Meshfree Approximation Methods with MATLAB

Transcription

1 Interdisciplinary Mathematical Sc Meshfree Approximation Methods with MATLAB Gregory E. Fasshauer Illinois Institute of Technology, USA Y f? World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONGKONG TAIPEI CHENNAI

2 Contents Preface vii 1. Introduction Motivation: Scattered Data Interpolation in M s The Scattered Data Interpolation Problem Example: Interpolation with Distance Matrices Some Historical Remarks Radial Basis Function Interpolation in MATLAB Radial (Basis) Functions Radial Basis Function Interpolation Positive Definite Functions Positive Definite Matrices and Functions Integral Characterizations for (Strictly) Positive Definite Functions Bochncr's Theorem Extensions to Strictly Positive Definite Functions Positive Definite Radial Functions Examples of Strictly Positive Definite Radial Functions Example 1: Gaussians Example 2: Laguerre-Gaussians Example 3: Poisson Radial Functions Example 4: Matern Functions Example 5: Generalized Inverse Multiquadrics Example 6: Truncated Power Functions Example 7: Potentials and Whittaker Radial Functions Example 8: Integration Against Strictly Positive Definite Kernels 45 xi

3 '- xii Meshfree Approximation Methods with MATLAB 4.9 Summary Completely Monotone and Multiply Monotone Functions Completely Monotone Functions Multiply Monotone Functions Scattered Data Interpolation with Polynomial Precision Interpolation with Multivariate Polynomials Example: Reproduction of Linear Functions Using Gaussian RBFs Scattered Data Interpolation with More General Polynomial Precision Conditionally Positive Definite Matrices and Reproduction of Constant Functions Conditionally Positive Definite Functions Conditionally Positive Definite Functions Defined Conditionally Positive Definite Functions and Generalized Fourier Transforms Examples of Conditionally Positive Definite Functions Example 1: Generalized Multiquadrics Example 2: Radial Powers Example 3: Thin Plate Splines Conditionally Positive Definite Radial Functions Conditionally Positive Definite Radial Functions and Completely Monotone Functions Conditionally Positive Definite Radial Functions and Multiply Monotone Functions Some Special Properties of Conditionally Positive Definite Functions of Order One Miscellaiieous Theory: Other Norms and Scattered Data Fitting on Manifolds Conditionally Positive Definite Functions and p-norms Scattered Data Fitting on Manifolds Remarks Compactly Supported Radial Basis Functions Operators for Radial Functions and Dimension Walks Wendland's Compactly Supported Functions 87

4 Contents xiii 11.3 Wu's Compactly Supported Functions Oscillatory Compactly Supported Functions Other Compactly Supported Radial Basis Functions Interpolation with Compactly Supported RBFs in MATLAB Asscmbly of the Sparse Interpolation Matrix Numerical Experiments with CSRBFs Reproducing Kernel Hubert Spaces and Native Spaces for Strictly Positive Definite Functions Reproducing Kernel Hubert Spaces Native Spaces for Strictly Positive Definite Functions Examples of Native Spaces for Populär Radial Basic Functions The Power Function and Native Space Error Estimates Fill Distance and Approximation Orders Lagrange Form of the Interpolant and Cardinal Basis Functions The Power Function Generic Error Estimates for Functions in A/$ ( 2) Error Estimates in Terms of the Fill Distance Refined and Improved Error Bounds Native Space Error Bounds for Specific Basis Functions Infinitely Smooth Basis Functions Basis Functions with Finite Smoothness Improvements for Native Space Error Bounds Error Bounds for Functions Outside the Native Space Error Bounds for Stationary Approximation Convergence with Respect to the Shape Parameter Polynomial Interpolation as the Limit of RBF Interpolation Stability and Trade-Off Principles Stability and Conditioning of Radial Basis Function Interpolants Trade-Off Principle I: Accuracy vs. Stability Trade-Off Principle II: Accuracy and Stability vs. Problem Size Trade-Off Principle III: Accuracy vs. Efficiency Numerical Evidence for Approximation Order Results Interpolation for e > Choosing a Good Shape Parameter via Trial and Error.. 142

5 ' - xiv Meshfree Approximation Methods with MATLAB The Power Function as Indicator for a Good Shape Parameter Choosing a Good Shape Parameter via Cross Validation The Contour-Pade Algorithm Summary Non-stationary Interpolation Stationary Interpolation The Optimality of RBF Interpolation The Connection to Optimal Recovery Orthogonality in Reproducing Kernel Hubert Spaces Optimality Theorem I Optimality Theorem II Optimality Theorem III Least Squares RBF Approximation with MATLAB Optimal Recovery Revisited Regularized Least Squares Approximation Least Squares Approximation When RBF Centers Differ from Data Sites Least Squares Smoothing of Noisy Data Theory for Least Squares Approximation Well-Posedness of RBF Least Squares Approximation Error Bounds for Least Squares Approximation Adaptive Least Squares Approximation Adaptive Least Squares using Knot Insertion Adaptive Least Squares using Knot Removal Some Numerical Examples Moving Least Squares Approximation Discrete Weighted Least Squares Approximation Standard Interpretation of MLS Approximation The Backus-Gilbert Approach to MLS Approximation Equivalence of the Two Formulations of MLS Approximation Duality and Bi-Orthogonal Bases Standard MLS Approximation as a Constrained Quadratic Optimization Problem Remarks Examples of MLS Generating Functions 205

6 : Contents xv 23.1 Shepard's Method MLS Approximation with Nontrivial Polynomial Reproduction MLS Approximation with MATLAB Approximation with Shepard's Method MLS Approximation with Linear Reproduction Plots of Basis-Dual Basis Pairs Error Bounds for Moving Least Squares Approximation Approximation Order of Moving Least Squares Approximate Moving Least Squares Approximation High-order Shepard Methods via Moment Conditions Approximate Approximation Construction of Generating Functions for Approximate MLS Approximation Numerical Experiments for Approximate MLS Approximation Univariate Experiments Bivariate Experiments Fast Fourier Transforms NFFT Approximate MLS Approximation via Non-uniform Fast Fourier Transforms Partition of Unity Methods Theory Partition of Unity Approximation with MATLAB Approximation of Point Cloud Data in 3D A General Approach via Implicit Surfaces An Illustration in 2D A Simplistic Implementation in 3D via Partition of Unity Approximation in MATLAB Fixed Level Residual Iteration Iterative Refmement Fixed Level Iteration Modifications of the Basic Fixed Level Iteration Algorithm Iterated Approximate MLS Approximation in MATLAB Iterated Shepard Approximation 274

7 ' xvi Meshfree Approximation Methods with MATLAB 32. Multilevel Iteration Stationary Multilevel Interpolation A MATLAB Implementation of Stationary Multilevel Interpolation Stationary Multilevel Approximation Multilevel Interpolation with Globally Supported RBFs Adaptive Iteration A Greedy Adaptive Algorithm The Faul-Powell Algorithm Improving the Condition Number of the Interpolation Matrix Preconditioning: Two Simple Examples Early Preconditioners Preconditioned GMRES via Approximate Cardinal Functions Change of Basis Effect of the "Better" Basis on the Condition Number of the Interpolation Matrix Effect of the "Better" Basis on the Accuracy of the Interpolant Other Efficient Numerical Methods The Fast Multipole Method Fast Tree Codes Domain Decornposition Generalized Hermite Interpolation The Generalized Hermite Interpolation Problem Motivation for the Symmetrie Formulation RBF Hermite Interpolation in MATLAB Solving Elliptic Partial Differential Equations via RBF Collocation Kansa's Approach An Hermite-based Approach Error Bounds for Symmetrie Collocation Other Issues Non-Symmetric RBF Collocation in MATLAB Kansa's Non-Symmetric Collocation Method Symmetrie RBF Collocation in MATLAB 365

8 Contents xvii 40.1 Symmetrie Collocation Method Summarizing Remarks on the Symmetrie and Non-Symmetric Collocation Methods Collocation with CSRBFs in MATLAB Collocation with Compactly Supported RBFs Multilevel RBF Collocation Using Radial Basis Functions in Pseudospectral Mode Differentiation Matrices PDEs with Boundary Conditions via Pseudospectral Methods A Non-Symmetric RBF-based Pseudospectral Method A Symmetrie RBF-based Pseudospectral Method A Unified Discussion Summary RBF-PS Methods in MATLAB Computing the RBF-Differentiation Matrix in MATLAB Solution of a 1-D Transport Equation Use of the Contour-Pade Algorithm with the PS Approach Solution of the 1D Transport Equation Revisited Computation of Higher-Order Derivatives Solution of the Allen-Cahn Equation Solution of a 2D Helmholtz Equation Solution of a 2D Laplace Equation with Piecewise Boundary Conditions Summary RBF Galerkin Methods An Elliptic PDE with Neumann Boundary Conditions A Convergence Estimate A Multilevel RBF Galerkin Algorithm RBF Galerkin Methods in MATLAB 423 Appendix A Useful Facts from Discrete Mathematics 427 A.l Haiton Points 427 A.2 kd-trees 428 Appendix B Useful Facts from Analysis 431 B.l Some Important Concepts from Measure Theory 431 B.2 A Brief Summary of Integral Transforms 432

9 - xviii Meshfree Approximation Methods with MATLAB B.3 The Schwartz Space and the Generalized Fourier Transform Appendix C Additional Computer Programs 435 C.l MATLAB Programs 435 C.2 Maple Programs 440 Appendix D Catalog of RBFs with Derivatives 443 D.l Generic Derivatives 443 D.2 Formulas for Specific Basic Functions 444 D.2.1 Globally Supported, Strictly Positive Definite Functions. 444 D.2.2 Globally Supported, Strictly Conditionally Positive Definite Functions of Order D.2.3 Globally Supported, Strictly Conditionally Positive Definite Functions of Order D.2.4 Globally Supported, Strictly Conditionally Positive Definite Functions of Order D.2.5 Globally Supported, Strictly Conditionally Positive Definite Functions of Order D.2.6 Globally Supported, Strictly Positive Definite and Oscillatory Functions 447 D.2.7 Compactly Supported, Strictly Positive Definite Functions 448 Bibliography 451 Index 491