Multiresolution Methods in Scattered Data Modelling

This application-oriented work concerns the design of efficient, robust and reliable algorithms for the numerical simulation of multiscale phenomena. To this end, various modern techniques from scattered data modelling, such as splines over triangulations and radial basis functions, are combined with customized adaptive strategies, which are developed individually in this work. The resulting multiresolution methods include thinning algorithms, multi­ levelapproximation schemes, and meshfree discretizations for transport equa­ tions. The utility of the proposed computational methods is supported by their wide range of applications, such as image compression, hierarchical sur­ face visualization, and multiscale flow simulation. Special emphasis is placed on comparisons between the various numerical algorithms developed in this work and comparable state-of-the-art methods. To this end, extensive numerical examples, mainly arising from real-world applications, are provided. This research monograph is arranged in 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 provides a preliminary discussion on basic concepts, tools and principles of multiresolution methods, scattered data modelling, multilevel methods and adaptive irregular sampling. Relevant algorithms and data structures, such as triangulation methods, heaps, and quadtrees, are then introduced in Chapter 2.

this Application-oriented Work Concerns The Design Of Efficient, Robust And Reliable Algorithms For The Numerical Simulation Of Multiscale Phenomena. To This End, Various Modern Techniques From Scattered Data Modelling, Such As Splines Over Triangulations And Radial Basis Functions, Are Combined With Customized Adaptive Strategies. The Resulting Multiresolution Methods Are Thinning Algorithms, Multilevel Approximation Schemes, And Meshfree Discretizations For Transport Equations. The Utility Of The Algorithmic Approach Taken In This Research Is Supported By The Wide Range Of Applications, Including Image Compression, Hierarchical Surface Visualization, And Multiscale Flow Simulation. Special Emphasis Is Placed On Comparisons Between The Various Numerical Algorithms Developed In This Work And Comparable State-of-the-art Methods.

This application-oriented work is concerned with the design of efficient, robust and reliable methods for modelling multiscale phenomena. To this end, various modern techniques from scattered data modelling are used, such as triangulation methods and radial basis functions. The new methods, proposed in this work, are thinning algorithms, multilevel approximation schemes, and meshless methods for transport equations. The utility of the algorithmic approach taken in this research is supported by the wide range of relevant applications, including digital image compression, hierarchical surface visualization, and fluid flow modelling Front Matter....Pages I-XII Introduction....Pages 1-6 Algorithms and Data Structures....Pages 7-30 Radial Basis Functions....Pages 31-65 Thinning Algorithms....Pages 67-125 Multilevel Approximation Schemes....Pages 127-141 Meshfree Methods for Transport Equations....Pages 143-169 Back Matter....Pages 171-188 Much of the following discussion in this work relies on standard tools from computational geometry, approximation theory, and computer programming.