Search

The guest editors discuss the highlights of the meeting Seminari Padovani di Analisi Numerica SPAN18 and briefly introduce the papers included in this special issue.

We investigate adaptivity issues for the approximation of Poisson equations via radial basis function-based partition of unity collocation. The adaptive residual subsampling approach is performed with quasi-uniform node sequences leading to a flexible tool which however might suffer from numerical instability due to ill-conditioning of the collocation matrices. We thus develop a hybrid method which makes use of the so-called variably scaled kernels. The proposed algorithm numerically ensures the convergence of the adaptive procedure

We investigate adaptivity issues for the approximation of Poisson equations via radial basis function-based partition of unity collocation. The adaptive residual subsampling approach is performed with quasi-uniform node sequences leading to a flexible tool which however might suffer from numerical instability due to ill-conditioning of the collocation matrices. We thus develop a hybrid method which makes use of the so-called variably scaled kernels. The proposed algorithm numerically ensures the convergence of the adaptive procedure. © 2018, Springer Science+Business Media, LLC, part of Springer Nature

In this work we extend some ideas about greedy algorithms, which are well-established tools for, e.g., kernel bases, and exponential-polynomial splines whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we develop some results on theoretically optimal interpolation points. Moreover, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals (f-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function ($$\lambda$$ λ -greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e., the spline nodes. While the f-greedy selection is tailored to one specific target function, the $$\lambda$$ λ -greedy algorithm enables us to define target-data-independent interpolation nodes.

Accurate interpolation and approximation techniques for functions with discontinuities are key tools in many applications, such as medical imaging. In this paper, we study a radial basis function type of 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 results of the conducted numerical experiments are in line with 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 in order to obtain edge-preserving image reconstructions in which ringing artifacts are reduced.

The data driven extrapolation requires the definition of a functional model depending on the available data and has the application scope of providing reliable predictions on the unknown dynamics. Since data might be scattered, we drive our attention towards kernel models that have the advantage of being meshfree. Precisely, the proposed numerical method makes use of the so-called Variably Scaled Kernels (VSKs), which are introduced to implement a feature augmentation-like strategy based on discrete data. Due to the possible uncertainty on the data and since we are interested in modelling the behaviour of the considered dynamics, we seek for a regularized solution by ridge regression. Focusing on polyharmonic splines, we investigate their implementation in the VSK setting and we provide error bounds in Beppo-Levi spaces. The performances of the method are then tested on functions which are common in the framework of the Laplace transform inversion. Comparisons with Support Vector Regression (SVR) are also carried out and show that the proposed method is effective particularly since it does not require to train complex architecture constructions.

An elliptic partial differential equation with a singular forcing term, describing a steady state flow determined by a pulse-like extraction at a constant volumetric rate, is approximated by a radial basis function approach which takes advantage of decomposing the original domain. The discretization error of such scheme is numerically estimated and we also face up to instability issues. This produces an effective tool for real applications, as confirmed by comparisons with classical grid-based approaches.