Rational RBF-based partition of unity method for efficiently and accurately approximating 3D objects

We consider the problem of reconstructing 3D objects via meshfree interpolation methods. In this framework, we usually deal with large data sets and thus we develop an efficient local scheme via the well-known Partition of Unity (PU) method. The main contribution in this paper consists in constructing the local interpolants for the implicit interpolation by means of Rational Radial Basis Functions (RRBFs). Numerical evidence confirms that the proposed method is particularly performing when 3D objects, or more in general implicit functions defined by scattered data, need to be approximated

Fast and flexible interpolation via PUM with applications in population dynamics

In this paper the Partition of Unity Method (PUM) is efficiently performed using Radial Basis Functions (RBFs) as local approximants. In particular, we present a new space-partitioning data structure extremely useful in applications because of its independence from the problem geometry. Moreover, we study, in the context of wild herbivores in forests, an application of such algorithm. This investigation shows that the ecosystem of the considered natural park is in a very delicate situation, for which the animal population could become extinguished. The determination of the so-called sensitivity surfaces, obtained with the new fast and flexible interpolation tool, indicates some possible preventive measures to the park administrators

Partition of Unity Interpolation on Multivariate Convex Domains

In this paper we present a new algorithm for multivariate interpolation of scattered data sets lying in convex domains \Omega \subseteq \RR^N, for any $N \geq 2$. To organize the points in a multidimensional space, we build a $kd$-tree space-partitioning data structure, which is used to efficiently apply a partition of unity interpolant. This global scheme is combined with local radial basis function approximants and compactly supported weight functions. A detailed description of the algorithm for convex domains and a complexity analysis of the computational procedures are also considered. Several numerical experiments show the performances of the interpolation algorithm on various sets of Halton data points contained in $\Omega$, where $\Omega$ can be any convex domain like a 2D polygon or a 3D polyhedron

Recursive POD expansion for the advection-diffusion-reaction equation

This paper deals with the approximation of advection-diffusion-reaction equation solution by reduced order methods. We use the Recursive POD approximation for multivariate functions introduced in [M. AZAÏEZ, F. BEN BELGACEM, T. CHACÓN REBOLLO, Recursive POD expansion for reactiondiffusion equation, Adv.Model. and Simul. in Eng. Sci. (2016) 3:3. DOI 10.1186/s40323-016-0060-1] and applied to the low tensor representation of the solution of the reaction-diffusion partial differential equation. In this contribution we extend the Recursive POD approximation for multivariate functions with an arbitrary number of parameters, for which we prove general error estimates. The method is used to approximate the solutions of the advection-diffusion-reaction equation. We prove spectral error estimates, in which the spectral convergence rate depends only on the diffusion interval, while the error estimates are affected by a factor that grows exponentially with the advection velocity, and are independent of the reaction rate if this lives in a bounded set. These error estimates are based upon the analyticity of the solution of these equations as a function of the parameters (advection velocity, diffusion, reaction rate). We present several numerical tests, strongly consistent with the theoretical error estimates.Ministerio de Economía y CompetitividadAgence nationale de la rechercheGruppo Nazionale per il Calcolo ScientificoUE ERA-PLANE

A new numerical method for processing longitudinal data: clinical applications

Background: Processing longitudinal data is a computational issue that arises in many applications, such as in aircraft design, medicine, optimal control and weather forecasting. Given some longitudinal data, i.e. scattered measurements, the aim consists in approximating the parameters involved in the dynamics of the considered process. For this problem, a large variety of well-known methods have already been developed. Results: Here, we propose an alternative approach to be used as effective and accurate tool for the parameters fitting and prediction of individual trajectories from sparse longitudinal data. In particular, our mixed model, that uses Radial Basis Functions (RBFs) combined with Stochastic Optimization Algorithms (SOMs), is here presented and tested on clinical data. Further, we also carry out comparisons with other methods that are widely used in this framework. Conclusions: The main advantages of the proposed method are the flexibility with respect to the datasets, meaning that it is effective also for truly irregularly distributed data, and its ability to extract reliable information on the evolution of the dynamics

analysis of a new class of rational rbf expansions

Abstract We propose a new method, namely an eigen-rational kernel-based scheme, for multivariate interpolation via mesh-free methods. It consists of a fractional radial basis function (RBF) expansion, with the denominator depending on the eigenvector associated to the largest eigenvalue of the kernel matrix. Classical bounds in terms of Lebesgue constants and convergence rates with respect to the mesh size of the eigen-rational interpolant are indeed comparable with those of classical kernel-based methods. However, the proposed approach takes advantage of rescaling the classical RBF expansion providing more robust approximations. Theoretical analysis, numerical experiments and applications support our findings