Sebastiano Seatzu

Multi-parameter regularization techniques for ill-conditioned linear systems

Summary. When a system of linear equations is ill-conditioned, regularization techniques provide a quite useful tool for trying to overcome the numerical inherent difficulties: the ill-conditioned system is replaced by another one whose solution depends on a regularization term formed by a scalar and a matrix which are to be chosen. In this paper, we consider the case of several regularizations terms added simultaneously, thus overcoming the problem of the best choice of the regularization matrix. The error of this procedure is analyzed and numerical results prove its efficiency.

Spectral factorization of Laurent polynomials

We analyse the performance of five numerical methods for factoring a Laurent polynomial, which is positive on the unit circle, as the modulus squared of a real algebraic polynomial. It is found that there is a wide disparity between the methods, and all but one of the methods are significantly influenced by the variation in magnitude of the coefficients of the Laurent polynomial, by the closeness of the zeros of this polynomial to the unit circle, and by the spacing of these zeros.

Error estimates for linear systems with applications to regularization

In this paper, we discuss several (old and new) estimates for the norm of the error in the solution of systems of linear equations, and we study their properties. Then, these estimates are used for approximating the optimal value of the regularization parameter in Tikhonov’s method for ill-conditioned systems. They are also used as a stopping criterion in iterative methods, such as the conjugate gradient algorithm, which have a regularizing effect. Several numerical experiments and comparisons with other procedures show the effectiveness of our estimates.

Error estimates for the regularization of least squares problems

The a posteriori estimate of the errors in the numerical solution of ill-conditioned linear systems with contaminated data is a complicated problem. Several estimates of the norm of the error have been recently introduced and analyzed, under the assumption that the matrix is square and nonsingular. In this paper we study the same problem in the case of a rectangular and, in general, rank-deficient matrix. As a result, a class of error estimates previously introduced by the authors (Brezinski et al., Numer Algorithms, in press, 2008) are extended to the least squares solution of consistent and inconsistent linear systems. Their application to various direct and iterative regularization methods are also discussed, and the numerical effectiveness of these error estimates is pointed out by the results of an extensive experimentation.

Extrapolation techniques for ill-conditioned linear systems

Summary. In this paper, the regularized solutions of an ill–conditioned system of linear equations are computed for several values of the regularization parameter

Sampling Expansions and Interpolation in Unitarily Translation Invariant Reproducing Kernel Hilbert Spaces

\lambda

A New Technique for Ill-Conditioned Linear Systems

. Then, these solutions are extrapolated at

LDU factorization results for bi-infinite and semi-infinite scalar and block Toeplitz matrices

\lambda=0

On the Cholesky Factorization of the Gram Matrix of Multivariate Functions

by various vector rational extrapolations techniques built for that purpose. These techniques are justified by an analysis of the regularized solutions based on the singular value decomposition and the generalized singular value decomposition. Numerical results illustrate the effectiveness of the procedures.

On the Cholesky factorization of the Gram matrix of locally supported functions

Sufficient conditions are established in order that, for a fixed infinite set of sampling points on the full line, a function satisfies a sampling theorem on a suitable closed subspace of a unitarily translation invariant reproducing kernel Hilbert space. A number of examples of such reproducing kernel Hilbert spaces and the corresponding sampling expansions are given. Sampling theorems for functions on the half-line are also established in RKHS using Riesz bases in subspaces of L2(R+).