Multiresolution methods in scattered data modelling

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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.