Multiresolution Methods in Scattered Data Modelling

by Iske, Armin

• ISBN13: 9783540204794
• ISBN10: 3540204792
• Format: Hardcover
• Copyright: 2004-05-14
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Introduction	1	(6)
Scattered Data Modelling	1	(2)
Multiresolution Methods	3	(2)
Multilevel Methods	5	(1)
Adaptive Irregular Sampling	6	(1)
Algorithms and Data Structures	7	(24)
Triangulation Methods	7	(2)
Delaunay Triangulations	9	(4)
Voronoi Diagrams	13	(2)
Data-Dependent Triangulations	15	(5)
Heaps and Priority Queues	20	(6)
Quadtrees	26	(5)
Radial Basis Functions	31	(36)
Interpolation	32	(5)
Conditionally Positive Definite Functions	37	(3)
Optimal Recovery	40	(2)
Pointwise Optimality	42	(4)
Error Estimates	46	(1)
Numerical Stability	47	(1)
Uncertainty Principle	48	(1)
Polyharmonic Splines	49	(10)
Optimal Point Sampling	59	(2)
Least Squares Approximation	61	(6)
Thinning Algorithms	67	(60)
Preliminary Remarks	68	(1)
Generic Formulation	69	(1)
Non-Adaptive Thinning	70	(8)
Scattered Data Filtering	78	(12)
Adaptive Thinning	90	(13)
Adaptive Thinning in Digital Image Compression	103	(24)
Multilevel Approximation Schemes	127	(16)
Generic Formulation	128	(1)
Multilevel Interpolation	129	(3)
Adaptive Multilevel Approximation	132	(4)
Hierarchical Surface Visualization	136	(7)
Meshfree Methods for Transport Equations	143	(28)
Transport Equations	144	(2)
Meshfree Method of Characteristics	146	(4)
Adaption Rules	150	(2)
Multiscale Flow Simulation	152	(19)
Bibliography	171	(10)
Index	181