Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial ...

220 Pages·2011·2.55 MB·English

Multiquadric Radial Basis Function
Approximation Methods for the Numerical
Solution of Partial Differential Equations
Scott A. Sarra
Marshall University
and
Edward J. Kansa
University of California, Davis
June 30, 2009
ii
Preface
Radial Basis Function (RBF) methods have become the primary tool
for interpolating multidimensional scattered data. RBF methods also have
become important tools for solving Partial Differential Equations (PDEs) in
complexly shaped domains.
Classical methods for the numerical solution of PDEs (finite difference,
finiteelement, finitevolume, andpseudospectral methods) arebasedonpoly-
nomial interpolation. Local polynomial based methods (finite difference, fi-
nite element, and finite volume) are limited by their algebraic convergence
rates. Numerical studies, such as the comparison of the MQ collocation
method with the finite element method in [136], have been done that il-
lustrate the superior accuracy of the MQ method when compared to local
polynomial methods. Global polynomial methods, such as spectral methods,
have exponential convergence rates but are limited by being tied to a fixed
grid. RBF methods are not tied to a grid and in turn belong to a category
of methods called meshless methods. The large number of recent books,
which include [4, 3, 42, 71, 92, 93, 94, 95, 133, 149, 150, 196], on meshfree
methods illustrates the popularity that the methods have recently enjoyed.
The global, non-polynomial, RBF methods may be successfully applied to
achieve exponential accuracy where traditional methods either have difficul-
ties or fail. An example is in multidimensional problems in non-rectangular
domains. RBF methods succeed in very general settings by composing a
univariate function with the Euclidean norm which turns a multidimensional
problem into one that is virtually one dimensional.
RBFmethodsareageneralizationoftheMultiquadric(MQ)RBFmethod
whichutilizes oneparticular RBF.TheMQ RBFmethodhasarichhistoryof
theoretical development and applications. The subject of this monograph is
the MQ RBF approximation method with a particular emphasis on using the
method to numerically solve partial differential equations. This monograph
differs from other recent books [31, 63, 179, 209] on meshless methods in
that it focuses only on the MQ RBF while others have focused on meshless
methods in general. It is hoped that this refined focus will result in a clear
and concise exposition of the area.
Matlab code that illustrates key ideas about the implementation of the
MQ method has been included in the text of the manuscript. The included
code, as well as additional Matlab code used to produce many of the numer-
ical examples, can be found on the web at
iii
www.ScottSarra.org/math/math.html
iv
Contents
1 Development and Overview of the MQ RBF Method 1
1.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Scattered Data Approximation 7
2.1 Alternate Definition of the MQ . . . . . . . . . . . . . . . . . 7
2.2 RBF Interpolation . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Invertibility of the Interpolation Matrix . . . . . . . . . . . . . 10
2.4 Convergence Rates . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . 18
2.6 Local versus Global Condition Numbers . . . . . . . . . . . . 20
2.7 Good Data Independent Center Locations . . . . . . . . . . . 24
2.8 Errors in Boundary Regions . . . . . . . . . . . . . . . . . . . 25
2.9 Approximating Derivatives . . . . . . . . . . . . . . . . . . . . 28
2.10 The Generalized MQ . . . . . . . . . . . . . . . . . . . . . . . 30
2.11 Least Squares mode . . . . . . . . . . . . . . . . . . . . . . . . 31
2.12 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Asymmetric Collocation 35
3.1 Steady Problems . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Neumann Boundary Conditions . . . . . . . . . . . . . 40
3.1.2 Nonlinear Boundary Value Problems . . . . . . . . . . 42
3.2 Time-Dependent PDEs . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Method of Lines . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Eigenvalue Stability . . . . . . . . . . . . . . . . . . . . 45
3.2.3 Linear Advection-Diffusion Equations . . . . . . . . . . 46
3.2.4 Nonlinear Equations . . . . . . . . . . . . . . . . . . . 51
3.2.5 Higher Dimensions . . . . . . . . . . . . . . . . . . . . 55
3.2.6 Hyperbolic PDEs . . . . . . . . . . . . . . . . . . . . . 57
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3.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Large N - small ε - small q 61
Ξ
4.1 Extended precision . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Contour-Pad´e algorithm . . . . . . . . . . . . . . . . . . . . . 63
4.3 SVD based methods . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.1 Truncated SVD (TSVD) . . . . . . . . . . . . . . . . . 67
4.3.2 Improved Truncated SVD (ITSVD) . . . . . . . . . . . 70
4.4 Affine Space Approach . . . . . . . . . . . . . . . . . . . . . . 72
4.5 GMRES iterative method . . . . . . . . . . . . . . . . . . . . 73
4.5.1 ACBF Preconditioning . . . . . . . . . . . . . . . . . . 75
4.6 Domain Decomposition . . . . . . . . . . . . . . . . . . . . . . 78
4.7 Greedy Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.8 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Additional Tools, Techniques, and Topics 87
5.1 Shape Parameter Selection . . . . . . . . . . . . . . . . . . . . 87
5.2 Variable Shape Parameter . . . . . . . . . . . . . . . . . . . . 89
5.3 Polynomial Connection . . . . . . . . . . . . . . . . . . . . . . 92
5.4 Connection to Wavelets . . . . . . . . . . . . . . . . . . . . . . 93
5.4.1 The Wavelet Optimized MQ method . . . . . . . . . . 96
5.5 Approximating Discontinuous Functions . . . . . . . . . . . . 98
5.5.1 Variable Shape Parameter . . . . . . . . . . . . . . . . 100
5.5.2 Digital Total Variation Filtering . . . . . . . . . . . . . 100
5.5.3 Gegenbauer Post-Processing . . . . . . . . . . . . . . . 104
5.6 Finite Difference Mode . . . . . . . . . . . . . . . . . . . . . . 105
5.7 Adaptive Center Locations . . . . . . . . . . . . . . . . . . . . 111
5.8 Integrated Multiquadric Methods . . . . . . . . . . . . . . . . 113
5.9 Boundary Collocation. . . . . . . . . . . . . . . . . . . . . . . 116
5.10 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . 117
6 Further Development of the MQ method 121
6.1 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1.1 The 2d Euler equations . . . . . . . . . . . . . . . . . . 122
6.1.2 Local transformations . . . . . . . . . . . . . . . . . . 123
6.1.3 Local translations . . . . . . . . . . . . . . . . . . . . . 124
6.2 Exact Time Integration . . . . . . . . . . . . . . . . . . . . . . 126
6.3 The MQ and the Level Set Method . . . . . . . . . . . . . . . 128
CONTENTS vii
7 New Frontiers: High dimensional PDEs 133
7.1 Physically important problems . . . . . . . . . . . . . . . . . . 133
7.2 Curse of dimensionality . . . . . . . . . . . . . . . . . . . . . . 134
7.3 Operator Splitting . . . . . . . . . . . . . . . . . . . . . . . . 134
7.4 Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.5 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.6 MQ high-d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.7 Reduction of discretization points . . . . . . . . . . . . . . . . 137
7.8 Effective dimensionality . . . . . . . . . . . . . . . . . . . . . 137
7.9 Transformations on the independent variables . . . . . . . . . 137
7.10 Dependent Variable Transformations . . . . . . . . . . . . . . 138
7.11 Translations to a moving node frame . . . . . . . . . . . . . . 139
7.12 Solution Space Enrichment . . . . . . . . . . . . . . . . . . . . 140
7.13 Parallel computer implementation . . . . . . . . . . . . . . . . 141
7.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8 Afterword 143
8.1 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.2 Promising Areas for RBF applications . . . . . . . . . . . . . 144
8.3 Other Applications and Developments . . . . . . . . . . . . . 145
A Matlab programs 147
B Additional Meshfree Method References 153
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List of Tables
5.1 Shape parameter strategies. . . . . . . . . . . . . . . . . . . . 89
5.2 Variable shape parameter strategies - interpolation . . . . . . 91
5.3 Variable shape parameter strategies - PDE . . . . . . . . . . . 91
5.4 Summary of IRBF properties. . . . . . . . . . . . . . . . . . . 114
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x LIST OF TABLES