Fast Multipole Methods for the Helmholtz Equation in Three Dimensions

Nail Gumerov
Institute of Advanced Computer Studies, University of Maryland, USA
Ramani Duraiswami
Department of Computer Science and Institute of Advanced Computer Studies, University of Maryland, USA

Series: Elsevier Series in Electromagnetism

ISBN: 0-08-044371-0 Book/Hardback

Measurements: 5 15/16 X 8 7/8 in
Pages: 426

Imprint: Elsevier Science

Publication Date: 15 March 2005

Contents

Preface

Acknowledgments

Outline of the Book

Chapter 1. Introduction

1.1. Helmholtz Equation

1.2. Boundary Conditions

1.3. Integral Theorems

1.4. What is Covered in This Book and What is Not

Chapter 2. Elementary Solutions

2.1. Elementary Solutions in Spherical Coordinates

2.2. Differentiation of Elementary Solutions

2.3. Sums of Elementary Solutions

2.4. Summary

Chapter 3. Translations and Rotations of Elementary Solutions

3.1. Expansions Over Spherical Basis Functions

3.2. Translations of Spherical Basis Functions

3.3. Rotations of Elementary Solutions

3.4. Summary

Chapter 4. Multipole Methods

4.1. Room Acoustics: Fast Summation of Sources

4.2. Scattering From a Single Sphere

4.3. Scattering From Two Spheres

4.4. Scattering From N Spheres

4.5. On Multiple Scattering from N Arbitrary Objects

Chapter 5. Fast Multipole Methods

5.1. Preliminary Ideas

5.2. Multilevel Fast Multipole Method

5.3. Data Structures and Efficient Implementation

5.4. Summary

Chapter 6. Complexity and Optimizations of the MLFMM

6.1. Model for Level Dependent Translation Parameters

6.2. Spatially Uniform Data

6.3. Error of MLFMM

6.4. Optimization

6.5. Non-Uniform Data

6.6. Adaptive MLFMM

6.7. Summary

Chapter 7. Fast Translations: O(p3) Methods

7.1. Representations of Translation and Rotation Operators

7.2. Rotation-Coaxial Translation Decomposition

7.3. Sparse Matrix Decomposition of Translation and Rotation Operators

7.4. Summary

Chapter 8. Faster Translation Methods

8.1. Fast Algorithms Based On Matrix Decompostions

8.2. Low and High Frequency Asymptotics

8.3. Diagonal Forms of Translation Operators

8.4. Summary

Chapter 9. Error Bounds

9.1. Truncation Errors for Expansions of Monopoles

9.2. Truncation Errors for Expansions of Multipoles

9.3. Translation Errors

9.4. Summary

Chapter 10. Fast Solution of Multiple Scattering Problems

10.1. Iterative Methods

10.2. Fast Multipole Method

10.3. Results of Computations

10.4. Summary

Bibliography

Index