Paper Info
Title
Algorithm 967: A Distributed-Memory Fast Multipole Method for Volume Potentials.
Abstract
The solution of a constant-coefficient elliptic Partial Differential Equation (PDE) can be computed using an integral transform: A convolution with the fundamental solution of the PDE, also known as a volume potential. We present a Fast Multipole Method (FMM) for computing volume potentials and use them to construct spatially adaptive solvers for the Poisson, Stokes, and low-frequency Helmholtz problems. Conventional N-body methods apply to discrete particle interactions. With volume potentials, one replaces the sums with volume integrals. Particle N-body methods can be used to accelerate such integrals. but it is more efficient to develop a special FMM. In this article, we discuss the efficient implementation of such an FMM. We use high-order piecewise Chebyshev polynomials and an octree data structure to represent the input and output fields and enable spectrally accurate approximation of the near-field and the Kernel Independent FMM (KIFMM) for the far-field approximation. For distributed-memory parallelism, we use space-filling curves, locally essential trees, and a hypercube-like communication scheme developed previously in our group. We present new near and far interaction traversals that optimize cache usage. Also, unlike particle N-body codes, we need a 2:1 balanced tree to allow for precomputations. We present a fast scheme for 2:1 balancing. Finally, we use vectorization, including the AVX instruction set on the Intel Sandy Bridge architecture to get better than 50% of peak floating-point performance. We use task parallelism to employ the Xeon Phi on the Stampede platform at the Texas Advanced Computing Center (TACC). We achieve about 600gflop/s of double-precision performance on a single node. Our largest run on Stampede took 3.5s on 16K cores for a problem with 18e+9 unknowns for a highly nonuniform particle distribution (corresponding to an effective resolution exceeding 3e+23 unknowns since we used 23 levels in our octree).
Year
DOI
Venue
2016
10.1145/2898349
ACM Trans. Math. Softw.
Keywords
Field
DocType
FMM,N-body problems,potential theory
Chebyshev polynomials,Volume integral,Xeon Phi,Theoretical computer science,Fast multipole method,Elliptic partial differential equation,Mathematical optimization,Convolution,Task parallelism,Parallel computing,Algorithm,Mathematics,Octree
Journal
Volume
Issue
ISSN
43
2
0098-3500
Citations 
PageRank 
References 
6
0.48
21
Authors
2
Name
Order
Citations
PageRank
Dhairya Malhotra1997.38
George Biros293877.86