Ornella Menchi

A framework for studying the regularizing properties of Krylov subspace methods

Krylov subspace iterative methods have recently received considerable attention as regularizing techniques for solving linear systems with a coefficient matrix of ill-determined rank and a right-hand side vector perturbed by noise. For many of them little is known from this point of view. In this paper, the regularizing properties of some methods of Krylov type (CGLS, GMRES, QMR, CGS, BiCG, Bi-CGSTAB) are examined. CGLS, for which a theoretical analysis is available, is taken as a reference method. Tools for measuring the regularization efficiency and the consistency with the discrepancy principle are introduced. An extensive experimentation validates the proposed measures for the studied methods.

Performance analysis of maximum likelihood methods for regularization problems with nonnegativity constraints

In many numerical applications, for instance in image deconvolution, the nonnegativity of the computed solution is required. When a problem of deconvolution is formulated in a statistical frame, the recorded image is seen as the realization of a random process, where the nature of the noise is taken into account. This formulation leads to the maximization of a likelihood function which depends on the statistical property assumed for the noise. In this paper we revisit, under this unifying statistical approach, some iterative methods coupled with suitable strategies for enforcing nonnegativity and other ones which instead naturally embed nonnegativity. For all these methods we carry out a comparative study taking into account several performance indicators. The reconstruction accuracy, the computational cost, the consistency with the discrepancy principle (a common technique for guessing the best regularization parameter) and the sensitivity to this choice are compared in a simulation context, by means of an extensive experimentation on both 1D and 2D problems.

Generalized Cross-Validation applied to Conjugate Gradient for discrete ill-posed problems

In this paper we propose a new method to apply the Generalized Cross-Validation (GCV) as a stopping rule for the Conjugate Gradient (CG). In general, to apply GCV to an iterative method, one must estimate the trace of the so-called influence matrix which appears in the denominator of the GCV function. In the case of CG, unlike what happens with stationary iterative methods, the regularized solution has a nonlinear dependence on the noise which affects the data of the problem. This fact is often pointed out as a cause of poor performance of GCV. To overcome this drawback, our proposal linearizes the dependence by computing the derivatives through iterative formulas. We compare the proposed method with other methods suggested in the literature by an extensive numerical experimentation on both 1D and 2D test problems.

On the numerical solution of a variational inequality connected with the hydrodynamic lubrication of a complete journal bearing

The theory of variational inequalities is applied to a problem of lubrication. A method for solving numerically the relevant variational inequality is presented.

Solution of infinite linear systems by automatic adaptive iterations

Abstract The problem of approximating the solution of infinite linear systems finitely expressed by a sparse coefficient matrix in block Hessenberg form is considered. The convergence of the solutions of a sequence of truncated problems to the infinite problem solution is investigated. A family of algorithms, some of which are adaptive, is introduced, based on the application of the Gauss–Seidel method to a sequence of truncated problems of increasing size n i with non-increasing tolerance 10 −t i . These algorithms do not require special structural properties of the coefficient matrix and they differ in the way the sequences {n i } and {t i } are generated. The testing has been performed on both infinite problems arising from the discretization of elliptical equations on unbounded domains and stochastic problems arising from queueing theory. Extensive numerical experiments permit the evaluation of the various strategies and suggest that the best trade-off between accuracy and computational cost is reached by some of the adaptive algorithms.

A Divide and Conquer Algorithm for the Superfast Solution of Toeplitz-like Systems

In this paper a new

Stability of the Levinson Algorithm for Toeplitz-Like Systems

O(N \,\log^3N)

A polynomial fit preconditioner for band Toeplitz matrices in image reconstruction

solver for

Recursive algorithms for unbalanced banded Toeplitz systems

N\times N

Separable Asymptotic Cost of Evaluating Elementary Functions

Toeplitz-like systems, based on a divide and conquer technique, is presented. Similarly to the superfast algorithm MBA for the inversion of a Toeplitz-like matrix [R. R. Bitmead and B. D. O. Anderson, Linear Algebra Appl., 34 (1980), pp. 103--116; M. Morf, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 1980, pp. 954--959], it exploits the displacement properties. In order to avoid the well-known numerical instability of the explicit inversion, the new algorithm relies on the triangular factorization and back-substitution formula for the system seen as a