Dario Fasino

An inverse Robin problem for Laplace's equation: theoretical results and numerical methods

We consider the problem of detecting corrosion damage on an inaccessible part of a metallic specimen. Electrostatic data are collected on an accessible part of the boundary. The adoption of a simplified model of corrosion appearance reduces our problem to recovering a functional coefficient in a Robin boundary condition for Laplaces equation. We review theoretical results and numerical methods based on the thin-plate approximation and the Galerkin method. Moreover, we introduce a numerical algorithm based on the quasi-reversibility method.

Orthogonal Rational Functions and Structured Matrices

The linear space of all proper rational functions with prescribed poles is considered. Given a set of points zi in the complex plane and the weights wi we define the discrete inner product

Existence of proper semihypergroups of type U on the right

\langle \phi,\psi \rangle := \sum_{i=0}^n |w_i|^2 \overline{\phi(z_i)} \psi(z_i).

Spectral properties of Hankel matrices and numerical solutions of finite moment problems

In this paper we derive a method to compute the coefficients of a recurrence relation generating a set of orthonormal rational basis functions with respect to the discrete inner product. We will show that these coefficients can be computed by solving an inverse eigenvalue problem for a matrix having a specific structure. In the case where all the points zi lie on the real line or on the unit circle, the computational complexity is reduced by an order of magnitude.

Direct and inverse eigenvalue problems for diagonal-plus-semiseparable matrices

Singular Spectrum Analysis is a quite recent technique for the analysis of experimental time series, based on the singular value decomposition of certain Hankel matrices. However, the mathematical and physical interpretation of the singular values in this kind of application is not fully clarified. In this paper, using asymptotic properties of the eigenvalues of Toeplitz matrices, we show that SSA is related to Fourier analysis. Indeed, the singular values provide information that can be interpreted, and estimated efficiently, by means of the power spectrum of the time series. We apply our results to the continuous seismic signal recorded at Stromboli volcano in order to highlight precursors of a paroxysmal volcanic eruption.

Spectral clustering properties of block multilevel Hankel matrices

We generalize the classical definition of hypergroups of type U on the right to semihypergroups, and we prove some properties of their subsemihypergroups and subhypergroups. In particular, we obtain that a finite proper semihypergroup of type U on the right can exist only if its order is at least 6. We prove that one such semihypergroup of order 6 actually exists. Moreover, we show that there exists a hypergroup of type U on the right of cardinality 9 containing a proper non-trivial subsemihypergroup. In this way, we solve a problem left open in [D. Freni, Sur les hypergroupes de type U et sous-hypergroupes engendres par un sous-ensemble, Riv. Mat. Univ. Parma 13 (1987) 29-41].

Orthogonal rational functions and diagonal-plus-semiseparable matrices

SUMMARY We prove that the unitary factor appearing in the QR factorization of a suitably dened rational Krylov matrix transforms a Hermitian matrix having pairwise distinct eigenvalues into a diagonal-plus- semiseparable form with prescribed diagonal term. This transformation is essentially uniquely dened by itsrst column. Furthermore, we prove that the set of Hermitian diagonal-plus-semiseparable matrices is invariant under QR iteration. These and other results are shown to be the rational counterpart of known facts involving structured matrices related to polynomial computations. Copyright ? 2005 John Wiley & Sons, Ltd.

A Lanczos‐type algorithm for the QR factorization of regular Cauchy matrices

Abstract After proving that any Hankel matrix generated by moments of positive functions is conditioned essentially the same as the Hilbert matrix of the same size, we show a preconditioning technique, i.e., a congruence transform of the original Hankel matrix that drastically reduces its ill-conditioning. Applications of this result to classical orthogonal polynomial sequences and to modified moment problems are given. Also, we outline an efficient algorithm for the computation of the function f ( x ) = w ( x ) exp ( p ( x )), where w ( x ) is positive and p ( x ) is a polynomial of degree n −1, from the knowledge of its first n moments.