Mira Bozzini

Adaptive Interpolation by Scaled Multiquadrics

We present an adaptive method to extract shape-preserving information from a univariate data sample. The behavior of the signal is obtained by interpolating at adaptively selected few data points by a linear combination of multiquadrics with variable scaling parameters. On the theoretical side, we give a sufficient condition for existence of the scaled multiquadric interpolant. On the practical side, we give various examples to show the applicability of the method.

Generalized Whittle---Matérn and polyharmonic kernels

This paper simultaneously generalizes two standard classes of radial kernels, the polyharmonic kernels related to the differential operator ( − Δ)m and the Whittle–Matérn kernels related to the differential operator ( − Δ + I)m. This is done by allowing general differential operators of the form

Non-regular Surface Approximation

\prod_{j=1}^m(-\Delta+\kappa_j^2I)

An approximation method of the local type. Application to a problem of heart potential mapping

with nonzero κj and calculating their associated kernels. It turns out that they can be explicity given by starting from scaled Whittle–Matérn kernels and taking divided differences with respect to their scale. They are positive definite radial kernels which are reproducing kernels in Hilbert spaces norm-equivalent to

Polyharmonic splines: An approximation method for noisy scattered data of extra-large size

W_2^m(\ensuremath{\mathbb{R}}^d)

Kernel B -splines and interpolation

. On the side, we prove that generalized inverse multiquadric kernels of the form

Radial kernels via scale derivatives

\prod_{j=1}^m(r^2+\kappa_j^2)^{-1}

A new method in order of determine the most significant members within a large sample, in problems of surface approximations

are positive definite, and we provide their Fourier transforms. Surprisingly, these Fourier transforms lead to kernels of Whittle–Matérn form with a variable scale κ(r) between κ1,...,κm. We also consider the case where some of the κj vanish. This leads to conditionally positive definite kernels that are linear combinations of the above variable-scale Whittle–Matérn kernels and polyharmonic kernels. Some numerical examples are added for illustration.

Approximation of multivariable functions with respect to random points less than 2k,k dimension of space

The aim of the paper is to provide a method for approximating non regular surfaces from a set of scattered data in a faithful way. The method we propose is effective and particularly well-suited for recovering geophysical surfaces with faults or drainage patterns. Some real examples will be presented.

Original article: Recovering functions: A method based on domain decomposition

In this paper a local type method is proposed to smooth noisy data. Criteria of convergence and error bounds are given.An applciation is also presented for a biomedical problem in ℝ3, which is usually solved only in ℝ2.ZusammenfassungIn dieser Arbeit wird eine lokale Methdoe vorgeschlagen, verrauschte Daten zu glätten. Konvergenzkriterien und Fehlerschranken werden angegeben.Insbesondere wird eine Anwendung auf ein biomedizinsiches Problem in ℝ3 dargestellt, welches üblicherweise nur im ℝ2 gelöst wird.