Fast Multipole Methods for the Helmholtz Equation in Three Dimensions:Fast Multipole Methods for the Helmholtz Equation in Three Dimensions (Elsevier Series in Electromagnetism) (Hardcover)by
Nail A Gumerov (Author),
Ramani Duraiswami (Author)
Hardcover: 426 pages
Publisher: Elsevier Science (February 10, 2005)
Language: English
ISBN-10: 0080443710
ISBN-13: 978-0080443713
Book Description
This is the only book which provides a comprehensive coverage of this topic in one volume
Product Description
This volume in the Elsevier Series in Electromagnetism presents a detailed, in-depth and self-contained treatment of the Fast Multipole Method and its applications to the solution of the Helmholtz equation in three dimensions. The Fast Multipole Method was pioneered by Rokhlin and Greengard in 1987 and has enjoyed a dramatic development and recognition during the past two decades. This method has been described as one of the best 10 algorithms of the 20th century. Thus, it is becoming increasingly important to give a detailed exposition of the Fast Multipole Method that will be accessible to a broad audience of researchers. This is exactly what the authors of this book have accomplished.
For this reason, it will be a valuable reference for a broad audience of engineers, physicists and applied mathematicians.
The Only book that provides comprehensive coverage of this topic in one location.
Presents a review of the basic theory of expansions of the Helmholtz equation solutions
Comprehensive description of both mathematical and practical aspects of the fast multipole method and it's applications to issues described by the Helmholtz equation
Contents
Preface
Acknowledgments
Outline of the Book
Chapter 1. Introduction
1.1 Helmholtz Equation
1.1.1 Acoustic waves
1.1.1.1 Barotropic fluids
1.1.1.2 Fourier and Laplace transforms
1.1.2 Scalar Helmholtz equations with complex k
1.1.2.1 Acoustic waves in complex media
1.1.2.2 Telegraph equation
1.1.2.3 Diffusion
1.1.2.4 Schrodinger equation
1.1.2.5 Klein-Gordan equation
1.1.3 Electromagnetic Waves
1.1.3.1 Maxwell's equations
1.1.3.2 Scalar potentials
1.2 Boundary Conditions
1.2.1 Conditions at infinity
1.2.1.1 Spherically symmetrical solutions
1.2.1.2 Somrnerfeld radiation condition
1.2.1.3 Complex wavenumber
1.2.1.4 Silver-Miiller radiation condition
1.2.2 Transmission conditions
1.2.2.1 Acoustic waves
1.2.2.2 Electromagnetic waves
xvii
xxiii
xxv
viii Contents
1.2.3 Conditions on the boundaries
1.2.3.1 Scalar Helmholtz equation
1.2.3.2 Maxwell equations
1.3 Integral Theorems
1.3.1 Scalar Helmholtz equation
1.3.1.1 Green's identity and formulae
1.3.1.2 Integral equation from Green's
formula for I)
1.3.1.3 Solution of the Helmholtz equation
as distribution of sources and dipoles
1.3.2 Maxwell equations
1.4 What is Covered in This Book and What is Not
Chapter 2. Elementary Solutions
2.1 Spherical Coordinates
2.1.1 Separation of variables
2.1.1.1 Equation with respect to the angle Q
2.1.1.2 Equation with respect to the angle 0
2.1.1.3 Equation with respect to the distance r
2.1.2 Special functions and properties
2.1 -2.1 Associated Legendre functions
2.1.2.2 Spherical Harmonics
2.1.2.3 Spherical Bessel and Hankel functions
2.1.3 Spherical basis functions
2.1.3.1 The case Im{k} = 0
2.1.3.2 The case Re{k) = 0
2.1.3.3 The case Im{k) > 0, Re{k} > 0
2.1.3.4 The case lm{k} < 0, Re{k} > 0
2.1.3.5 Basis functions
2.2 Differentiation of Elementary Solutions
2.2.1 Differentiation theorems
2.2.2 Multipole solutions
2.3 Sums of Elementary Solutions
2.3.1 Plane waves
2.3.2 Representation of solutions as series
2.3.3 Far field expansions
2.3.3.1 Asymptotic expansion
2.3.3.2 Relation to expansion over singular
spherical basis functions
2.3.4 Local expansions
2.3.4.1 Asymptotic expansion
Contents ix
2.3.4.2 Relation to expansion over
regular spherical basis functions 83
2.3.5 Uniqueness 86
2.4 Summary 86
Chapter 3. Translations and Rotations of Elementary Solutions
3.1 Expansions over Spherical Basis Functions
3.1.1 Translations
3.1.2 Rotations
3.2 Translations of Spherical Basis Functions
3.2.1 Structure of translation coefficients
3.2.1.1 Relation to spherical basis functions
3.2.1.2 Addition theorems for spherical
basis functions
3.2.1.3 Relation to Clebsch-Gordan coefficients
3.2.1.4 Symmetries of translation coefficients
3.2.2 Recurrence relations for translation coefficients
3.2.2.1 Sectorial coefficients
3.2.2.2 Computation of translation coefficients
3.2.3 Coaxial translation coefficients
3.2.3.1 Recurrences
3.2.3.2 Symmetries
3.2.3.3 Computations
3.3 Rotations of Elementary Solutions
3.3.1 Angles of rotation
3.3.2 Rotation coefficients
3.3.3 Structure of rotation coefficients
3.3.3.1 Symmetries of rotation coefficients
3.3.3.2 Relation to Clebsch-Gordan coefficients
3.3.4 Recurrence relations for rotation coefficients
3.3.4.1 Computational procedure
3.4 Summary
Chapter 4. Multipole Methods
4.1 Room Acoustics: Fast Summation of Sources
4.1.1 Formulation
4.1.2 Solution
4.1.3 Computations and discussion
4.2 Scattering from a Single Sphere
4.2.1 Formulation
4.2.2 Solution
Contents
4.2.2.1 Determination of expansion coefficients
4.2.2.2 Surface function
4.2.3 Computations and discussion
4.3 Scattering from Two Spheres
4.3.1 Formulation
4.3.2 Solution
4.3.2.1 Determination of expansion coefficients
4.3.2.2 Surface function
4.3.3 Computations and discussion
4.4 Scattering from N Spheres
4.4.1 Formulation
4.4.2 Solution
4.4.3 Computations and discussion
4.5 On Multiple Scattering from N Arbitrary Objects
4.5.1 A method for computation of the T-matrix
4.6 Summary
Chapter 5. Fast Multipole Methods 171
5.1 Preliminary Ideas 171
5.1.1 Factorization (Middleman method) 172
5.1.2 Space partitioning (modified Middleman method) 173
5.1.2.1 Space partitioning with respect to
evaluation set 1 74
5.1.2.2 Space partitioning with respect to
source set 177
5.1.3 Translations (SLFMM) 179
5.1.4 Hierarchical space partitioning (MLFMM) 183
5.1.5 Truncation number dependence 184
5.1.5.1 Geometrically decaying error 185
5.1.5.2 Dependence of the truncation number
on the box size 186
5.1.6 Multipole summations 189
5.1.7 Function representations 190
5.1.7.1 Concept 190
5.1.7.2 FMM operations 192
5.1.7.3 SLFMM 194
5.2 Multilevel Fast Multipole Method 196
5.2.1 Setting up the hierarchical data structure 196
5.2.1.1 Generalized octrees (2d trees) 196
5.2.1.2 Data hierarchies 199
5.2.1.3 Hierarchical spatial domains 200
Contents
5.2.1.4 Spatial scaling and size of neighborhood
5.2.2 MLFMM procedure
5.2.2.1 Upward pass
5.2.2.2 Downward pass
5.2.2.3 Final summation
5.3 Data Structures and Efficient Implementation
5.3.1 Indexing
5.3.2 Spatial ordering
5.3.2.1 Scaling
5.3.2.2 Ordering in one dimension
(binary ordering)
5.3.2.3 Ordering in d dimensions
5.3.3 Structuring data sets
5.3.3.1 Ordering of d-dimensional data
5.3.3.2 Determination of the threshold level
5.3.3.3 Search procedures and operations
on point sets
5.4 Summary
Chapter 6. Complexity and Optimizations of the MLFMM
6.1 Model for Level-Dependent Translation Parameters
6.2 Spatially Uniform Data
6.2.1 Upward pass
6.2.1.1 Step 1
6.2.1.2 Step 2
6.2.1.3 Step 3
6.2.2 Downward pass
6.2.2.1 Step 1
6.2.2.2 Step 2
6.2.3 Final summation
6.2.4 Total complexity of the MLFMM
6.3 Error of MLFMM
6.4 Optimization
6.4.1 Lower frequencies or larger number of
sources and receivers
6.4.2 Higher frequencies or smaller number of sources
and receivers
6.4.2.1 Volume element methods
6.4.2.2 Some numerical tests
6.5 Non-uniform Data
xii Contents
6.5.1 Use of data hierarchies 248
6.5.2 Surface distributions of sources and receivers:
simple objects 249
6.5.2.1 Complexity of MLFMM 250
6.5.2.2 Error of MLFMM 253
6.5.2.3 Optimization for lower frequencies
or larger number of sources and receivers 253
6.5.2.4 Optimization for higher frequencies
or smaller number of sources and receivers 255
6.5.2.5 Boundary element methods 257
6.5.3 Surface distributions of sources and
receivers: complex objects 258
6.5.4 Other distributions 263
6.6 Adaptive MLFMM 264
6.6.1 Setting up the hierarchical data structure 265
6.6.1.1 General idea 265
6.6.1.2 Determination of the target box
levels /numbers 266
6.6.1.3 Construction of the D-tree 267
6.6.1.4 Construction of the D-tree 268
6.6.1.5 Construction of the C-forest 268
6.6.2 Procedure 270
6.6.2.1 Upward pass 270
6.6.2.2 Downward pass 272
6.6.2.3 Final summation 273
6.6.3 Complexity and optimization of the
adaptive MLFMM 273
6.6.3.1 Data distributions 274
6.6.3.2 High frequencies 28 1
6.7 Summary 283
Chapter 7. Fast Translations: Basic Theory and 0 ( p 3 ) Methods
7.1 Representations of Translation and Rotation Operators
7.1.1 Functions and operators
7.1.1.1 Linear vector spaces
7.1.1.2 Linear operators
7.1.1.3 Groups of transforms
7.1.1.4 Representations of groups
7.1.2 Representations of translation operators
using signature functions
7.1.2.1 (RIR) translation
Contents
7.1.2.2 (S IS) translation 298
7.1.2.3 SIR translation 301
7.1.2.4 Coaxial translations 304
7.1.2.5 Rotations 305
7.2 Rotational-coaxial translation decomposition 306
7.2.1 Rotations 308
7.2.2 Coaxial translation 310
7.2.3 Decomposition of translation 311
7.3 Sparse matrix decomposition of translation
and rotation operators 313
7.3.1 Matrix representations of differential operators 314
7.3.1.1 Operator D, 316
7.3.1.2 Operator D,+iy 317
7.3.1.3 Operator D,-iy 319
7.3.1.4 Operator Di 320
7.3.1.5 Matrix form of the Helmholtz equation 321
7.3.2 Spectra of differential and translation operators 322
7.3.2.1 Continuous spectra of differential operators 322
7.3.2.2 Continuous spectra of translation operators 323
7.3.3 Integral representations of differential operators 325
7.3.4 Sparse matrix decomposition of
translation operators 326
7.3.4.1 Matrix exponential 326
7.3.4.2 Legendre series 329
7.3.5 Sparse matrix decomposition of rotation operators 330
7.3.5.1 Infinitesimal rotations 334
7.3.5.2 Decomposition of the rotation operator
for Euler angle P 336
7.4 Summary 338
Chapter 8. Asymptotically Faster Translation Methods
8.1 Fast Algorithms Based on Matrix Decompositions
8.1.1 Fast rotation transform
8.1.1.1 Toeplitz and Hankel matrices
8.1.1.2 Decomposition of rotation into
product of Toeplitz and diagonal matrices
8.1.2 Fast coaxial translation
8.1.2.1 Decomposition of translation matrix
8.1.2.2 Legendre transform
8.1.2.3 Extension and truncation operators
8.1.2.4 Fast coaxial translation algorithm
Contents
8.1.2.5 Precomputation of diagonal matrices
8.1.3 Fast general translation
8.1.3.1 Decomposition of the translation matrix
8.1.3.2 Fast spherical transform
8.1.3.3 Precomputation of diagonal matrices
8.2 Low- and High-Frequency Asymptotics
8.2.1 Low frequencies
8.2.1.1 Exponential sparse matrix decomposition
of the R I R matrix
8.2.1.2 Toeplitz/Hankel matrix representations
8.2.1.3 Renormalization
8.2.2 High frequencies
8.2.2.1 Surface delta-function
8.2.2.2 Principal term of the SIR translation
8.2.2.3 Non-uniform and uniform asymptotic
expansions
8.2.2.4 Expansion of coaxial SIR matrix
8.2.2.5 RI R translation
8.3 Diagonal Forms of Translation Operators
8.3.1 Representations using the far-field
signature function
8.3.1.1 Spherical cubatures
8.3.1.2 Signature functions for multipoles
8.3.2 Translation procedures
8.3.2.1 Algorithm using band-unlimited
functions
8.3.2.2 Numerical tests and discussion
8.3.2.3 Deficiencies of the signature
function method
8.3.2.4 Algorithms using band-limited functions
8.3.3 Fast spherical filtering
8.3.3.1 Integral representation of spherical filter
8.3.3.2 Separation of variables
8.3.3.3 Legendre filter
8.4 Summary
Chapter 9. Error Bounds
9.1 Truncation Errors for Expansions of Monopoles
9.1.1 Behavior of spherical Hankel functions
9.1.2 Low frequency error bounds and
series convergence
Contents
9.1.3 High frequency asymptotics
9.1.4 Transition region and combined approximation
9.2 Truncation Errors for Expansions of Multipoles
9.2.1 Low frequency error bounds and series
convergence
9.2.2 High frequency asymptotics
9.3 Translation Errors
9.3.1 S IS translations
9.3.1.1 Problem
9.3.1.2 Solution
9.3.2 Multipole-to-local S 1 R translations
9.3.2.1 Problem
9.3.2.2 Solution
9.3.3 Local-to-local RI R translations
9.3.3.1 Problem
9.3.3.2 Solution
9.3.4 Some remarks
9.3.5 FMM errors
9.3.5.1 Low and moderate frequencies
9.3.5.2 Higher frequencies
9.4 Summary
Chapter 10. Fast Solution of Multiple Scattering
Problems
10.1 Iterative Methods
10.1.1 Reflection method
10.1.2 Generalized minimal residual and other
iterative methods
10.1.2.1 Preconditioners
10.1.2.2 Flexible GMRES
10.2 Fast Multipole Method
10.2.1 Data structures
10.2.2 Decomposition of the field
10.2.3 Algorithm for matrix-vector multiplication
10.2.4 Complexity of the FMM
10.2.4.1 Complexity and translation
methods for large problems
10.2.4.2 Smaller problems or low frequencies
10.2.5 Truncation numbers
Contents
10.2.6 Use of the FMM for preconditioning in the
GMRES
10.3 Results of Computations
10.3.1 Typical pictures and settings
10.3.1.1 FMM for spatial imaging/field
calculation
10.3.1.2 Surface imaging
10.3.2 A posteriori error evaluation
10.3.3 Convergence
10.3.4 Performance study
10.4 Summary
Color Plates
Bibliography
Index
这是一本好书，期待共享
期待共享
Fast Multipole Methods for the Helmholtz Equation in Three Dimensions.part1
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[ 本帖最后由 drjiachen 于 2008-12-15 19:57 编辑 ]
Fast Multipole Methods for the Helmholtz Equation in Three Dimensions.part2-6
[ 本帖最后由 drjiachen 于 2008-12-15 19:57 编辑 ]
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Fast Multipole Methods for the Helmholtz Equation in Three Dimensions: Fast Multipole Methods for the Helmholtz Equation in Three Dimensions.part5.rar
