case, in the setting of preconditioning large linear systems with Toeplitz structure, into the infinite dimensional context of operators acting on separable Hilbert spaces. The asymptotic of these pre-conditioners are obtained and analyzed using the concept of completely positive maps. It is observed that any two limit points of the same sequence of pre-conditioners are the same modulo compact operators. Finally, we prove the generalized versions of the Korovkin type theorems in [23]. Keywords: Completely positive maps, Frobenius norm, Pre-conditioners.

math.FA↗

arXiv2010

Fast Preconditioners for Total Variation Deblurring with Anti-Reflective Boundary Conditions

In recent works several authors have proposed the use of precise boundary conditions (BCs) for blurring models and they proved that the resulting choice (Neumann or reflective, anti-reflective) leads to fast algorithms both for deblurring and for detecting the regularization parameters in presence of noise. When considering a symmetric point spread function, the crucial fact is that such BCs are related to fast trigonometric transforms. In this paper we combine the use of precise BCs with the Total Variation (TV) approach in order to preserve the jumps of the given signal (edges of the given image) as much as possible. We consider a classic fixed point method with a preconditioned Krylov method (usually the conjugate gradient method) for the inner iteration. Based on fast trigonometric transforms, we propose some preconditioning strategies which are suitable for reflective and anti-reflective BCs. A theoretical analysis motivates the choice of our preconditioners and an extensive numerical experimentation is reported and critically discussed. The latter shows that the TV regularization with anti-reflective BCs implies not only a reduced analytical error, but also a lower computational cost of the whole restoration procedure over the other BCs.

math.NA↗

arXiv2010

Multigrid methods for Toeplitz linear systems with different size reduction

Starting from the spectral analysis of g-circulant matrices, we consider a new multigrid method for circulant and Toeplitz matrices with given generating function. We assume that the size n of the coefficient matrix is divisible by g \geq 2 such that at the lower level the system is reduced to one of size n/g by employing g-circulant based projectors. We perform a rigorous two-grid convergence analysis in the circulant case and we extend experimentally the results to the Toeplitz setting, by employing structure preserving projectors. The optimality of the proposed two-grid method and of the multigrid method is proved, when the number theta \in N of recursive calls is such that 1 < theta < g. The previous analysis is used also to overcome some pathological cases, in which the generating function has zeros located at "mirror points" and the standard two-grid method with g = 2 is not optimal. The numerical experiments show the correctness and applicability of the proposed ideas both for circulant and Toeplitz matrices.

math.NA↗

arXiv2010

Multigrid and preconditioning strategies for implicit PDE solvers for degenerate parabolic equations

The novel contribution of this paper relies in the proposal of a fully implicit numerical method designed for nonlinear degenerate parabolic equations, in its convergence/stability analysis, and in the study of the related computational cost. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative or multi-iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures: in particular we investigate the mutual benefit of combining in various ways suitable preconditioners with V-cycle algorithms. Numerical experiments in one and two spatial dimensions for the validation of our multi-facet analysis complement this contribution.

math.NA↗

arXiv2009

Spectral features and asymptotic properties for alpha-circulants and alpha-Toeplitz sequences: theoretical results and examples

For a given nonnegative integer alpha, a matrix A_{n} of size n is called alpha-Toeplitz if its entries obey the rule A_{n}=[a_{r-alpha*s}]_{r,s=0}^{n-1}. Analogously, a matrix A_{n} again of size n is called alpha-circulant if A_{n}= [a_{(r-alpha*s)mod n}]_{r,s=0}^{n-1}. Such kind of matrices arises in wavelet analysis, subdivision algorithms and more generally when dealing with multigrid/multilevel methods for structured matrices and approximations of boundary value problems. In this paper we study the singular values of alpha-circulants and we provide an asymptotic analysis of the distribution results for the singular values of alpha-Toeplitz sequences in the case where {a_{k}} can be interpreted as the sequence of Fourier coeffcients of an integrable function f over the domain (-pi;pi). Some generalizations to the block, multilevel case, amounting to choose f multivariate and matrix valued, are briefly considered.

math.NA↗

arXiv2005

The asymptotic properties of the spectrum of non symmetrically perturbed Jacobi matrix sequences

Under the mild trace-norm assumptions we show that the eigenvalues of a generic (non Hermitian) complex perturbation of a Jacobi matrix sequence (not necessarily real) are still distributed as the real-valued function $2\cos t$ on $[0,π]$, which characterizes the nonperturbed case. In this way the real interval $[-2,2]$ is still a cluster for the asymptotic joint spectrum and, moreover, $[-2,2]$ attracts strongly (with infinite order) the perturbed matrix sequence. The results follow in a straightforward way from more general facts that we prove in an asymptotic linear algebra framework and are plainly generalized to the case of matrix-valued symbol, which arises when dealing with orthogonal polynomials with asymptotically periodic recurrence coefficients.

math.SP↗