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E-grāmata: Introduction to the Fast Multipole Method: Topics in Computational Biophysics, Theory, and Implementation

  • Formāts: 460 pages
  • Izdošanas datums: 06-Dec-2019
  • Izdevniecība: CRC Press Inc
  • ISBN-13: 9781439839065
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  • Formāts: 460 pages
  • Izdošanas datums: 06-Dec-2019
  • Izdevniecība: CRC Press Inc
  • ISBN-13: 9781439839065

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Introduction to the Fast Multipole Method introduces the reader to the theory and computer implementation of the Fast Multipole Method. It covers the topics of Laplaces equation, spherical harmonics, angular momentum, the Wigner matrix, the addition theorem for solid harmonics, and lattice sums for periodic boundary conditions, along with providing a complete, self-contained explanation of the math of the method, so that anyone having an undergraduate grasp of calculus should be able to follow the material presented. The authors derive the Fast Multipole Method from first principles and systematically construct the theory connecting all the parts.

Key Features





Introduces each topic from first principles





Derives every equation presented, and explains each step in its derivation





Builds the necessary theory in order to understand, develop, and use the method





Describes the conversion from theory to computer implementation





Guides through code optimization and parallelization
Preface ix
1 Legendre Polynomials
1(26)
1.1 Potential of a Point Charge Located on the z-Axis
1(2)
1.2 Laplace's Equation
3(2)
1.3 Solution of Laplace's Equation in Cartesian Coordinates
5(2)
1.4 Laplace's Equation in Spherical Polar Coordinates
7(4)
1.5 Orthogonality and Normalization of Legendre Polynomials
11(4)
1.6 Expansion of an Arbitrary Function in Legendre Series
15(2)
1.7 Recurrence Relations for Legendre Polynomials
17(4)
1.8 Analytic Expressions for First Few Legendre Polynomials
21(1)
1.9 Symmetry Properties of Legendre Polynomials
21(6)
2 Associated Legendre Functions
27(24)
2.1 Generalized Legendre Equation
28(4)
2.2 Associated Legendre Functions
32(2)
2.3 Orthogonality and Normalization of Associated Legendre Functions
34(3)
2.4 Recurrence Relations for Associated Legendre Functions
37(5)
2.5 Derivatives of Associated Legendre Functions
42(2)
2.6 Analytic Expression for First Few Associated Legendre Functions
44(1)
2.7 Symmetry Properties of Associated Legendre Functions
44(7)
3 Spherical Harmonics
51(20)
3.1 Spherical Harmonics Functions
52(1)
3.2 Orthogonality and Normalization of Spherical Harmonics
53(3)
3.3 Symmetry Properties of Spherical Harmonics
56(3)
3.4 Recurrence Relations for Spherical Harmonics
59(4)
3.5 Analytic Expression for the First Few Spherical Harmonics
63(1)
3.6 Nodal Properties of Spherical Harmonics
63(8)
4 Angular Momentum
71(38)
4.1 Rotation Matrices
72(5)
4.2 Unitary Matrices
77(5)
4.3 Rotation Operator
82(4)
4.4 Commutative Properties of the Angular Momentum
86(5)
4.5 Eigenvalues of the Angular Momentum
91(4)
4.6 Angular Momentum Operator in Spherical Polar Coordinates
95(6)
4.7 Eigenvectors of the Angular Momentum Operator
101(1)
4.8 Characteristic Vectors of the Rotation Operator
102(3)
4.9 Rotation of Eigenfunctions of Angular Momentum
105(4)
5 Wigner Matrix
109(38)
5.1 The Euler Angles
109(4)
5.2 Wigner Matrix for j = 1
113(7)
5.3 Wigner Matrix for j = 1/2
120(8)
5.4 General Form of the Wigner Matrix Elements
128(13)
5.5 Addition Theorem for Spherical Harmonics
141(6)
6 Clebsch-Gordan Coefficients
147(30)
6.1 Addition of Angular Momenta
147(6)
6.2 Evaluation of Clebsch-Gordan Coefficients
153(13)
6.3 Addition of Angular Momentum and Spin
166(5)
6.4 Rotation of the Coupled Eigenstates of Angular Momentum
171(6)
7 Recurrence Relations for Wigner Matrix
177(12)
7.1 Recurrence Relations with Increment in Index m
177(5)
7.2 Recurrence Relations with Increment in Index k
182(7)
8 Solid Harmonics
189(26)
8.1 Regular and Irregular Solid Harmonics
189(2)
8.2 Regular Multipole Moments
191(1)
8.3 Irregular Multipole Moments
192(1)
8.4 Computation of Electrostatic Energy via Multipole Moments
193(1)
8.5 Recurrence Relations for Regular Solid Harmonics
194(2)
8.6 Recurrence Relations for Irregular Solid Harmonics
196(2)
8.7 Generating Functions for Solid Harmonics
198(3)
8.8 Addition Theorem for Regular Solid Harmonics
201(4)
8.9 Addition Theorem for Irregular Solid Harmonics
205(5)
8.10 Transformation of the Origin of Irregular Harmonics
210(2)
8.11 Vector Diagram Approach to Multipole Translations
212(3)
9 Electrostatic Force
215(10)
9.1 Gradient of Electrostatic Potential
215(2)
9.2 Differentiation of Multipole Expansion
217(1)
9.3 Differentiation of Regular Solid Harmonics in Spherical Polar Coordinates
218(2)
9.4 Differentiation of Spherical Polar Coordinates
220(2)
9.5 Differentiation of Regular Solid Harmonics in Cartesian Coordinates
222(2)
9.6 FMM Force in Cartesian Coordinates
224(1)
10 Scaling of Solid Harmonics
225(26)
10.1 Optimization of Expansion of Inverse Distance Function
225(2)
10.2 Scaling of Associated Legendre Functions
227(2)
10.3 Recurrence Relations for Scaled Regular Solid Harmonics
229(3)
10.4 Recurrence Relations for Scaled Irregular Solid Harmonics
232(3)
10.5 First Few Terms of Scaled Solid Harmonics
235(1)
10.6 Design of Computer Code for Computation of Solid Harmonics
236(1)
10.7 Program Code for Computation of Multipole Expansions
237(5)
10.8 Computation of Electrostatic Force Using Scaled Solid Harmonics
242(2)
10.9 Program Code for Computation of Force
244(7)
11 Scaling of Multipole Translations
251(24)
11.1 Scaling of Multipole Translation Operations
251(2)
11.2 Program Code for M2M Translation
253(8)
11.3 Program Code for M2L Translation
261(4)
11.4 Program Code for L2L Translation
265(10)
12 Fast Multipole Method
275(28)
12.1 Near and Far Fields: Prerequisites for the Use of the Fast Multipole Method
275(2)
12.2 Series Convergence and Truncation of Multipole Expansion
277(7)
12.3 Hierarchical Division of Boxes in the Fast Multipole Method
284(4)
12.4 Far Field
288(1)
12.5 Near Field and Far Field Pair Counts
289(5)
12.6 FMM Algorithm
294(6)
12.7 Accuracy Assessment of Multipole Operations
300(3)
13 Multipole Translations along the z-Axis
303(22)
13.1 M2M Translation along the z-Axis
303(7)
13.2 L2L Translation along the z-Axis
310(6)
13.3 M2L Translation along the z-Axis
316(9)
14 Rotation of Coordinate System
325(36)
14.1 Rotation of Coordinate System to Align the z-axis with the Axis of Translation
325(3)
14.2 Rotation Matrix
328(2)
14.3 Computation of Scaled Wigner Matrix Elements with Increment in Index m
330(7)
14.4 Computation of Scaled Wigner Matrix Elements with Increment in Index k
337(5)
14.5 Program Code for Computation of Scaled Wigner Matrix Elements Based on the k-set
342(12)
14.6 Program Code for Computation of Scaled Wigner Matrix Elements Based on the m-set
354(7)
15 Rotation-Based Multipole Translations
361(14)
15.1 Assembly of Rotation Matrix
361(4)
15.2 Rotation-Based M2M Operation
365(3)
15.3 Rotation-Based M2L Operation
368(2)
15.4 Rotation-Based L2L Operation
370(5)
16 Periodic Boundary Conditions
375(32)
16.1 Principles of Periodic Boundary Conditions
375(2)
16.2 Lattice Sum for Energy in Periodic FMM
377(2)
16.3 Multipole Moments of the Central Super-Cell
379(5)
16.4 Far-Field Contribution to the Lattice Sum for Energy
384(6)
16.5 Contribution of the Near-Field Zone into the Central Unit Cell
390(1)
16.6 Derivative of Electrostatic Energy on Particles in the Central Unit Cell
391(1)
16.7 Stress Tensor
392(2)
16.8 Analytic Expression for Stress Tensor
394(4)
16.9 Lattice Sum for Stress Tensor
398(9)
Appendix 407(34)
Bibliography 441(2)
Index 443
Victor Anisimov is an Application Performance Engineer at the Argonne Leadership Computing Facility. He holds a Ph.D. degree in Physical Chemistry from the Institute of Chemical Physics of the Russian Academy of Sciences (1997), which was followed by 5 years of computational chemistry software development with Fujitsu, where his team developed the linear scaling semi-empirical quantum chemistry code LocalSCF that expanded the limits of the approximate electronic structure theory to millions of atoms. He performed postdoctoral work at the University of Maryland at Baltimore (2003-2008), and at the University of Texas at Houston (2008-2011), improving molecular dynamics methods and contributing to the CHARMM code. In the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign (2011-2019), Dr. Anisimov held the position of Senior Research Scientist, conducted application support for petascale resource allocation teams on the Blue Waters supercomputer, optimized various application codes, and improved the performance and scaling profiles of coupled cluster singles and doubles electronic structure method in NWChem code. Dr. Anisimov works on the faithful representation of long-range electrostatic interactions in large-scale molecular simulations, near-neighbor communication algorithms, and linear-scaling methods. He specializes in performance optimization and fidelity improvements of electronic structure and soft matter simulation application codes on exascale platforms.

James J. P. Stewart pioneered the use of semiempirical quantum chemistry methods in research and teaching. After teaching at the University of Strathclyde in Glasgow, Scotland, he became a researcher at the United States Air Force Academy, then taught as an adjoint professor at the University of Colorado. For the past 30 years, his company, Stewart Computational Chemistry, has been marketing his program, MOPAC, which now has over 30,000 licensed users and groups worldwide. Dr. Stewart has authored over 150 research papers and his works have been cited over 38,000 times.