Architecting the finite element method pipeline for the GPU

Abstract

The finite element method (FEM) is a widely employed numerical technique for approximating the solution of partial differential equations (PDEs) in various science and engineering applications. Many of these applications benefit from fast execution of the FEM pipeline. One way to accelerate the FEM pipeline is by exploiting advances in modern computational hardware, such as the many-core streaming processors like the graphical processing unit (GPU). In this paper, we present the algorithms and data-structures necessary to move the entire FEM pipeline to the GPU. First we propose an efficient GPU-based algorithm to generate local element information and to assemble the global linear system associated with the FEM discretization of an elliptic PDE. To solve the corresponding linear system efficiently on the GPU, we implement a conjugate gradient method preconditioned with a geometry-informed algebraic multi-grid (AMG) method preconditioner. We propose a new fine-grained parallelism strategy, a corresponding multigrid cycling stage and efficient data mapping to the many-core architecture of GPU. Comparison of our on-GPU assembly versus a traditional serial implementation on the CPU achieves up to an 87 × speedup. Focusing on the linear system solver alone, we achieve a speedup of up to 51 × versus use of a comparable state-of-the-art serial CPU linear system solver. Furthermore, the method compares favorably with other GPU-based, sparse, linear solvers.

Simultaneous head tissue conductivity and EEG source location estimation

“…Since the computational intensity (the ratio of mathematical operations to size of input data) for the integral calculations is fairly high, a significant speedup should be achievable (Wolters et al, 2002; Ataseven et al, 2008). In the literature, an 87× speed-up has been reported using a FEM GPU implementation (Fu et al, 2014). A comparable speed-up applied here would reduce the computation time required for a single iteration to only 7.5 minutes, and could give SCALE convergence in 1 hour or less.…”

Efficient matrix-free GPU implementation of Fixed Grid Finite Element Analysis

“…The principal disadvantage of these strategies relies on the fact that some preprocessing is required, which is time consuming and hardly parallelizable. Recent studies [28] have proposed a compact sparse-matrix data structure and an agglomeration strategy for the assembly step, which is based on atomic addition operations in the device memory. This strategy permits to reduce the memory footprint and avoid the preprocessing.…”

Locality optimized unstructured mesh algorithms on GPUs

“…Instead they use a duplication method similar to that of LULESH and miniAero, as described below. Fu et al [23] also create contiguous patches (blocks) in the mesh to be loaded into shared memory, although they partition the nodes (the to-set) but not the elements (the from-set of the mapping). Furthermore, they do not load all data into shared memory, only what is inside the patch.…”