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Kurzbeschreibung
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. This is the only book that provides comprehensive coverage of this topic in one location. It presents a review of the basic theory of expansions of the Helmholtz equation solutions. It also contains comprehensive description of both mathematical and practical aspects of the fast multipole method and it's applications to issues described by the Helmholtz equation.
Synopsis
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. This is the only book that provides comprehensive coverage of this topic in one location. It presents a review of the basic theory of expansions of the Helmholtz equation solutions. It also contains comprehensive description of both mathematical and practical aspects of the fast multipole method and it's applications to issues described by the Helmholtz equation.
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Some additional content from the book
3. Januar 2006
Format:Gebundene Ausgabe
Link to the table of contents, book outline and preface
http://www.umiacs.umd.edu/~ramani/FMM_HELMHOLTZ/prelims.pdf
Link to Chapter 1 of the book
http://www.umiacs.umd.edu/~ramani/FMM_HELMHOLTZ/Introduction.pdf
Link to the cover
http://www.umiacs.umd.edu/~ramani/FMM_HELMHOLTZ/GumerovWT.pdf
Short table of contents
Preface xvii
Acknowledgments xxiii
Outline of the Book xxv
Chapter 1. Introduction 1
1.1 Helmholtz Equation 1
1.2 Boundary Conditions 15
1.3 Integral Theorems 25
1.4 What is Covered in This Book and What is Not 37
Chapter 2. Elementary Solutions 39
2.1 Spherical Coordinates 39
2.2 Differentiation of Elementary Solutions 65
2.3 Sums of Elementary Solutions 73
2.4 Summary 86
Chapter 3. Translations and Rotations of Elementary Solutions 89
3.1 Expansions over Spherical Basis Functions 89
3.2 Translations of Spherical Basis Functions 93
3.3 Rotations of Elementary Solutions 117
3.4 Summary 137
Chapter 4. Multipole Methods 139
4.1 Room Acoustics: Fast Summation of Sources 139
4.2 Scattering from a Single Sphere 143
4.3 Scattering from Two Spheres 150
4.4 Scattering from N Spheres 161
4.5 On Multiple Scattering from N Arbitrary Objects 168
4.6 Summary 170
Chapter 5. Fast Multipole Methods 171
5.1 Preliminary Ideas 171
5.2 Multilevel Fast Multipole Method 196
5.3 Data Structures and Efficient Implementation 207
5.4 Summary 222
Chapter 6. Complexity and Optimizations of the MLFMM 225
6.1 Model for Level-Dependent Translation Parameters 225
6.2 Spatially Uniform Data 227
6.3 Error of MLFMM 236
6.4 Optimization 238
6.5 Non-uniform Data 248
6.6 Adaptive MLFMM 264
6.7 Summary 283
Chapter 7. Fast Translations: Basic Theory and O(p^3) Methods 285
7.1 Representations of Translation and Rotation Operators 286
7.2 Rotational-coaxial translation decomposition 306
7.3 Sparse matrix decomposition of translation and rotation operators 313
7.4 Summary 338
Chapter 8. Asymptotically Faster Translation Methods 339
8.1 Fast Algorithms Based on Matrix Decompositions 340
8.2 Low- and High-Frequency Asymptotics 361
8.3 Diagonal Forms of Translation Operators 387
8.4 Summary 416
Chapter 9. Error Bounds 417
9.1 Truncation Errors for Expansions of Monopoles 417
9.2 Truncation Errors for Expansions of Multipoles 432
9.3 Translation Errors 439
9.4 Summary 462
Chapter 10. Fast Solution of Multiple Scattering
Problems 465
10.1 Iterative Methods 466
10.2 Fast Multipole Method 472
10.3 Results of Computations 487
10.4 Summary 498
Color Plates 499
Bibliography 509
Index 515
http://www.umiacs.umd.edu/~ramani/FMM_HELMHOLTZ/prelims.pdf
Link to Chapter 1 of the book
http://www.umiacs.umd.edu/~ramani/FMM_HELMHOLTZ/Introduction.pdf
Link to the cover
http://www.umiacs.umd.edu/~ramani/FMM_HELMHOLTZ/GumerovWT.pdf
Short table of contents
Preface xvii
Acknowledgments xxiii
Outline of the Book xxv
Chapter 1. Introduction 1
1.1 Helmholtz Equation 1
1.2 Boundary Conditions 15
1.3 Integral Theorems 25
1.4 What is Covered in This Book and What is Not 37
Chapter 2. Elementary Solutions 39
2.1 Spherical Coordinates 39
2.2 Differentiation of Elementary Solutions 65
2.3 Sums of Elementary Solutions 73
2.4 Summary 86
Chapter 3. Translations and Rotations of Elementary Solutions 89
3.1 Expansions over Spherical Basis Functions 89
3.2 Translations of Spherical Basis Functions 93
3.3 Rotations of Elementary Solutions 117
3.4 Summary 137
Chapter 4. Multipole Methods 139
4.1 Room Acoustics: Fast Summation of Sources 139
4.2 Scattering from a Single Sphere 143
4.3 Scattering from Two Spheres 150
4.4 Scattering from N Spheres 161
4.5 On Multiple Scattering from N Arbitrary Objects 168
4.6 Summary 170
Chapter 5. Fast Multipole Methods 171
5.1 Preliminary Ideas 171
5.2 Multilevel Fast Multipole Method 196
5.3 Data Structures and Efficient Implementation 207
5.4 Summary 222
Chapter 6. Complexity and Optimizations of the MLFMM 225
6.1 Model for Level-Dependent Translation Parameters 225
6.2 Spatially Uniform Data 227
6.3 Error of MLFMM 236
6.4 Optimization 238
6.5 Non-uniform Data 248
6.6 Adaptive MLFMM 264
6.7 Summary 283
Chapter 7. Fast Translations: Basic Theory and O(p^3) Methods 285
7.1 Representations of Translation and Rotation Operators 286
7.2 Rotational-coaxial translation decomposition 306
7.3 Sparse matrix decomposition of translation and rotation operators 313
7.4 Summary 338
Chapter 8. Asymptotically Faster Translation Methods 339
8.1 Fast Algorithms Based on Matrix Decompositions 340
8.2 Low- and High-Frequency Asymptotics 361
8.3 Diagonal Forms of Translation Operators 387
8.4 Summary 416
Chapter 9. Error Bounds 417
9.1 Truncation Errors for Expansions of Monopoles 417
9.2 Truncation Errors for Expansions of Multipoles 432
9.3 Translation Errors 439
9.4 Summary 462
Chapter 10. Fast Solution of Multiple Scattering
Problems 465
10.1 Iterative Methods 466
10.2 Fast Multipole Method 472
10.3 Results of Computations 487
10.4 Summary 498
Color Plates 499
Bibliography 509
Index 515
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