## Multiresolution Methods in Scattered Data ModellingThis 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 |

180 | |

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### Common terms and phrases

adaptive thinning algorithm Algorithm 16 AMMoC anticipated error approximation error approximation quality basis function interpolation bounds cell Chapter coding computational costs construction convex triangulation covering radius CPDd(m data hierarchy data structure Delaunay triangulation denote discussion edge equation exchange Algorithm ﬁeld ﬁner ﬂow ﬂuid FrontSim greedy thinning heap condition heapify heapsort Hurrungane image compression initial interpolation scheme irregular sampling Iske let us ﬁrst locally optimal M.S. Floater meshfree method minimal Moreover multilevel approximation schemes multilevel interpolation multiscale node non-adaptive thinning Note numerical stability point pairs polyharmonic spline polyharmonic spline interpolation polynomial priority queue problem PSNR quadtree radial basis functions recursively reﬁnement removable point removal criterion resulting ry,X satisﬁes satisfying scattered data ﬁltering scattered data modelling Section semi-Lagrangian signiﬁcance value signiﬁcant pixels simulation speciﬁc SPIHT spline interpolation splitting subset Y C X Theorem thin plate spline tion update Voronoi diagram Voronoi tile wavelets