简介
Meshfree approximation methods are a relatively new area of research, and there are only a few books covering it at present. Whereas other works focus almost entirely on theoretical aspects or applications in the engineering field, this book provides the salient theoretical results needed for a basic understanding of meshfree approximation methods. The emphasis here is on a hands-on approach that includes MATLAB routines for all basic operations. Meshfree approximation methods, such as radial basis function and moving least squares method, are discussed from a scattered data approximation and partial differential equations point of view. A good balance is supplied between the necessary theory and implementation in terms of many MATLAB programs, with examples and applications to illustrate key points. Used as class notes for graduate courses at Northwestern University, Illinois Institute of Technology, and Vanderbilt University, this book will appeal to both mathematics and engineering graduate students.
目录
Table Of Contents:
Preface vii
1. Introduction 1
1.1 Motivation: Scattered Data Interpolation in Rs 2
1.1.1 The Scattered Data Interpolation Problem 2
1.1.2 Example: Interpolation with Distance Matrices 4
1.2 Some Historical Remarks 13
2. Radial Basis Function Interpolation in MATLAB 17
2.1 Radial (Basis) Functions 17
2.2 Radial Basis Function Interpolation 19
3. Positive Definite Functions 27
3.1 Positive Definite Matrices and Functions 27
3.2 Integral Characterizations for (Strictly) Positive Definite Functions 31
3.2.1 Bochner's Theorem 31
3.2.2 Extensions to Strictly Positive Definite Functions 32
3.3 Positive Definite Radial Functions 33
4. Examples of Strictly Positive Definite Radial Functions 37
4.1 Example 1: Gaussians 37
4.2 Example 2: Laguerre-Gaussians 38
4.3 Example 3: Poisson Radial Functions 39
4.4 Example 4: Mat茅rn Functions 41
4.5 Example 5: Generalized Inverse Multiquadrics 41
4.6 Example 6: Truncated Power Functions 42
4.7 Example 7: Potentials and Whittaker Radial Functions 43
4.8 Example 8: Integration Against Strictly Positive Definite Kernels 45
4.9 Summary 45
5. Completely Monotone and Multiply Monotone Functions 47
5.1 Completely Monotone Functions 47
5.2 Multiply Monotone Functions 49
6. Scattered Data Interpolation with Polynomial Precision 53
6.1 Interpolation with Multivariate Polynomials 53
6.2 Example: Reproduction of Linear Functions Using Gaussian RBFs 55
6.3 Scattered Data Interpolation with More General Polynomial Precision 57
6.4 Conditionally Positive Definite Matrices and Reproduction of Constant Functions 59
7. Conditionally Positive Definite Functions 63
7.1 Conditionally Positive Definite Functions Defined 63
7.2 Conditionally Positive Definite Functions and Generalized Fourier Transforms 65
8. Examples of Conditionally Positive Definite Functions 67
8.1 Example 1: Generalized Multiquadrics 67
8.2 Example 2: Radial Powers 69
8.3 Example 3: Thin Plate Splines 70
9. Conditionally Positive Definite Radial Functions 73
9.1 Conditionally Positive Definite Radial Functions and Completely Monotone Functions 73
9.2 Conditionally Positive Definite Radial Functions and Multiply Monotone Functions 75
9.3 Some Special Properties of Conditionally Positive Definite Functions of Order One 76
10. Miscellaneous Theory: Other Norms and Scattered Data Fitting on Manifolds 79
10.1 Conditionally Positive Definite Functions and p-Norms 79
10.2 Scattered Data Fitting on Manifolds 83
10.3 Remarks 83
11. Compactly Supported Radial Basis Functions 85
11.1 Operators for Radial Functions and Dimension Walks 85
11.2 Wendland's Compactly Supported Functions 87
11.3 Wu's Compactly Supported Functions 88
11.4 Oscillatory Compactly Supported Functions 90
11.5 Other Compactly Supported Radial Basis Functions 92
12. Interpolation with Compactly Supported RBFs in MATLAB 95
12.1 Assembly of the Sparse Interpolation Matrix 95
12.2 Numerical Experiments with CSRBFs 99
13. Reproducing Kernel Hilbert Spaces and Native Spaces for Strictly Positive Definite Functions 103
13.1 Reproducing Kernel Hilbert Spaces 103
13.2 Native Spaces for Strictly Positive Definite Functions 105
13.3 Examples of Native Spaces for Popular Radial Basic Functions 108
14. The Power Function and Native Space Error Estimates 111
14.1 Fill Distance and Approximation Orders 111
14.2 Lagrange Form of the Interpolant and Cardinal Basis Functions 112
14.3 The Power Function 115
14.4 Generic Error Estimates for Functions in NΦ (Ω) 117
14.5 Error Estimates in Terms of the Fill Distance 119
15. Refined and Improved Error Bounds 125
15.1 Native Space Error Bounds for Specific Basis Functions 125
15.1.1 Infinitely Smooth Basis Functions 125
15.1.2 Basis Functions with Finite Smoothness 126
15.2 Improvements for Native Space Error Bounds 127
15.3 Error Bounds for Functions Outside the Native Space 128
15.4 Error Bounds for Stationary Approximation 130
15.5 Convergence with Respect to the Shape Parameter 132
15.6 Polynomial Interpolation as the Limit of RBF Interpolation 133
16. Stability and Trade-Off Principles 135
16.1 Stability and Conditioning of Radial Basis Function Interpolants 135
16.2 Trade-Off Principle I: Accuracy vs. Stability 138
16.3 Trade-Off Principle II: Accuracy and Stability vs. Problem Size 140
16.4 Trade-Off Principle III: Accuracy vs. Efficiency 140
17. Numerical Evidence for Approximation Order Results 141
17.1 Interpolation for epsilon -> 0 141
17.1.1 Choosing a Good Shape Parameter via Trial and Error 142
17.1.2 The Power Function as Indicator for a Good Shape Parameter 142
17.1.3 Choosing a Good Shape Parameter via Cross Validation 146
17.1.4 The Contour-Pad茅 Algorithm 151
17.1.5 Summary 152
17.2 Non-stationary Interpolation 153
17.3 Stationary Interpolation 155
18. The Optimality of REF Interpolation 159
18.1 The Connection to Optimal Recovery 159
18.2 Orthogonality in Reproducing Kernel Hilbert Spaces 160
18.3 Optimality Theorem I 162
18.4 Optimality Theorem II 163
18.5 Optimality Theorem III 164
19. Least Squares RBF Approximation with MATLAB 165
19.1 Optimal Recovery Revisited 165
19.2 Regularized Least Squares Approximation 166
19.3 Least Squares Approximation When RBF Centers Differ from Data Sites 168
19.4 Least Squares Smoothing of Noisy Data 170
20. Theory for Least Squares Approximation 177
20.1 Well-Posedness of REF Least Squares Approximation 177
20.2 Error Bounds for Least Squares Approximation 179
21. Adaptive Least Squares Approximation 181
21.1 Adaptive Least Squares using Knot Insertion 181
21.2 Adaptive Least Squares using Knot Removal 184
21.3 Some Numerical Examples 188
22. Moving Least Squares Approximation 191
22.1 Discrete Weighted Least Squares Approximation 191
22.2 Standard Interpretation of MLS Approximation 192
22.3 The Backus-Gilbert Approach to MLS Approximation 194
22.4 Equivalence of the Two Formulations of MLS Approximation 198
22.5 Duality and Bi-Orthogonal Bases 199
22.6 Standard MLS Approximation as a Constrained Quadratic Optimization Problem 202
22.7 Remarks 202
23. Examples of MLS Generating Functions 205
23.1 Shepard's Method 205
23.2 MLS Approximation with Nontrivial Polynomial Reproduction 207
24. MLS Approximation with MATLAB 211
24.1 Approximation with Shepard's Method 211
24.2 MLS Approximation with Linear Reproduction 216
24.3 Plots of Basis-Dual Basis Pairs 222
25. Error Bounds for Moving Least Squares Approximation 225
25.1 Approximation Order of Moving Least Squares 225
26. Approximate Moving Least Squares Approximation 229
26.1 High-order Shepard Methods via Moment Conditions 229
26.2 Approximate Approximation 230
26.3 Construction of Generating Functions for Approximate MLS Approximation 232
27. Numerical Experiments for Approximate MLS Approximation 237
27.1 Univariate Experiments 237
27.2 Bivariate Experiments 241
28. Fast Fourier Transforms 243
28.1 NFFT 243
28.2 Approximate MLS Approximation via Non-uniform Fast Fourier Transforms 245
29. Partition of Unity Methods 249
29.1 Theory 249
29.2 Partition of Unity Approximation with MATLAB 251
30. Approximation of Point Cloud Data in 3D 255
30.1 A General Approach via Implicit Surfaces 255
30.2 An Illustration in 2D 257
30.3 A Simplistic Implementation in 3D via Partition of Unity Approximation in MATLAB 260
31. Fixed Level Residual Iteration 265
31.1 Iterative Refinement 265
31.2 Fixed Level Iteration 267
31.3 Modifications of the Basic Fixed Level Iteration Algorithm 269
31.4 Iterated Approximate MLS Approximation in MATLAB 270
31.5 Iterated Shepard Approximation 274
32. Multilevel Iteration 277
32.1 Stationary Multilevel Interpolation 277
32.2 A MATLAB Implementation of Stationary Multilevel Interpolation 279
32.3 Stationary Multilevel Approximation 283
32.4 Multilevel Interpolation with Globally Supported RBFs 287
33. Adaptive Iteration 291
33.1 A Greedy Adaptive Algorithm 291
33.2 The Faul-Powell Algorithm 298
34. Improving the Condition Number of the Interpolation Matrix 303
34.1 Preconditioning: Two Simple Examples 304
34.2 Early Preconditioners 305
34.3 Preconditioned GMRES via Approximate Cardinal Functions 309
34.4 Change of Basis 311
34.5 Effect of the "Better" Basis on the Condition Number of the Interpolation Matrix 314
34.6 Effect of the "Better" Basis on the Accuracy of the Interpolant 316
35. Other Efficient Numerical Methods 321
35.1 The Fast Multipole Method 321
35.2 Fast Tree Codes 327
35.3 Domain Decomposition 331
36. Generalized Hermite Interpolation 333
36.1 The Generalized Hermite Interpolation Problem 333
36.2 Motivation for the Symmetric Formulation 335
37. RBF Hermite Interpolation in MATLAB 339
38. Solving Elliptic Partial Differential Equations via RBF Collocation 345
38.1 Kansa's Approach 345
38.2 An Hermite-based Approach 348
38.3 Error Bounds for Symmetric Collocation 349
38.4 Other Issues 350
39. Non-Symmetric RBF Collocation in MATLAB 353
39.1 Kansa's Non-Symmetric Collocation Method 353
40. Symmetric RBF Collocation in MATLAB 365
40.1 Symmetric Collocation Method 365
40.2 Summarizing Remarks on the Symmetric and Non-Symmetric Collocation Methods 372
41. Collocation with CSRBFs in MATLAB 375
41.1 Collocation with Compactly Supported RBFs 375
41.2 Multilevel RBF Collocation 380
42. Using Radial Basis Functions in Pseudospectral Mode 387
42.1 Differentiation Matrices 388
42.2 PDEs with Boundary Conditions via Pseudospectral Methods 390
42.3 A Non-Symmetric RBF-based Pseudospectral Method 391
42.4 A Symmetric RBF-based Pseudospectral Method 394
42.5 A Unified Discussion 396
42.6 Summary 398
43. RBF-PS Methods in MATLAB 401
43.1 Computing the RBF-Differentiation Matrix in MATLAB 401
43.1.1 Solution of a 1-D Transport Equation 403
43.2 Use of the Contour-Pad茅 Algorithm with the PS Approach 405
43.2.1 Solution of the 1D Transport Equation Revisited 405
43.3 Computation of Higher-Order Derivatives 407
43.3.1 Solution of the Allen-Cahn Equation 409
43.4 Solution of a 2D Helmholtz Equation 411
43.5 Solution of a 2D Laplace Equation with Piecewise Boundary Conditions 415
43.6 Summary 416
44. RBF Galerkin Methods 419
44.1 An Elliptic PDE with Neumann Boundary Conditions 419
44.2 A Convergence Estimate 420
44.3 A Multilevel RBF Galerkin Algorithm 421
45 RBF Galerkin Methods in MATLAB 423
Appendix A Useful Facts from Discrete Mathematics 427
A.1 Halton Points 427
A.2 kd- Trees 428
Appendix B Useful Facts from Analysis 431
B.1 Some Important Concepts from Measure Theory 431
B.2 A Brief Summary of Integral Transforms 432
B.3 The Schwartz Space and the Generalized Fourier Transform 433
Appendix C Additional Computer Programs 435
C.1 MATLAB Programs 435
C.2 Maple Programs 440
Appendix D Catalog of RBFs with Derivatives 443
D.1 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 1 445
D.2.3 Globally Supported, Strictly Conditionally Positive Definite Functions of Order 2 446
D.2.4 Globally Supported, Strictly Conditionally Positive Definite Functions of Order 3 446
D.2.5 Globally Supported, Strictly Conditionally Positive Definite Functions of Order 4 447
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
Preface vii
1. Introduction 1
1.1 Motivation: Scattered Data Interpolation in Rs 2
1.1.1 The Scattered Data Interpolation Problem 2
1.1.2 Example: Interpolation with Distance Matrices 4
1.2 Some Historical Remarks 13
2. Radial Basis Function Interpolation in MATLAB 17
2.1 Radial (Basis) Functions 17
2.2 Radial Basis Function Interpolation 19
3. Positive Definite Functions 27
3.1 Positive Definite Matrices and Functions 27
3.2 Integral Characterizations for (Strictly) Positive Definite Functions 31
3.2.1 Bochner's Theorem 31
3.2.2 Extensions to Strictly Positive Definite Functions 32
3.3 Positive Definite Radial Functions 33
4. Examples of Strictly Positive Definite Radial Functions 37
4.1 Example 1: Gaussians 37
4.2 Example 2: Laguerre-Gaussians 38
4.3 Example 3: Poisson Radial Functions 39
4.4 Example 4: Mat茅rn Functions 41
4.5 Example 5: Generalized Inverse Multiquadrics 41
4.6 Example 6: Truncated Power Functions 42
4.7 Example 7: Potentials and Whittaker Radial Functions 43
4.8 Example 8: Integration Against Strictly Positive Definite Kernels 45
4.9 Summary 45
5. Completely Monotone and Multiply Monotone Functions 47
5.1 Completely Monotone Functions 47
5.2 Multiply Monotone Functions 49
6. Scattered Data Interpolation with Polynomial Precision 53
6.1 Interpolation with Multivariate Polynomials 53
6.2 Example: Reproduction of Linear Functions Using Gaussian RBFs 55
6.3 Scattered Data Interpolation with More General Polynomial Precision 57
6.4 Conditionally Positive Definite Matrices and Reproduction of Constant Functions 59
7. Conditionally Positive Definite Functions 63
7.1 Conditionally Positive Definite Functions Defined 63
7.2 Conditionally Positive Definite Functions and Generalized Fourier Transforms 65
8. Examples of Conditionally Positive Definite Functions 67
8.1 Example 1: Generalized Multiquadrics 67
8.2 Example 2: Radial Powers 69
8.3 Example 3: Thin Plate Splines 70
9. Conditionally Positive Definite Radial Functions 73
9.1 Conditionally Positive Definite Radial Functions and Completely Monotone Functions 73
9.2 Conditionally Positive Definite Radial Functions and Multiply Monotone Functions 75
9.3 Some Special Properties of Conditionally Positive Definite Functions of Order One 76
10. Miscellaneous Theory: Other Norms and Scattered Data Fitting on Manifolds 79
10.1 Conditionally Positive Definite Functions and p-Norms 79
10.2 Scattered Data Fitting on Manifolds 83
10.3 Remarks 83
11. Compactly Supported Radial Basis Functions 85
11.1 Operators for Radial Functions and Dimension Walks 85
11.2 Wendland's Compactly Supported Functions 87
11.3 Wu's Compactly Supported Functions 88
11.4 Oscillatory Compactly Supported Functions 90
11.5 Other Compactly Supported Radial Basis Functions 92
12. Interpolation with Compactly Supported RBFs in MATLAB 95
12.1 Assembly of the Sparse Interpolation Matrix 95
12.2 Numerical Experiments with CSRBFs 99
13. Reproducing Kernel Hilbert Spaces and Native Spaces for Strictly Positive Definite Functions 103
13.1 Reproducing Kernel Hilbert Spaces 103
13.2 Native Spaces for Strictly Positive Definite Functions 105
13.3 Examples of Native Spaces for Popular Radial Basic Functions 108
14. The Power Function and Native Space Error Estimates 111
14.1 Fill Distance and Approximation Orders 111
14.2 Lagrange Form of the Interpolant and Cardinal Basis Functions 112
14.3 The Power Function 115
14.4 Generic Error Estimates for Functions in NΦ (Ω) 117
14.5 Error Estimates in Terms of the Fill Distance 119
15. Refined and Improved Error Bounds 125
15.1 Native Space Error Bounds for Specific Basis Functions 125
15.1.1 Infinitely Smooth Basis Functions 125
15.1.2 Basis Functions with Finite Smoothness 126
15.2 Improvements for Native Space Error Bounds 127
15.3 Error Bounds for Functions Outside the Native Space 128
15.4 Error Bounds for Stationary Approximation 130
15.5 Convergence with Respect to the Shape Parameter 132
15.6 Polynomial Interpolation as the Limit of RBF Interpolation 133
16. Stability and Trade-Off Principles 135
16.1 Stability and Conditioning of Radial Basis Function Interpolants 135
16.2 Trade-Off Principle I: Accuracy vs. Stability 138
16.3 Trade-Off Principle II: Accuracy and Stability vs. Problem Size 140
16.4 Trade-Off Principle III: Accuracy vs. Efficiency 140
17. Numerical Evidence for Approximation Order Results 141
17.1 Interpolation for epsilon -> 0 141
17.1.1 Choosing a Good Shape Parameter via Trial and Error 142
17.1.2 The Power Function as Indicator for a Good Shape Parameter 142
17.1.3 Choosing a Good Shape Parameter via Cross Validation 146
17.1.4 The Contour-Pad茅 Algorithm 151
17.1.5 Summary 152
17.2 Non-stationary Interpolation 153
17.3 Stationary Interpolation 155
18. The Optimality of REF Interpolation 159
18.1 The Connection to Optimal Recovery 159
18.2 Orthogonality in Reproducing Kernel Hilbert Spaces 160
18.3 Optimality Theorem I 162
18.4 Optimality Theorem II 163
18.5 Optimality Theorem III 164
19. Least Squares RBF Approximation with MATLAB 165
19.1 Optimal Recovery Revisited 165
19.2 Regularized Least Squares Approximation 166
19.3 Least Squares Approximation When RBF Centers Differ from Data Sites 168
19.4 Least Squares Smoothing of Noisy Data 170
20. Theory for Least Squares Approximation 177
20.1 Well-Posedness of REF Least Squares Approximation 177
20.2 Error Bounds for Least Squares Approximation 179
21. Adaptive Least Squares Approximation 181
21.1 Adaptive Least Squares using Knot Insertion 181
21.2 Adaptive Least Squares using Knot Removal 184
21.3 Some Numerical Examples 188
22. Moving Least Squares Approximation 191
22.1 Discrete Weighted Least Squares Approximation 191
22.2 Standard Interpretation of MLS Approximation 192
22.3 The Backus-Gilbert Approach to MLS Approximation 194
22.4 Equivalence of the Two Formulations of MLS Approximation 198
22.5 Duality and Bi-Orthogonal Bases 199
22.6 Standard MLS Approximation as a Constrained Quadratic Optimization Problem 202
22.7 Remarks 202
23. Examples of MLS Generating Functions 205
23.1 Shepard's Method 205
23.2 MLS Approximation with Nontrivial Polynomial Reproduction 207
24. MLS Approximation with MATLAB 211
24.1 Approximation with Shepard's Method 211
24.2 MLS Approximation with Linear Reproduction 216
24.3 Plots of Basis-Dual Basis Pairs 222
25. Error Bounds for Moving Least Squares Approximation 225
25.1 Approximation Order of Moving Least Squares 225
26. Approximate Moving Least Squares Approximation 229
26.1 High-order Shepard Methods via Moment Conditions 229
26.2 Approximate Approximation 230
26.3 Construction of Generating Functions for Approximate MLS Approximation 232
27. Numerical Experiments for Approximate MLS Approximation 237
27.1 Univariate Experiments 237
27.2 Bivariate Experiments 241
28. Fast Fourier Transforms 243
28.1 NFFT 243
28.2 Approximate MLS Approximation via Non-uniform Fast Fourier Transforms 245
29. Partition of Unity Methods 249
29.1 Theory 249
29.2 Partition of Unity Approximation with MATLAB 251
30. Approximation of Point Cloud Data in 3D 255
30.1 A General Approach via Implicit Surfaces 255
30.2 An Illustration in 2D 257
30.3 A Simplistic Implementation in 3D via Partition of Unity Approximation in MATLAB 260
31. Fixed Level Residual Iteration 265
31.1 Iterative Refinement 265
31.2 Fixed Level Iteration 267
31.3 Modifications of the Basic Fixed Level Iteration Algorithm 269
31.4 Iterated Approximate MLS Approximation in MATLAB 270
31.5 Iterated Shepard Approximation 274
32. Multilevel Iteration 277
32.1 Stationary Multilevel Interpolation 277
32.2 A MATLAB Implementation of Stationary Multilevel Interpolation 279
32.3 Stationary Multilevel Approximation 283
32.4 Multilevel Interpolation with Globally Supported RBFs 287
33. Adaptive Iteration 291
33.1 A Greedy Adaptive Algorithm 291
33.2 The Faul-Powell Algorithm 298
34. Improving the Condition Number of the Interpolation Matrix 303
34.1 Preconditioning: Two Simple Examples 304
34.2 Early Preconditioners 305
34.3 Preconditioned GMRES via Approximate Cardinal Functions 309
34.4 Change of Basis 311
34.5 Effect of the "Better" Basis on the Condition Number of the Interpolation Matrix 314
34.6 Effect of the "Better" Basis on the Accuracy of the Interpolant 316
35. Other Efficient Numerical Methods 321
35.1 The Fast Multipole Method 321
35.2 Fast Tree Codes 327
35.3 Domain Decomposition 331
36. Generalized Hermite Interpolation 333
36.1 The Generalized Hermite Interpolation Problem 333
36.2 Motivation for the Symmetric Formulation 335
37. RBF Hermite Interpolation in MATLAB 339
38. Solving Elliptic Partial Differential Equations via RBF Collocation 345
38.1 Kansa's Approach 345
38.2 An Hermite-based Approach 348
38.3 Error Bounds for Symmetric Collocation 349
38.4 Other Issues 350
39. Non-Symmetric RBF Collocation in MATLAB 353
39.1 Kansa's Non-Symmetric Collocation Method 353
40. Symmetric RBF Collocation in MATLAB 365
40.1 Symmetric Collocation Method 365
40.2 Summarizing Remarks on the Symmetric and Non-Symmetric Collocation Methods 372
41. Collocation with CSRBFs in MATLAB 375
41.1 Collocation with Compactly Supported RBFs 375
41.2 Multilevel RBF Collocation 380
42. Using Radial Basis Functions in Pseudospectral Mode 387
42.1 Differentiation Matrices 388
42.2 PDEs with Boundary Conditions via Pseudospectral Methods 390
42.3 A Non-Symmetric RBF-based Pseudospectral Method 391
42.4 A Symmetric RBF-based Pseudospectral Method 394
42.5 A Unified Discussion 396
42.6 Summary 398
43. RBF-PS Methods in MATLAB 401
43.1 Computing the RBF-Differentiation Matrix in MATLAB 401
43.1.1 Solution of a 1-D Transport Equation 403
43.2 Use of the Contour-Pad茅 Algorithm with the PS Approach 405
43.2.1 Solution of the 1D Transport Equation Revisited 405
43.3 Computation of Higher-Order Derivatives 407
43.3.1 Solution of the Allen-Cahn Equation 409
43.4 Solution of a 2D Helmholtz Equation 411
43.5 Solution of a 2D Laplace Equation with Piecewise Boundary Conditions 415
43.6 Summary 416
44. RBF Galerkin Methods 419
44.1 An Elliptic PDE with Neumann Boundary Conditions 419
44.2 A Convergence Estimate 420
44.3 A Multilevel RBF Galerkin Algorithm 421
45 RBF Galerkin Methods in MATLAB 423
Appendix A Useful Facts from Discrete Mathematics 427
A.1 Halton Points 427
A.2 kd- Trees 428
Appendix B Useful Facts from Analysis 431
B.1 Some Important Concepts from Measure Theory 431
B.2 A Brief Summary of Integral Transforms 432
B.3 The Schwartz Space and the Generalized Fourier Transform 433
Appendix C Additional Computer Programs 435
C.1 MATLAB Programs 435
C.2 Maple Programs 440
Appendix D Catalog of RBFs with Derivatives 443
D.1 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 1 445
D.2.3 Globally Supported, Strictly Conditionally Positive Definite Functions of Order 2 446
D.2.4 Globally Supported, Strictly Conditionally Positive Definite Functions of Order 3 446
D.2.5 Globally Supported, Strictly Conditionally Positive Definite Functions of Order 4 447
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
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