Meshfree approximation methods with MATLAB:
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| Format: | Buch |
| Sprache: | English |
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Hackensack, NJ [u.a.]
World Scientific
2007
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| Schriftenreihe: | Interdisciplinary mathematical sciences
6 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 500 S. Ill., graph. Darst. 1 CD-ROM (12 cm) |
| ISBN: | 9789812706331 9789812706348 981270633X 9812706348 |
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MARC
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| 020 | |a 9789812706348 |9 978-981-270-634-8 | ||
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| 100 | 1 | |a Fasshauer, Gregory E. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Meshfree approximation methods with MATLAB |c Gregory E. Fasshauer |
| 264 | 1 | |a Hackensack, NJ [u.a.] |b World Scientific |c 2007 | |
| 300 | |a XVIII, 500 S. |b Ill., graph. Darst. |e 1 CD-ROM (12 cm) | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Interdisciplinary mathematical sciences |v 6 | |
| 650 | 0 | 7 | |a MATLAB |0 (DE-588)4329066-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Gitterfreie Methode |0 (DE-588)4796173-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Radiale Basisfunktion |0 (DE-588)4380647-8 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Methode der kleinsten Quadrate |0 (DE-588)4038974-1 |D s |
| 689 | 0 | 2 | |a Radiale Basisfunktion |0 (DE-588)4380647-8 |D s |
| 689 | 0 | 3 | |a MATLAB |0 (DE-588)4329066-8 |D s |
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| 830 | 0 | |a Interdisciplinary mathematical sciences |v 6 |w (DE-604)BV035420471 |9 6 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016497923 | ||
Datensatz im Suchindex
| _version_ | 1804137648826089472 |
|---|---|
| adam_text | Contents
Preface
vii
1.
Introduction
1
1.1
Motivation:
Scattered
Data Interpolation in
К.
.......... 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:
Matérn
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
xii
Meshfree
Approximation
Methods with
Matlab
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
Contents xiii
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 Hubert Spaces and Native Spaces for
Strictly Positive Definite Functions
103
13.1
Reproducing Kernel Hubert 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
..............
Ill
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
Λ/φ (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
ε
—> 0........................ 141
17.1.1
Choosing a Good Shape Parameter via Trial and Error
. . 142
xiv
Meshfree
Approximation
Methods with
Matlab
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-Padé
Algorithm
................ 151
17.1.5
Summary
........................... 152
17.2
Non-stationary Interpolation
..................... 153
17.3
Stationary Interpolation
....................... 155
18.
The Optimality of RBF Interpolation
159
18.1
The Connection to Optimal Recovery
................ 159
18.2
Orthogonality in Reproducing Kernel Hubert 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 RBF 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
Contents xv
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
Divariate
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
xvi
Meshfree Approximation Methods with
Matlab
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 Preconditioned
......................... 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
Contents xvij
40.1 Symmetrie
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
......................
З8О
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-Padé
Algorithm with the PS Approach
.... 405
43.2.1
Solution of the ID 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.I
Halton
Points
.............................
427
A.2
Ы
-Trees
................................ 428
Appendix
В
Useful Facts from Analysis
431
B.I Some Important Concepts from Measure Theory
.......... 431
B.2 A Brief Summary of Integral Transforms
.............. 432
xviii
Meshfree
Approximation
Methods with
Matlab
В.
3
The Schwartz Space and the Generalized Fourier Transform
. . . 433
Appendix
С
Additional Computer Programs
435
Cl Matlab
Programs
.......................... 435
C.2 Maple Programs
............................ 440
Appendix
D
Catalog of RBFs with Derivatives
443
D.I 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
|
| adam_txt |
Contents
Preface
vii
1.
Introduction
1
1.1
Motivation:
Scattered
Data Interpolation in
К."
. 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:
Matérn
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
xii
Meshfree
Approximation
Methods with
Matlab
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
Contents xiii
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 Hubert Spaces and Native Spaces for
Strictly Positive Definite Functions
103
13.1
Reproducing Kernel Hubert 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
.
Ill
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
Λ/φ (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
ε
—> 0. 141
17.1.1
Choosing a Good Shape Parameter via Trial and Error
. . 142
xiv
Meshfree
Approximation
Methods with
Matlab
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-Padé
Algorithm
. 151
17.1.5
Summary
. 152
17.2
Non-stationary Interpolation
. 153
17.3
Stationary Interpolation
. 155
18.
The Optimality of RBF Interpolation
159
18.1
The Connection to Optimal Recovery
. 159
18.2
Orthogonality in Reproducing Kernel Hubert 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 RBF 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
Contents xv
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
Divariate
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
xvi
Meshfree Approximation Methods with
Matlab
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 Preconditioned
. 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
Contents xvij
40.1 Symmetrie
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
.
З8О
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-Padé
Algorithm with the PS Approach
. 405
43.2.1
Solution of the ID 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.I
Halton
Points
.
427
A.2
Ы
-Trees
. 428
Appendix
В
Useful Facts from Analysis
431
B.I Some Important Concepts from Measure Theory
. 431
B.2 A Brief Summary of Integral Transforms
. 432
xviii
Meshfree
Approximation
Methods with
Matlab
В.
3
The Schwartz Space and the Generalized Fourier Transform
. . . 433
Appendix
С
Additional Computer Programs
435
Cl Matlab
Programs
. 435
C.2 Maple Programs
. 440
Appendix
D
Catalog of RBFs with Derivatives
443
D.I 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 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Fasshauer, Gregory E. |
| author_facet | Fasshauer, Gregory E. |
| author_role | aut |
| author_sort | Fasshauer, Gregory E. |
| author_variant | g e f ge gef |
| building | Verbundindex |
| bvnumber | BV023313703 |
| classification_rvk | SK 905 |
| ctrlnum | (OCoLC)255597710 (DE-599)HBZHT015141100 |
| dewey-full | 518.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.2 |
| dewey-search | 518.2 |
| dewey-sort | 3518.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV023313703 |
| illustrated | Illustrated |
| index_date | 2024-07-02T20:51:17Z |
| indexdate | 2024-07-09T21:15:38Z |
| institution | BVB |
| isbn | 9789812706331 9789812706348 981270633X 9812706348 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016497923 |
| oclc_num | 255597710 |
| open_access_boolean | |
| owner | DE-703 DE-20 DE-83 DE-29T DE-188 DE-384 |
| owner_facet | DE-703 DE-20 DE-83 DE-29T DE-188 DE-384 |
| physical | XVIII, 500 S. Ill., graph. Darst. 1 CD-ROM (12 cm) |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | World Scientific |
| record_format | marc |
| series | Interdisciplinary mathematical sciences |
| series2 | Interdisciplinary mathematical sciences |
| spelling | Fasshauer, Gregory E. Verfasser aut Meshfree approximation methods with MATLAB Gregory E. Fasshauer Hackensack, NJ [u.a.] World Scientific 2007 XVIII, 500 S. Ill., graph. Darst. 1 CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Interdisciplinary mathematical sciences 6 MATLAB (DE-588)4329066-8 gnd rswk-swf Gitterfreie Methode (DE-588)4796173-9 gnd rswk-swf Radiale Basisfunktion (DE-588)4380647-8 gnd rswk-swf Methode der kleinsten Quadrate (DE-588)4038974-1 gnd rswk-swf Gitterfreie Methode (DE-588)4796173-9 s Methode der kleinsten Quadrate (DE-588)4038974-1 s Radiale Basisfunktion (DE-588)4380647-8 s MATLAB (DE-588)4329066-8 s DE-604 Interdisciplinary mathematical sciences 6 (DE-604)BV035420471 6 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016497923&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fasshauer, Gregory E. Meshfree approximation methods with MATLAB Interdisciplinary mathematical sciences MATLAB (DE-588)4329066-8 gnd Gitterfreie Methode (DE-588)4796173-9 gnd Radiale Basisfunktion (DE-588)4380647-8 gnd Methode der kleinsten Quadrate (DE-588)4038974-1 gnd |
| subject_GND | (DE-588)4329066-8 (DE-588)4796173-9 (DE-588)4380647-8 (DE-588)4038974-1 |
| title | Meshfree approximation methods with MATLAB |
| title_auth | Meshfree approximation methods with MATLAB |
| title_exact_search | Meshfree approximation methods with MATLAB |
| title_exact_search_txtP | Meshfree approximation methods with MATLAB |
| title_full | Meshfree approximation methods with MATLAB Gregory E. Fasshauer |
| title_fullStr | Meshfree approximation methods with MATLAB Gregory E. Fasshauer |
| title_full_unstemmed | Meshfree approximation methods with MATLAB Gregory E. Fasshauer |
| title_short | Meshfree approximation methods with MATLAB |
| title_sort | meshfree approximation methods with matlab |
| topic | MATLAB (DE-588)4329066-8 gnd Gitterfreie Methode (DE-588)4796173-9 gnd Radiale Basisfunktion (DE-588)4380647-8 gnd Methode der kleinsten Quadrate (DE-588)4038974-1 gnd |
| topic_facet | MATLAB Gitterfreie Methode Radiale Basisfunktion Methode der kleinsten Quadrate |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016497923&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035420471 |
| work_keys_str_mv | AT fasshauergregorye meshfreeapproximationmethodswithmatlab |