Multiresolution Methods in Scattered Data Modelling

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Springer Science & Business Media, Apr 1, 2004 - Mathematics - 188 pages
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.
 

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Contents

Introduction
1
12 Multiresolution Methods
3
13 Multilevel Methods
5
14 Adaptive Irregular Sampling
6
Algorithms and Data Structures
7
22 Delaunay Triangulations
9
23 Voronoi Diagrams
13
24 DataDependent Triangulations
15
310 Least Squares Approximation
61
Thinning Algorithms
67
41 Preliminary Remarks
68
42 Generic Formulation
69
43 NonAdaptive Thinning
70
44 Scattered Data Filtering
78
45 Adaptive Thinning
90
46 Adaptive Thinning in Digital Image Compression
103

25 Heaps and Priority Queues
20
26 Quadtrees
26
Radial Basis Functions
31
31 Interpolation
32
32 Conditionally Positive Definite Functions
37
33 Optimal Recovery
40
34 Pointwise Optimality
42
35 Error Estimates
46
36 Numerical Stability
47
37 Uncertainty Principle
48
38 Polyharmonic Splines
49
39 Optimal Point Sampling
59
Multilevel Approximation Schemes
127
51 Generic Formulation
128
52 Multilevel Interpolation
129
53 Adaptive Multilevel Approximation
132
54 Hierarchical Surface Visualization
136
Meshfree Methods for Transport Equations
143
61 Transport Equations
144
62 Meshfree Method of Characteristics
146
63 Adaption Rules
150
64 Multiscale Flow Simulation
152
Index
180
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