Mathematical and Computational Methods for Modelling, Approximation and Simulation

In this paper, we present recent solutions to the problem of approximating functions by polynomials for reducing in a substantial manner two well-known phenomena: Runge and Gibbs. The main idea is to avoid resampling the function or data and relies on the mapped polynomials or “fake” nodes approach. This technique turns out to be effective for stability by reducing the Runge effect and also in the presence of discontinuities by almost cancelling the Gibbs phenomenon. The technique is very general and can be applied to any approximant of the underlying function to be reconstructed: polynomials, rational functions or any other basis. A “natural” application is then quadrature, that we started recently to investigate and we propose here some results. In the case of jumps or discontinuities, where the Gibbs phenomenon appears, we propose a different approach inspired by approximating functions by kernels, in particular Radial Basis Functions (RBF). We use the so called Variably Scaled Discontinuous Kernels (VSDK) as an extension of the Variably Scaled Kernels (VSK) firstly introduced in Bozzini et al. (IMA J Numer Anal 35:199–219, 2015). VSDK show to be a very flexible tool suitable to substantially reducing the Gibbs phenomenon in reconstructing functions with jumps. As an interesting application we apply VSDK in Magnetic Particle Imaging which is a recent non-invasive tomographic technique that detects super-paramagnetic nanoparticle tracers and finds applications in diagnostic imaging and material science. In fact, the image generated by the MPI scanners are usually discontinuous and sampled at scattered data points, making the reconstruction problem affected by the Gibbs phenomenon. We show that VSDK are well suited in MPI image reconstruction also for identifying image discontinuities.

After some general comments on the tremendous difficulties associated with non-linear systems of PDEs, we will focus on two such illustrative situations where the underlying model depends on the highly nonlinear behavior coming from a system of PDEs. The first one is motivated by inverse problems in conductivity and the process to recover an unknown conductivity coefficient from measurements in the boundary; the second focuses on an optimal control problem for soft robots in which the underlying model comes from hyper-elasticity where the state system models the behavior of non-linear elastic materials capable of undergoing large deformations.

We consider two-stage scattered data fitting with truncated hierarchical B-splines (THB-splines) for the adaptive reconstruction of industrial models. The first stage of the scheme is devoted to the computation of local least squares variational spline approximations, exploiting a simple fairness functional to handle data distributions with a locally varying density of points. Hierarchical spline quasi-interpolation based on THB-splines is considered in the second stage of the method to construct the adaptive spline surface approximating the whole scattered data set and a suitable strategy to guide the adaptive refinement is introduced. A selection of examples on geometric models representing components of aircraft turbine blades highlights the performances of the scheme. The tests include a scattered data set with voids and the adaptive reconstruction of a cylinder-like surface.

We present a method for constructing differential and integral spline quasi-interpolants defined on uniform partitions of the real line. It is based on a expression to the quasi-interpolation error for a enough regular function and involves the errors for the non-reproduced monomials. From it, a minimization problem is proposed whose solution is calculated progressively. It is characterized in terms of some splines which do not depend on the linear functional defining the quasi-interpolation operators. The resulting quasi-interpolants are compared with other well-known schemes to show the efficiently of this construction.

In this paper, we analyse the Nyström method based on a sextic spline quasi-interpolant for approximating the solution of a linear Fredholm integral equation of the second kind. For a sufficiently smooth kernel the method is shown to have convergence of order 8 and the Richardson extrapolation is used to further improve this order to 9. Numerical examples are given to confirm the theoretical estimates.

In this paper we propose two collocation methods, based on superconvergent cubic spline interpolant and quasi-interpolant, for approximating the solution of the second kind Fredholm integral equations. Convergence analysis is established. Some numerical examples are given to show the validity of the presented methods.

The main interest of this paper is to state a new method which allows to adjust the statistical difficulty when the statistical series are spliced. Hence, we study the scope of different splicing methods in the literature. We present an approximation method for statistical splicing of economic data by using smoothing quadratic splines. Finally, we show the effectiveness of our method by presenting a complete data of Gross Domestic Product for Venezuela by productive economic activity from 1950 to 2005, expressed at prices of the base year of 1997, also by showing the results of some data of Morocco for different economics activities such as the Gross Domestic Product, the agriculture, the trade and the electricity generation from petroleum sources of Morocco between 1971 and 2015.

We discuss, in Menger spaces, the notion of convexity using the convex structure introduced by Takahashi (Kodai Math Sem Rep 22:142–149, 1970), then we develop some geometric and topological properties. Furthermore, we introduce the notion of strong convex structure and we compare it with the Takahashi convex structure. At the end, we prove the existence and uniqueness of a solution for a Volterra type integral equation.

In this paper we use the developed B-spline representation to construct a cubic super-superconvergent quasi-interpolant with an optimal approximation order which improves the efficiency and accuracy over traditional methods.

The analysis of poverty measures has been receiving increased attention in recent years. This paper contributes to the literature by developing percentile ratio estimators when there are missing data. Calibration adjustment is used for treating the non-response bias. Variances of the proposed estimators could be not expressible by simple formulae and resampling techniques are investigated for obtaining variance estimators. A numerical example based on data from the Spanish Household Panel Survey is taken up to illustrate how suggested procedures can perform better than existing ones.

In this paper, we propose collocation type method, its iterated version and Nyström method based on discrete spline quasi-interpolating operators to solve Hammerstein integral equation. We present an error analysis of the approximate solutions and we show that the iterated solution of collocation type exhibits a superconvergence as in the case of the Galerkin method. Finally, we provide numerical tests, that confirm the theoretical results.