Fast Multipole Method for Coulomb Interaction and Other
Fast Multipole Method for Coulomb Interaction and Other Non-Oscillating Interaction Using Cartesian Tensor and Differential Algebra He Zhang 1, He Huang 2, Rui Li 1, Jie Chen 1, Li-Shi Luo 2 1. Jefferson Lab 2. Old Dominion University AAC 2016, 08/02/2016 National Harbor, MD
Outline q Introduction of the FMM q The FMM using traceless totally symmetric Cartesian tensor q Numerical noise q Boundary condition q Example of tracking simulation using FMM q FMM for a general non-oscillating function
Introduction of FMM He Zhang ---3 ---
Strategy of FMM q Hierarchical decomposition of space Ø Finer level for higher charge density Ø Empty boxes ignored Ø Tree structure of boxes q Five different relations between boxes Ø Touch each other (1) Ø Well separated (2) Ø Ill separated (3, 4) Ø No interaction (5) He Zhang ---4 ---
Steps of FMM q Hierarchical tree construction q Upwards. Calculate the multipole expansion of all boxes q Downwards. Calculate the local expansion and transfer them to the childless boxes Ø Converted from multipole expansions (relation 2) Ø Inherited from ancestor boxes Ø Calculated from the source charges (relation 4) q Potential/Field calculation Ø Near region field by Coulomb formula (relation 1) Ø Far region field from local expansions Ø Far region field from multipole expansions (relation 3) He Zhang ---5 ---
Algorithms in the FMM Family q For Coulomb potential Ø Spherical harmonic functions (L. Greengard, V. Roklin) Ø Totally symmetric tensor (B. Shanker, H. Huang) Ø Differential algebra (H. Zhang, M. Berz) Ø Traceless totally symmetric tensor (H. Huang, H. Zhang) q For a general non-oscillating kernel function Ø Kernel independent FMM (L. Ying) Ø Black box FMM (E. Darve) Ø Totally symmetric tensor and differential algebra He Zhang ---6 ---
Traceless Totally Symmetric Tensor FMM He Zhang ---7 ---
Traceless Totally Symmetric Tensor FMM Totally symmetric tensor (n+1)(n+2)/2 terms n In nth order Total terms 0 1 1 1 1 3 4 3 4 2 9 13 6 10 5 9 3 27 40 10 20 7 16 4 81 121 15 35 9 25 5 243 364 21 56 11 36 6 729 1093 28 84 13 49 He Zhang ---8 ---
Formulas q Taylor expansion: q Multipole expansion and local expansion: q Translate a multipole expansion (child box to parent box) q Convert a multipole expansion into a local expansion (well separated boxes) q Translate a local expansion (parent box to child box) q Field He Zhang ---9 ---
Numerical Result q 3 D space charge field calculation for 1 million electrons with Gaussian distribution on a laptop with Intel i 7 -3630 QM processor (3. 4 GHz) Potential Order Time (s) 2 11. 13 4 29. 90 6 76. 46 8 181. 13 Relative error Field Order Time (s) 2 17. 20 4 39. 63 6 87. 86 10 407. 19 Relative error q Linear scale with particle number He Zhang ---10 ---
Numerical Noise He Zhang ---11 ---
Boundary Condition • Boundary value problem: • Boundary integral equation (Green’s function G and its derivative F): • Linear system: • Solve it iteratively: FMMBEM [1] Fast Multipole Boundary Element Method - Theory and Applications in Engineering, by Dr. Yijun Liu, Cambridge University Press He Zhang ---12 ---
Parallelization • Parallelization of the FMM is challenging due to the unbalanced tree structure. • Great achievement in the previous decade. • Work from the two groups: • exa. FMM: Yokoda & Barba, http: //www. bu. edu/exafmm/ https: //github. com/exafmm Parallel efficiency is 93% on 2048 processes for 10 8 particles [1] • p. KIFMM: Biros, Mang, Ying, et al. , http: //padas. ices. utexas. edu/software/ 90 billion unknowns on 32, 767 MPI processes, Gordon Bell Prize 2010 [2] • Both runs on CPU-GPU hybrid cluster • The FMM framework are separated from the kernel. For a new interaction, only need to revise the kernel code. [1] “A tuned and scalable fast multipole method as a preeminent algorithm for exascale systems”, Rio Yokota and Lorena A Barba, Int. J. High-perf. Comput. , online 24 Jan. 2012, doi: 10. 1177/1094342011429952 [2] “Petascale direct numerical simulation of blood flow on 200 K cores and heterogeneous architectures”, Abtin Rahimian, Ilya Lashuk, et al. acm/ieee scxy conference series, pp. 1– 11, 2010, He Zhang ---13 ---
Example of Tracking Simulation using FMM Simulation of the Photoemission process: • 2, 000 3 d Gaussian macro-particle. • 8 th order Runge-Kutta integrator 70 ps 90 ps He Zhang 100 ps ---14 ---
Example of Tracking Simulation using FMM q The first 120 fs, the electron bunch is generated by the laser pulse. He Zhang --15 --
Example of Tracking Simulation using FMM q The following 115 ps, the electron bunch leaves the surface. He Zhang --16 --
Other Non-Oscillating Interactions Order Relative error 2 4 6 8 10 He Zhang ---17 ---
Summary q The FMM scales linearly with the particle number q Grid-free, adaptive to any distribution and geometry q The traceless property helps to reduce the element number to 2 n+1 q Numerical noise can be reduced using soft macro-particles q The boundary value problem can be solved by the FMMBEM q The FMM has been successfully using in particle tracking simulation for photoemission process q It is possible to extend the method for a general nonoscillating function with the help of DA He Zhang ---18 ---
He Zhang --19 --
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