A parallel fast boundary element method using cyclic graph decompositions

Abstract

We propose a method of a parallel distribution of densely populated matrices arising in boundary element discretizations of partial differential equations. In our method the underlying boundary element mesh consisting of n elements is decomposed into N submeshes. The related N ×N submatrices are assigned to N concurrent processes to be assembled. Additionally we require each process to hold exactly one diagonal submatrix, since its assembling is typically most time consuming when applying fast boundary elements. We obtain a class of such optimal parallel distributions of the submeshes and corresponding submatrices by cyclic decompositions of undirected complete graphs. It results in a method the theoretical complexity of which is O((n/ √ N) log(n/ √ N)) in terms of time for the setup, assembling, matrix action, as well as memory consumption per process. Nevertheless, numerical experiments up to n = 2744832 and N = 273 on a real-world geometry document that the method exhibits superior parallel scalability O((n/N) log n) of the overall time, while the memory consumption scales accordingly to the theoretical estimate.

Matching Point Clouds with STL Models by Using the Principle Component Analysis and a Decomposition into Geometric Primitives

“…Our work on hierarchical PCA was inspired by [4,5], where the authors used the method for clustering of 3D surface meshes into binary trees. The method is invariant with respect to rigid body modes, which may make it superior when compared, for instance, to oct-tree algorithms.…”

“…We refer to [3], which focuses on machine learning. The PCA also has good application for sparsification of densely populated matrices arising in boundary element methods [4,5].…”

Parallel Adaptive Cross Approximation for the Multi-trace Formulation of Scattering Problems

“…Due to a limited scope of this paper, we refer the reader to [3,16] for more details. The parallelization of the method based on the cyclic graph decomposition was presented in [13] where only certain special numbers of processors were discussed. In [12] we further extended the approach to support general number of processors.…”

“…Afterwards, the P 2 blocks are assembled in parallel via P MPI processes. The workload distribution follows the so-called cyclic graph decomposition introduced in [13] for special values of P and later generalized in [12]. This way, each MPI process requires O(n/ √ P ) mesh data for the assembly of the blocks and actively works with O(n/ √ P ) degrees of freedom during matrix-vector multiplication, which has a positive effect on the efficiency of the required MPI synchronization phase.…”

“…In Sect. 3 we propose a strategy to parallelize the assembly of the MTF matrix blocks and their application in an iterative solver based on the approach presented in [11][12][13] for single domain problems. Except for the distributed parallelism, the method takes full advantage of the BEM4I library [14,20,21] and its assemblers parallelized in shared memory and vectorized by OpenMP.…”