arXiv subjects

Stefano De Marchi

Publications and source records attributed to Stefano De Marchi.

31 records · Page 2Linked to original sources

arXiv2021

A shape preserving quasi-interpolation operator based on a new transcendental RBF

It is well-known that the univariate Multiquadric quasi-interpolation operator is constructed based on the piecewise linear interpolation by |x|. In this paper, we first introduce a new transcendental RBF based on the hyperbolic tangent function as a smooth approximant to f(r)=r with higher accuracy and better convergence properties than the multiquadric. Then Wu-Schaback's quasi-interpolation formula is rewritten using the proposed RBF. It preserves convexity and monotonicity. We prove that the proposed scheme converges with a rate of O(h^2). So it has a higher degree of smoothness. Some numerical experiments are given in order to demonstrate the efficiency and accuracy of the method.

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arXiv2021

Stable discontinuous mapped bases: the Gibbs-Runge-Avoiding Stable Polynomial Approximation (GRASPA) method

The mapped bases or Fake Nodes Approach (FNA), introduced in [10], allows to change the set of nodes without the need of resampling the function. Such scheme has been successfully applied in preventing the appearance of the Gibbs phenomenon when interpolating discontinuous functions. However, the originally proposed S-Gibbs map suffers of a subtle instability when the interpolant is constructed at equidistant nodes, due to the Runge's phenomenon. Here, we propose a novel approach, termed Gibbs-Runge-Avoiding Stable Polynomial Approximation (GRASPA), where both Runge's and Gibbs phenomena are mitigated. After providing a theoretical analysis of the Lebesgue constant associated to the mapped nodes, we test the new approach by performing different numerical experiments which confirm the theoretical findings.

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arXiv2021

Oversampling errors in multimodal medical imaging are due to the Gibbs effect

To analyse multimodal 3-dimensional medical images, interpolation is required for resampling which - unavoidably - introduces an interpolation error. In this work we consider three segmented 3-dimensional images resampled with three different neuroimaging software tools for comparing undersampling and oversampling strategies and to identify where the oversampling error lies. The results indicate that undersampling to the lowest image size is advantageous in terms of mean value per segment errors and that the oversampling error is larger where the gradient is steeper, showing a Gibbs effect.

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arXiv2021

On $(β,γ)$-Chebyshev functions and points of the interval

In this paper, we introduce the class of $(β,γ)$-Chebyshev functions and corresponding points, which can be seen as a family of {\it generalized} Chebyshev polynomials and points. For the $(β,γ)$-Chebyshev functions, we prove that they are orthogonal in certain subintervals of $[-1,1]$ with respect to a weighted arc-cosine measure. In particular we investigate the cases where they become polynomials, deriving new results concerning classical Chebyshev polynomials of first kind. Besides, we show that subsets of Chebyshev and Chebyshev-Lobatto points are instances of $(β,γ)$-Chebyshev points. We also study the behavior of the Lebesgue constants of the polynomial interpolant at these points on varying the parameters $β$ and $γ$.

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arXiv2019

Shape-Driven Interpolation with Discontinuous Kernels: Error Analysis, Edge Extraction and Applications in MPI

Accurate interpolation and approximation techniques for functions with discontinuities are key tools in many applications as, for instance, medical imaging. In this paper, we study an RBF type method for scattered data interpolation that incorporates discontinuities via a variable scaling function. For the construction of the discontinuous basis of kernel functions, information on the edges of the interpolated function is necessary. We characterize the native space spanned by these kernel functions and study error bounds in terms of the fill distance of the node set. To extract the location of the discontinuities, we use a segmentation method based on a classification algorithm from machine learning. The conducted numerical experiments confirm the theoretically derived convergence rates in case that the discontinuities are a priori known. Further, an application to interpolation in magnetic particle imaging shows that the presented method is very promising.

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arXiv2016

A rescaled method for RBF approximation

In the recent paper [8], a new method to compute stable kernel-based interpolants has been presented. This \textit{rescaled interpolation} method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allow us to consider its error and stability properties.

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arXiv2015

Approximating basins of attraction for dynamical systems via stable radial bases

In applied sciences, such as physics and biology, it is often required to model the evolution of populations via dynamical systems. In this paper, we focus on the problem of approximating the basins of attraction of such models in case of multi-stability. We propose to reconstruct the domains of attraction via an implicit interpolant using stable radial bases, obtaining the surfaces by partitioning the phase space into disjoint regions. An application to a competition model presenting jointly three stable equilibria is considered.

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arXiv2015

Trivariate polynomial approximation on Lissajous curves

We study Lissajous curves in the 3-cube, that generate algebraic cubature formulas on a special family of rank-1 Chebyshev lattices. These formulas are used to construct trivariate hyperinterpolation polynomials via a single 1-d Fast Chebyshev Transform (by the Chebfun package), and to compute discrete extremal sets of Fekete and Leja type for trivariate polynomial interpolation. Applications could arise in the framework of Lissajous sampling for MPI (Magnetic Particle Imaging).

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arXiv2014

On the constrained mock-Chebyshev least-squares

The algebraic polynomial interpolation on uniformly distributed nodes is affected by the Runge phenomenon, also when the function to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which is an interpolation made on a subset of the given nodes whose elements mimic as well as possible the Chebyshev-Lobatto points. In this work we use the simultaneous approximation theory to combine the previous technique with a polynomial regression in order to increase the accuracy of the approximation of a given analytic function. We give indications on how to select the degree of the simultaneous regression in order to obtain polynomial approximant good in the uniform norm and provide a sufficient condition to improve, in that norm, the accuracy of the mock-Chebyshev interpolation with a simultaneous regression. Numerical results are provided.

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arXiv2014

On certain multivariate Vandermonde determinants whose variables separate

We prove that for almost square tensor product grids and certain sets of bivariate polynomials the Vandermonde determinant can be factored into a product of univariate Vandermonde determinants. This result generalizes the conjecture [Lemma 1, L. Bos et al. (2009), Dolomites Research Notes on Approximation, 2:1-15]. As a special case, we apply the result to Padua and Padua-like points.

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arXiv2011

Polynomial approximation and cubature at approximate Fekete and Leja points of the cylinder

The paper deals with polynomial interpolation, least-square approximation and cubature of functions defined on the rectangular cylinder, $K=D\times [-1,1]$, with $D$ the unit disk. The nodes used for these processes are the {\it Approximate Fekete Points} (AFP) and the {\it Discrete Leja Points} (DLP) extracted from suitable {\it Weakly Admissible Meshes} (WAMs) of the cylinder. From the analysis of the growth of the Lebesgue constants, approximation and cubature errors, we show that the AFP and the DLP extracted from WAM are good points for polynomial approximation and numerical integration of functions defined on the cylinder.

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arXiv2008

New cubature formulae and hyperinterpolation in three variables

A new algebraic cubature formula of degree $2n+1$ for the product Chebyshev measure in the $d$-cube with $\approx n^d/2^{d-1}$ nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree $n$ in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpolant provides a new Clenshaw-Curtis type cubature formula in the 3-cube.

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arXiv2006

Bivariate Lagrange interpolation at the Padua points: the ideal theory approach

Padua points is a family of points on the square $[-1,1]^2$ given by explicit formulas that admits unique Lagrange interpolation by bivariate polynomials. The interpolation polynomials and cubature formulas based on the Padua points are studied from an ideal theoretic point of view, which leads to the discovery of a compact formula for the interpolation polynomials. The $L^p$ convergence of the interpolation polynomials is also studied.

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