Focusing on the theory and numerics of approximation methods, this textbook merges classical concepts with contemporary advancements in mathematical signal processing. It emphasizes a constructive approach, highlighting the development of numerical algorithms essential for data analysis. This blend of traditional and modern techniques makes it a valuable resource for understanding both foundational and cutting-edge topics in the field.
Various challenges concerning the production workflow are discussed in detail.The thirteen chapters of this joint work, authored by international experts from academic and industrial institutions, include survey papers of expository character as well as original research articles.
Approximation methods are vital in many challenging applications of computational science and engineering. This is a collection of papers from world experts in a broad variety of relevant applications, including pattern recognition, machine learning, multiscale modelling of fluid flow, metrology, geometric modelling, tomography, signal and image processing. It documents recent theoretical developments which have lead to new trends in approximation, it gives important computational aspects and multidisciplinary applications, thus making it a perfect fit for graduate students and researchers in science and engineering who wish to understand and develop numerical algorithms for the solution of their specific problems. An important feature of the book is that it brings together modern methods from statistics, mathematical modelling and numerical simulation for the solution of relevant problems, with a wide range of inherent scales. Contributions of industrial mathematicians, including representatives from Microsoft and Schlumberger, foster the transfer of the latest approximation methods to real-world applications.
This application-oriented work focuses on designing efficient, robust, and reliable algorithms for numerical simulation of multiscale phenomena. It integrates modern techniques from scattered data modeling, including splines over triangulations and radial basis functions, with customized adaptive strategies developed within the text. The resulting multiresolution methods encompass thinning algorithms, multilevel approximation schemes, and meshfree discretizations for transport equations. The proposed computational methods demonstrate utility across various applications, such as image compression, hierarchical surface visualization, and multiscale flow simulation. A significant emphasis is placed on comparing the developed numerical algorithms with state-of-the-art methods, supported by extensive numerical examples from real-world scenarios. The research is structured into six chapters: 1. Introduction; 2. Algorithms and Data Structures; 3. Radial Basis Functions; 4. Thinning Algorithms; 5. Multilevel Approximation Schemes; 6. Meshfree Methods for Transport Equations. Chapter 1 introduces basic concepts, tools, and principles of multiresolution methods, scattered data modeling, multilevel methods, and adaptive irregular sampling, while Chapter 2 presents relevant algorithms and data structures, including triangulation methods, heaps, and quadtrees.
This is the only textbook available on multiresolution methods in geometric modeling, a central topic in visualization, which is of great importance for industrial applications. Written in tutorial form, the book is introductory in character, and includes supporting exercises. Other supplementary material and software can be downloaded from the website www.ma.tum.de/primus 2001/.
Dieses Lehrbuch bietet eine anschauliche Einführung in die Theorie und Numerik der Approximation mit Bezügen zu aktuellen Anwendungen der Datenanalyse. Dabei werden klassische Themen der Approximation mit relevanten Methoden der mathematischen Signalverarbeitung verknüpft und gut nachvollziehbar erklärt. Bei den Herleitungen der verschiedenen Approximationsmethoden werden konstruktive Zugänge bevorzugt. Dies führt direkt zu numerischen Algorithmen, deren Implementierung im Detail erklärt wird. Weiterhin illustriert eine Vielzahl an Beispielen die theoretischen und numerischen Grundlagen. Das Lehrbuch behandelt u. a. folgende Themen: Bestapproximationen in normierten linearen Räumen Approximation in euklidischen Räumen Tschebyscheff-Approximation Asymptotische Resultate der Approximation Kern-basierte Approximation mit gitterfreien Methoden Approximationsmethoden der Computertomographie Neben zahlreichen Beispielen sind für die weitere Vertiefung der Kernthemen auch viele Übungsaufgaben mit Lösungshinweisen enthalten.
