Mathematical and Computational Modelling, Approximation and Simulation

Wavelets have been playing a prominent role in approximation theory, signal processing, and many other different contexts for several decades. A lot of effort has been spent on designing wavelet bases, in particular in the multivariate setting. Nevertheless, general procedures for their construction remain challenging tasks. This paper aims to illustrate and summarize the research carried out in the last few years by the authors and collaborators in the direction of exploring new strategies for realizing novel wavelet filterbank systems but also of unifying different existing approaches, both in the univariate and in the bivariate cases.

Fractal analysis for the characterization of complex structures is widely used in many fields. Among the fractal models, the fractional Brownian motion (fBm) is well suited for the characterization of images involving a varying range of gray levels. This chapter focuses on the use of the fBm for image analysis. Starting from the classical fBm of unique parameter H, different stochastic models are defined and discussed. The fractal anisotropy is considered and different methods emphasizing this property will be discussed. An application of these models to the field of biomedical imaging is also presented.

Some old and new results relating accuracy, stability and optimal bases are revisited. We focus on the relationship of these fields in the framework of totally positive bases and matrices. In fact, for some subclasses of totally positive matrices, which are often collocation, wronskian or gramian matrices of totally positive bases, high relative accurate algebraic algorithms have been found. Moreover, for spaces with a totally positive basis, the concept of B-bases corresponds to the optimally stable bases, and these bases also satisfy many other optimality properties, which play an important role in Approximation Theory and Computer Aided Geometric Design.

Many spaces of multivariate piecewise functions (or splines) with prescribed global smoothness exhibit additional, unprescribed, or intrinsic supersmoothness at certain faces where the pieces join together. Intrinsic supersmoothness is a consequence of the geometry of the underlying partition. In this survey, we provide insights into this phenomenon, review and generalize known results, and demonstrate how intrinsic supersmoothness affects other properties of spline spaces.

This paper is a preliminary work for the analysis of strategies to valuate weather derivatives using a complete Italian archive of average temperature data. We set a model to forecast average temperatures over relatively small time intervals (up to one year) and describe the related parameters estimation. The analysis ends up with the valuation of a Call on HDD index.

This paper is concerned with the approximation order of ENO-SR piecewise polynomial interpolatory techniques for piecewise smooth functions with isolated singularities. When the interpolatory pieces are of degree r, we propose a detection procedure based on differences of order r that allows us to prove that the global order of accuracy of the interpolatory reconstruction is improved by a factor of h relative to the linear methods, where h is the sampling rate, for isolated discontinuities in \(f^{(q)}\) and \(q \le r\). In addition, we show that full accuracy is recovered for h below a critical threshold, that depends on the jump in \(f^{(q)}\) at the isolated singularities.

Splines over triangulation are a fundamental tool in finite element analysis (FEM), due to their ability to perform local refinements and accurately represent complex geometries. An important example is the \(C^1\) quadratic splines on the Powell-Sabin (PS) 6-split triangulations. This spline space is characterized by specifying discrete values and first derivative values at the vertices of the triangulation on which it is defined. However, a common challenge arises in many applications where only functional evaluations are known at equally spaced nodes, making the direct application of the Powell-Sabin finite element method impractical. To address this limitation in this setting, we present a novel technique for approximating derivative values at the set of vertices, essential for effectively defining the Powell-Sabin finite element. The idea of this technique lies in leveraging the provided functional evaluations and approximating the partial derivatives at the data points using an interpolation-regression operator.

We address the problem of existence, uniqueness and approximation of the solution of a large class of fractional integro-differential equations with boundary conditions. Based on fixed point techniques with Boyd-Wong’s type conditions, the existence and uniqueness of the solution is studied. This, together with the use of certain types of biorthogonal systems, allows us to propose a method to approximate the solution that is tested with several examples.

For the purpose of approximating the solution of second-kind Volterra integral equations with algebraic singularities at the diagonal and endpoints, this work explores a modified Nyström approach based on the zeros of the Jacobi polynomials. The technique employs the interpolating operator in suitable weighted spaces and the associated product integration formulae. The method is shown to be stable and more accurate than the standard Nyström scheme, and the precision of the solution can further be enhanced by using the Sloan iteration. The convergence and stability results are confirmed by several numerical experiments.

In this paper, we present a simple method based on \(C^1\) quadratic polynomial splines that provides the numerical solution of Love’s integral equation. This technique consists in approximating the univariate right section of the kernel of the considered equation by a quadratic quasi-interpolant (QI). The approximate solution is obtained by solving a linear system where the matrix coefficients are computed explicitly. We study the convergence analysis and we show that the method is of order four. Finally, we give some numerical examples that illustrate the theoretical results and the validity of the presented method.

This paper deals with the study of the interpolation of a bidirectional B-spline curve network by both biquadratic tensor product B-spline surfaces and quadratic B-spline surfaces on criss-cross triangulations, with the aim of simulating discontinuities in the gradient and even in the surface itself. An algorithm for such a construction is proposed and graphical results show the performance of the method.

The purpose of this research is to solve the Sturm-Liouville-type differential equation with Dirichlet boundary conditions. We employ a newly constructed locally supported wavelet basis based on Hermite \({\mathcal {C}}^1\)-quadratic splines over a regular grid. The new Hermite quadratic wavelet basis has been carefully adapted to the interval [0, 1] for improved computational simplicity. We validate our approach by presenting numerical examples at the end of this paper.

Medical image processing is one of the most challenging tasks in computer vision. Many obstacles are encountered when performing automatic lesion segmentation in medical images, such as: shadows, artifacts, image quality, etc. This work aims to automatically segment a variety of medical images, overcoming all such drawbacks. To achieve this goal, we propose a combination of two convolutional neural networks and a deformable model using the spline approximation functions. More specifically, we introduce an automatic parametric active contour method based on an energy functional that combines the auto-encoder model for texture feature analysis and the U-net model for edge feature localization and extraction. The robustness of the proposed deep snake model is evaluated using three different datasets of brain MRI images, breast ultrasound images, and Foot Ulcer Segmentation Challenge 2021. Obtained results confirm that our proposed model improves segmentation performance and is robust to medical image segmentation challenges.