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Fast Multipole Methods For The Helmholtz Equation In Three Dimensions : Elsevier Series in Electromagnetism - Nail A. Gumerov

Fast Multipole Methods For The Helmholtz Equation In Three Dimensions

Elsevier Series in Electromagnetism

Hardcover Published: 27th January 2005
ISBN: 9780080443713
Number Of Pages: 426

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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

Prefacep. xvii
Acknowledgmentsp. xxiii
Outline of the Bookp. xxv
Introductionp. 1
Helmholtz Equationp. 1
Acoustic wavesp. 1
Scalar Helmholtz equations with complex kp. 5
Electromagnetic Wavesp. 10
Boundary Conditionsp. 15
Conditions at infinityp. 15
Transmission conditionsp. 21
Conditions on the boundariesp. 23
Integral Theoremsp. 25
Scalar Helmholtz equationp. 26
Maxwell equationsp. 34
What is Covered in This Book and What is Notp. 37
Elementary Solutionsp. 39
Spherical Coordinatesp. 39
Separation of variablesp. 40
Special functions and propertiesp. 44
Spherical basis functionsp. 57
Differentiation of Elementary Solutionsp. 65
Differentiation theoremsp. 66
Multipole solutionsp. 71
Sums of Elementary Solutionsp. 73
Plane wavesp. 73
Representation of solutions as seriesp. 74
Far field expansionsp. 75
Local expansionsp. 82
Uniquenessp. 86
Summaryp. 86
Translations and Rotations of Elementary Solutionsp. 89
Expansions over Spherical Basis Functionsp. 89
Translationsp. 90
Rotationsp. 92
Translations of Spherical Basis Functionsp. 93
Structure of translation coefficientsp. 97
Recurrence relations for translation coefficientsp. 105
Coaxial translation coefficientsp. 113
Rotations of Elementary Solutionsp. 117
Angles of rotationp. 117
Rotation coefficientsp. 121
Structure of rotation coefficientsp. 126
Recurrence relations for rotation coefficientsp. 131
Summaryp. 137
Multipole Methodsp. 139
Room Acoustics: Fast Summation of Sourcesp. 139
Formulationp. 139
Solutionp. 141
Computations and discussionp. 142
Scattering from a Single Spherep. 143
Formulationp. 143
Solutionp. 144
Computations and discussionp. 148
Scattering from Two Spheresp. 150
Formulationp. 150
Solutionp. 152
Computations and discussionp. 157
Scattering from N Spheresp. 161
Formulationp. 161
Solutionp. 161
Computations and discussionp. 165
On Multiple Scattering from N Arbitrary Objectsp. 168
A method for computation of the T-matrixp. 169
Summaryp. 170
Fast Multipole Methodsp. 171
Preliminary Ideasp. 171
Factorization (Middleman method)p. 172
Space partitioning (modified Middleman method)p. 173
Translations (SLFMM)p. 179
Hierarchical space partitioning (MLFMM)p. 183
Truncation number dependencep. 184
Multipole summationsp. 189
Function representationsp. 190
Multilevel Fast Multipole Methodp. 196
Setting up the hierarchical data structurep. 196
MLFMM procedurep. 203
Data Structures and Efficient Implementationp. 207
Indexingp. 208
Spatial orderingp. 211
Structuring data setsp. 219
Summaryp. 222
Complexity and Optimizations of the MLFMMp. 225
Model for Level-Dependent Translation Parametersp. 225
Spatially Uniform Datap. 227
Upward passp. 228
Downward passp. 231
Final summationp. 233
Total complexity of the MLFMMp. 233
Error of MLFMMp. 236
Optimizationp. 238
Lower frequencies or larger number of sources and receiversp. 239
Higher frequencies or smaller number of sources and receiversp. 241
Non-uniform Datap. 248
Use of data hierarchiesp. 248
Surface distributions of sources and receivers: simple objectsp. 249
Surface distributions of sources and receivers: complex objectsp. 258
Other distributionsp. 263
Adaptive MLFMMp. 264
Setting up the hierarchical data structurep. 265
Procedurep. 270
Complexity and optimization of the adaptive MLFMMp. 273
Summaryp. 283
Fast Translations: Basic Theory and O(p[superscript 3]) Methodsp. 285
Representations of Translation and Rotation Operatorsp. 286
Functions and operatorsp. 286
Representations of translation operators using signature functionsp. 295
Rotational-coaxial translation decompositionp. 306
Rotationsp. 308
Coaxial translationp. 310
Decomposition of translationp. 311
Sparse matrix decomposition of translation and rotation operatorsp. 313
Matrix representations of differential operatorsp. 314
Spectra of differential and translation operatorsp. 322
Integral representations of differential operatorsp. 325
Sparse matrix decomposition of translation operatorsp. 326
Sparse matrix decomposition of rotation operatorsp. 330
Summaryp. 338
Asymptotically Faster Translation Methodsp. 339
Fast Algorithms Based on Matrix Decompositionsp. 340
Fast rotation transformp. 340
Fast coaxial translationp. 348
Fast general translationp. 356
Low- and High-Frequency Asymptoticsp. 361
Low frequenciesp. 362
High frequenciesp. 371
Diagonal Forms of Translation Operatorsp. 387
Representations using the far-field signature functionp. 389
Translation proceduresp. 395
Fast spherical filteringp. 408
Summaryp. 416
Error Boundsp. 417
Truncation Errors for Expansions of Monopolesp. 417
Behavior of spherical Hankel functionsp. 420
Low frequency error bounds and series convergencep. 423
High frequency asymptoticsp. 427
Transition region and combined approximationp. 431
Truncation Errors for Expansions of Multipolesp. 432
Low frequency error bounds and series convergencep. 436
High frequency asymptoticsp. 438
Translation Errorsp. 439
S[vertical line]S translationsp. 439
Multipole-to-local S/R translationsp. 446
Local-to-local R/R translationsp. 452
Some remarksp. 456
FMM errorsp. 458
Summaryp. 462
Fast Solution of Multiple Scattering Problemsp. 465
Iterative Methodsp. 466
Reflection methodp. 466
Generalized minimal residual and other iterative methodsp. 470
Fast Multipole Methodp. 472
Data structuresp. 473
Decomposition of the fieldp. 475
Algorithm for matrix-vector multiplicationp. 477
Complexity of the FMMp. 479
Truncation numbersp. 482
Use of the FMM for preconditioning in the GMRESp. 485
Results of Computationsp. 487
Typical pictures and settingsp. 487
A posteriori error evaluationp. 490
Convergencep. 493
Performance studyp. 495
Summaryp. 498
Color Platesp. 499
Bibliographyp. 509
Indexp. 515
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780080443713
ISBN-10: 0080443710
Series: Elsevier Series in Electromagnetism
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 426
Published: 27th January 2005
Publisher: Elsevier Science & Technology
Country of Publication: GB
Dimensions (cm): 15.2 x 22.9  x 2.72
Weight (kg): 1.0