Nonlinear numerical techniques for the processing of data with discontinuities

[SPA] En esta tesis de doctorado, hemos intentado diseñar algoritmos capaces de manejar datos discontinuos. Hemos centrado nuestra atención en tres aplicaciones principales: • Integración numérica más términos de corrección. En esta parte de la tesis, construimos y analizamos una nueva técnica no lineal que permite obtener integraciones numéricas precisas de cualquier orden utilizando datos que contienen discontinuidades, y cuando el integrando solo se conoce en puntos de la malla. La novedad de la técnica consiste en la inclusión de términos de corrección con una expresión cerrada que depende del tamaño de los saltos de la función y sus derivadas en las discontinuidades, cuya posición se supone conocida. La adición de estos términos permite recuperar la precisión de las formulas clásicas de integración numérica cerca de las discontinuidades, ya que estos términos de corrección tienen en cuenta el error que cometen las formulas clásicas de integración hasta su precisión en las zonas de suavidad de los datos. Por lo tanto, los términos de corrección se pueden agregar durante la integración o como un post-proceso, lo cual es útil si el cálculo principal de la integral ya se ha realizado utilizando fórmulas clásicas. Durante nuestra investigación, logramos concluir varios experimentos numéricos que confirmaron las conclusiones teóricas alcanzadas. Los resultados de esta parte de la tesis se incluyeron en el artículo [1], publicado en la revista Mathematics and Computers in Simulation, una revista internacional que pertenece al primer cuartil del Journal of Citation Reports. • Interpolación de Hermite más términos de corrección. Esta técnica (sin términos de corrección) se utiliza clásicamente para reconstruir datos suaves cuando la función y sus derivadas de primer orden están disponibles en ciertos nodos. Si las derivadas de primer orden no están disponibles, es fácil establecer un sistema de ecuaciones imponiendo algunas condiciones de regularidad sobre los nodos. Este proceso conduce a la construcción de un spline de Hermite. El problema del spline de Hermite descrito es que se pierde la precisión si los datos contienen singularidades (nos centraremos fundamentalmente en discontinuidades en la función o en la primera derivada, aunque también analizaremos que ocurre cuando hay discontinuidades en la segunda derivada). La consecuencia es la aparición de oscilaciones, si hay una discontinuidad abrupta en la función, que afecta globalmente la precisión del spline, o el suavizado de las singularidades, si las discontinuidades están en las derivadas de la función. Nuestro objetivo en esta parte de la tesis es la construcción y análisis de una nueva técnica que permite el cálculo preciso de derivadas de primer orden de una función cerca de las singularidades utilizando un spline cúbico de Hermite. La idea es corregir el sistema de ecuaciones del spline para alcanzar la precisión deseada incluso cerca de las singularidades. Una vez que hemos calculado las derivadas de primer orden con suficiente precisión, se agrega un término de corrección al spline de Hermite en los intervalos que contienen una singularidad. El objetivo es reconstruir funciones suaves a trozos con precisión O(h 4 ) incluso cerca de las singularidades. El proceso de adaptación requerirá algún conocimiento sobre la posición del salto, así como del tamaño de los saltos en la función y algunas derivadas en dicha posición. Esta técnica puede usarse como post-proceso, donde agregamos un término de corrección al spline cúbico de Hermite clásico. Durante nuestra investigación, obtuvimos pruebas para la precisión y regularidad del spline corregido y sus derivadas. También analizamos el mecanismo que elimina el fenómeno Gibbs cerca del salto en la función. Además, también realizamos varios experimentos numéricos que confirmaron los resultados teóricos obtenidos. Los resultados de esta parte de la tesis se incluyeron en el artículo [2], publicado en la revista Journal of Scientific Computing, una revista internacional que pertenece al primer cuartil del Journal of Citation Reports. • Super resolución. Aunque se presenta en ´ultima posición, este tema marcó el comienzo de esta tesis, donde centramos nuestra atención en algoritmos de multiresolución. La super resolución busca mejorar la calidad de imágenes y videos con baja resolución agregando detalles más finos, lo que resulta en una salida más nítida y clara. Esta parte de la tesis es muy breve y solo trata de reflejar el trabajo que se realizó para obtener el D.E.A., ya que poco después centramos nuestra atención en otras líneas de investigación que aparentaban ser algo más prometedoras para la elaboración de esta tesis.[ENG] In this PhD thesis we have tried to design algorithms capable of dealing with discontinuous data. We have centred our attention in three main applications: • Numerical integration plus correction terms. In this part of the thesis we constructed and analyzed a new nonlinear technique that allows obtaining accurate numerical integrations of any order using data that contains discontinuities, and when the integrand is only known at grid points. The novelty of the technique consists in the inclusion of correction terms with a closed expression that depends on the size of the jumps of the function and its derivatives at the discontinuities, that are supposed to be known. The addition of these terms allows recovering the accuracy of classical numerical integration formulas close to the discontinuities, as these correction terms account for the error that the classical integration formulas commit up to their accuracy at smooth zones. Thus, the correction terms can be added during the integration or as post-processing, which is useful if the main calculation of the integral has been already done using classical formulas. During our research, we managed to conclude several numerical experiments that confirmed the theoretical conclusions reached. The results of this part of the thesis were included in the article [1] published in the journal Mathematics and Computers in Simulation, an international journal that belongs to the first quartile of the Journal of Citations Report. • Hermite interpolation plus correction terms. This technique (without correction terms) is classically used to reconstruct smooth data when the function and its first order derivatives are available at certain nodes. If first order derivatives are not available, it is easy to set a system of equations imposing some regularity conditions at the data nodes in order to obtain them. This process leads to the construction of a Hermite spline. The problem of the described Hermite splines is that the accuracy is lost if the data contains singularities (we will center our attention on discontinuities in the function or in the first derivative, although we will also analyze what happens when there are discontinuities in the second derivative). The consequence is the appearance of oscillations, if there is a jump discontinuity in the function, that globally a↵ects the accuracy of the spline, or the smearing of singularities, if the discontinuities are in the derivatives of the function.Our objective in this part of the thesis is devoted to the construction and analysis of a new technique that allows for the computation of accurate first order derivatives of a function close to singularities using a cubic Hermite spline. The idea is to correct the system of equations of the spline in order to attain the desired accuracy even close to the singularities. Once we have computed the first order derivatives with enough accuracy, a correction term is added to the Hermite spline in the intervals that contain a singularity. The aim is to reconstruct piecewise smooth functions with O(h 4 ) accuracy even close to the singularities. The process of adaption will require some knowledge about the position of the singularity and the jumps of the function and some of its derivatives at the singularity. The whole process can be used as a post-processing, where a correction term is added to the classical cubic Hermite spline. During our research, we obtained proofs for the accuracy and regularity of the corrected spline and its derivatives. We also analysed the mechanism that eliminates the Gibbs phenomenon close to jump discontinuities in the function. In addition, we also performed several numerical experiments that confirmed the theoretical results obtained. The results of this part of the thesis were included in the article [2] published in the journal Journal of Scientific Computing, an international journal that belongs to the first quartile of the Journal of Citations Report. • Super resolution. While it is presented in the last position, this marked the beginning of this thesis, where we focused our attention on multi-resolution algorithms. Super resolution seeks to enhance the quality of low-resolution images and videos by adding finer details, resulting in a sharper and clearer output. These algorithms operate by analyzing different levels of image data and combining them to create a higher-resolution version. Applications for these algorithms can be found across industries, including surveillance, medical imaging, and media, to improve visual fidelity. Although the study of super resolution was the starting point of the thesis, we soon shifted our focus to the study of other algorithms in the context of numerical approximation. These alternative approaches proved to be more promising in terms of results that could be published. Nevertheless, this first part of the research served to obtain the D.E.A.Escuela Internacional de Doctorado de la Universidad Politécnica de CartagenaUniversidad Politécnica de CartagenaPrograma Doctorado en Tecnologías Industriale

An extended Filon--Clenshaw--Curtis method for high-frequency wave scattering problems in two dimensions

We study the efficient approximation of integrals involving Hankel functions of the first kind which arise in wave scattering problems on straight or convex polygonal boundaries. Filon methods have proved to be an effective way to approximate many types of highly oscillatory integrals, however finding such methods for integrals that involve non-linear oscillators and frequency-dependent singularities is subject to a significant amount of ongoing research. In this work, we demonstrate how Filon methods can be constructed for a class of integrals involving a Hankel function of the first kind. These methods allow the numerical approximation of the integral at uniform cost even when the frequency $\omega$ is large. In constructing these Filon methods we also provide a stable algorithm for computing the Chebyshev moments of the integral based on duality to spectral methods applied to a version of Bessel's equation. Our design for this algorithm has significant potential for further generalisations that would allow Filon methods to be constructed for a wide range of integrals involving special functions. These new extended Filon methods combine many favourable properties, including robustness in regard to the regularity of the integrand and fast approximation for large frequencies. As a consequence, they are of specific relevance to applications in wave scattering, and we show how they may be used in practice to assemble collocation matrices for wavelet-based collocation methods and for hybrid oscillatory approximation spaces in high-frequency wave scattering problems on convex polygonal shapes

Wavelet Methods for the Solutions of Partial and Fractional Differential Equations Arising in Physical Problems

The subject of fractional calculus has gained considerable popularity and importance during the past three decades or so, mainly due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It deals with derivatives and integrals of arbitrary orders. The fractional derivative has been occurring in many physical problems, such as frequency-dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PI D controller for the control of dynamical systems etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, control theory, neutron point kinetic model, anomalous diffusion, Brownian motion, signal and image processing, fluid dynamics and material science are well described by differential equations of fractional order. Generally, nonlinear partial differential equations of fractional order are difficult to solve. So for the last few decades, a great deal of attention has been directed towards the solution (both exact and numerical) of these problems. The aim of this dissertation is to present an extensive study of different wavelet methods for obtaining numerical solutions of mathematical problems occurring in disciplines of science and engineering. This present work also provides a comprehensive foundation of different wavelet methods comprising Haar wavelet method, Legendre wavelet method, Legendre multi-wavelet methods, Chebyshev wavelet method, Hermite wavelet method and Petrov-Galerkin method. The intension is to examine the accuracy of various wavelet methods and their efficiency for solving nonlinear fractional differential equations. With the widespread applications of wavelet methods for solving difficult problems in diverse fields of science and engineering such as wave propagation, data compression, image processing, pattern recognition, computer graphics and in medical technology, these methods have been implemented to develop accurate and fast algorithms for solving integral, differential and integro-differential equations, especially those whose solutions are highly localized in position and scale. The main feature of wavelets is its ability to convert the given differential and integral equations to a system of linear or nonlinear algebraic equations, which can be solved by numerical methods. Therefore, our main focus in the present work is to analyze the application of wavelet based transform methods for solving the problem of fractional order partial differential equations. The introductory concept of wavelet, wavelet transform and multi-resolution analysis (MRA) have been discussed in the preliminary chapter. The basic idea of various analytical and numerical methods viz. Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM), First Integral Method (FIM), Optimal Homotopy Asymptotic Method (OHAM), Haar Wavelet Method, Legendre Wavelet Method, Chebyshev Wavelet Method and Hermite Wavelet Method have been presented in chapter 1. In chapter 2, we have considered both analytical and numerical approach for solving some particular nonlinear partial differential equations like Burgers’ equation, modified Burgers’ equation, Huxley equation, Burgers-Huxley equation and modified KdV equation, which have a wide variety of applications in physical models. Variational Iteration Method and Haar wavelet Method are applied to obtain the analytical and numerical approximate solution of Huxley and Burgers-Huxley equations. Comparisons between analytical solution and numerical solution have been cited in tables and also graphically. The Haar wavelet method has also been applied to solve Burgers’, modified Burgers’, and modified KdV equations numerically. The results thus obtained are compared with exact solutions as well as solutions available in open literature. Error of collocation method has been presented in this chapter. Methods like Homotopy Perturbation Method (HPM) and Optimal Homotopy Asymptotic Method (OHAM) are very powerful and efficient techniques for solving nonlinear PDEs. Using these methods, many functional equations such as ordinary, partial differential equations and integral equations have been solved. We have implemented HPM and OHAM in chapter 3, in order to obtain the analytical approximate solutions of system of nonlinear partial differential equation viz. the Boussinesq-Burgers’ equations. Also, the Haar wavelet method has been applied to obtain the numerical solution of BoussinesqBurgers’ equations. Also, the convergence of HPM and OHAM has been discussed in this chapter. The mathematical modeling and simulation of systems and processes, based on the description of their properties in terms of fractional derivatives, naturally leads to differential equations of fractional order and the necessity to solve such equations. The mathematical preliminaries of fractional calculus, definitions and theorems have been presented in chapter 4. Next, in this chapter, the Haar wavelet method has been analyzed for solving fractional differential equations. The time-fractional Burgers-Fisher, generalized Fisher type equations, nonlinear time- and space-fractional Fokker-Planck equations have been solved by using two-dimensional Haar wavelet method. The obtained results are compared with the Optimal Homotopy Asymptotic Method (OHAM), the exact solutions and the results available in open literature. Comparison of obtained results with OHAM, Adomian Decomposition Method (ADM), VIM and Operational Tau Method (OTM) has been demonstrated in order to justify the accuracy and efficiency of the proposed schemes. The convergence of two-dimensional Haar wavelet technique has been provided at the end of this chapter. In chapter 5, the fractional differential equations such as KdV-Burger-Kuramoto (KBK) equation, seventh order KdV (sKdV) equation and Kaup-Kupershmidt (KK) equation have been solved by using two-dimensional Legendre wavelet and Legendre multi-wavelet methods. The main focus of this chapter is the application of two-dimensional Legendre wavelet technique for solving nonlinear fractional differential equations like timefractional KBK equation, time-fractional sKdV equation in order to demonstrate the efficiency and accuracy of the proposed wavelet method. Similarly in chapter 6, twodimensional Chebyshev wavelet method has been implemented to obtain the numerical solutions of the time-fractional Sawada-Kotera equation, fractional order Camassa-Holm equation and Riesz space-fractional sine-Gordon equations. The convergence analysis has been done for these wavelet methods. In chapter 7, the solitary wave solution of fractional modified Fornberg-Whitham equation has been attained by using first integral method and also the approximate solutions obtained by optimal homotopy asymptotic method (OHAM) are compared with the exact solutions acquired by first integral method. Also, the Hermite wavelet method has been implemented to obtain approximate solutions of fractional modified Fornberg-Whitham equation. The Hermite wavelet method is implemented to system of nonlinear fractional differential equations viz. the fractional Jaulent-Miodek equations. Convergence of this wavelet methods has been discussed in this chapter. Chapter 8 emphasizes on the application of Petrov-Galerkin method for solving the fractional differential equations such as the fractional KdV-Burgers’ (KdVB) equation and the fractional Sharma-TassoOlver equation with a view to exhibit the capabilities of this method in handling nonlinear equation. The main objective of this chapter is to establish the efficiency and accuracy of Petrov-Galerkin method in solving fractional differential equtaions numerically by implementing a linear hat function as the trial function and a quintic B-spline function as the test function. Various wavelet methods have been successfully employed to numerous partial and fractional differential equations in order to demonstrate the validity and accuracy of these procedures. Analyzing the numerical results, it can be concluded that the wavelet methods provide worthy numerical solutions for both classical and fractional order partial differential equations. Finally, it is worthwhile to mention that the proposed wavelet methods are promising and powerful methods for solving fractional differential equations in mathematical physics. This work also aimed at, to make this subject popular and acceptable to engineering and science community to appreciate the universe of wonderful mathematics, which is in between classical integer order differentiation and integration, which till now is not much acknowledged, and is hidden from scientists and engineers. Therefore, our goal is to encourage the reader to appreciate the beauty as well as the usefulness of these numerical wavelet based techniques in the study of nonlinear physical system

A practical method for computing with piecewise Chebyshevian splines

A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial splines, the related parametric curves exhibit the desired properties of convex hull inclusion, variation diminution and intuitive relation between the curve shape and the location of the control points. For a good-for-design space, in this paper we construct a set of functions, called transition functions, which allow for efficient computation of the B-spline basis, even in the case of nonuniform and multiple knots. Moreover, we show how the spline coefficients of the representations associated with a refined knot partition and with a raised order can conveniently be expressed by means of transition functions. This result allows us to provide effective procedures that generalize the classical knot insertion and degree raising algorithms for polynomial splines. We further discuss how the approach can straightforwardly be generalized to deal with geometrically continuous piecewise Chebyshevian splines as well as with splines having section spaces of different dimensions. From a numerical point of view, we show that the proposed evaluation method is easier to implement and has higher accuracy than other existing algorithms

Cubic hat-functions approximation for linear and nonlinear fractional integral-differential equations with weakly singular kernels

In the current study, a new numerical algorithm is presented to solve a class of nonlinear fractional integral-differential equations with weakly singular kernels. Cubic hat functions (CHFs) and their properties are introduced for the first time. A new fractional-order operational matrix of integration via CHFs is presented. Utilizing the operational matrices of CHFs, the main problem is transformed into a number of trivariate polynomial equations. Error analysis and the convergence of the proposed method are evaluated, and the convergence rate is addressed. Ultimately, three examples are provided to illustrate the precision and capabilities of this algorithm. The numerical results are presented in some tables and figures

Wavelet Analysis on the Sphere

The goal of this monograph is to develop the theory of wavelet harmonic analysis on the sphere. By starting with orthogonal polynomials and functional Hilbert spaces on the sphere, the foundations are laid for the study of spherical harmonics such as zonal functions. The book also discusses the construction of wavelet bases using special functions, especially Bessel, Hermite, Tchebychev, and Gegenbauer polynomials