Parallel finite element technique using Gaussian belief propagation

Abstract

a b s t r a c tThe computational efficiency of Finite Element Methods (FEMs) on parallel architectures is severely limited by conventional sparse iterative solvers. Conventional solvers are based on a sequence of global algebraic operations that limits their parallel efficiency. Traditionally, sophisticated programming techniques tailored to specific CPU architectures are used to improve the poor performance of sparse algebraic kernels. The introduced FEM Multigrid Gaussian Belief Propagation (FMGaBP) algorithm is a novel technique that eliminates all global algebraic operations and sparse data-structures. The algorithm is based on reformulating the FEM into a distributed variational inference problem on graphical models. We present new formulations for FMGaBP, which enhance its computation and communication complexities. A Helmholtz problem is used to validate the FMGaBP formulation for 2D, 3D and higher FEM degrees. Implementation techniques for multicore architectures that exploit the parallel features of FMGaBP are presented showing speedups compared to open-source libraries, specifically deal.II and Trilinos. FMGaBP is also implemented on manycore architectures in this work; Speedups of 4.8X, 2.3X and 1.5X are achieved on an NVIDIA Tesla C2075 compared to the parallel CPU implementation of FMGaBP on dual-core, quad-core and 12-core CPUs respectively.

Parallel Multigrid Acceleration for the Finite-Element Gaussian Belief Propagation Algorithm

“…The FGaBP algorithm solves the FEM in parallel element by element without the need to assemble a global sparse matrix. FGaBP can be shown empirically to reach high parallel efficiency as the scale of the FEM problem increases [6]. However, like most of the iterative solvers, the FGaBP convergence rate tends to stall when executed on fine meshes.…”

“…However, such algorithms, which are derived based on pairwise interconnect assumptions on the underlying graphical model, suffer mostly from lack of convergence when diagonal dominance properties are not met. To address these convergence shortcomings and improve the parallel efficiency of the FEM computation the FEM-GaBP (FGaBP) algorithm was recently introduced, which was derived directly from a FEM variational formulation [6]. The FGaBP algorithm solves the FEM in parallel element by element without the need to assemble a global sparse matrix.…”

Solving finite-element time-domain problems with GaBP

“…This approach can be considered as a competitor to the discontinuous Galerkin time-domain (DGTD) method [5], which solves the problem element-wise and the elements communicate with each other by passing numerical flux values. An important advantage of the proposed method over the DGTD is its unconditional stability.…”

Acceleration of the Finite-Element Gaussian Belief Propagation Solver Using Minimum Residual Techniques

“…Distributed message schedules can flexibly be varied on the structure of the FEM-FG graph, such as element merging, allowing adaptable memory bandwidth utilization and enhanced overall parallel efficiency [4]. Fig.…”

“…ARALLEL methods, such as the recently introduced finite-element Gaussian belief propagation (FGaBP) method [1], address the challenging problem of attaining high computational scalability on manycore computing architectures used in high-performance computing platforms. The FGaBP algorithm, when adapted in a multigrid setting [2], demonstrated considerable scalability over the conventional finite-element method (FEM) software.…”