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

Abstract

We introduce a novel parallel multigrid algorithm, referred to as the finite-element multigrid Gaussian belief propagation (FMGaBP), to accelerate the convergence of the recently introduced finite-element Gaussian belief propagation solver. The FMGaBP algorithm processes the FEM computation in a fully distributed and parallel manner, with stencil-like element-by-element operations, demonstrating high parallel efficiency. The results for both sequential as well as parallel message scheduling versions of FMGaBP demonstrate high convergence rates independent of the scale of discretization on the finest mesh. In comparison with the multigrid preconditioned conjugate gradient (MG-PCG) solver, the FMGaBP algorithm demonstrates considerable iteration reductions as tested by Laplace benchmark problems. In addition, the parallel implementation of FMGaBP shows a speedup of 2.9 times over the parallel implementation of MG-PCG using eight CPU cores.

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

“…The FGaBP algorithm, as shown in [2], can be restarted from an arbitrary approximate solution by correspondingly approximating its intermediate messages along the FEM-FG graph. At the IR step, the method needs to perform global operations, such as SMVM and dot products; however, in between the IR iterations, the FGaBP can perform a number of update sweeps maintaining its distributed nature.…”

“…Since the FGaBP messages take Gaussian forms, each message is composed of two parameters: 1) a first-order parameter (β) and 2) a second-order parameter (α). In [2], it was shown that the FGaBP can be adapted into a completely distributed and stationary multigrid process resulting in high computational scalability. In essence, the FGaBP exploits the inherent structure of the FEM problem resulting in localized computational operations on small matrices of size n or less, where n represents the number of densely connected variables in the resulting computational structure.…”

“…The FGaBP algorithm, when adapted in a multigrid setting [2], demonstrated considerable scalability over the conventional finite-element method (FEM) software. This was a direct consequence of reformulating the FEM as a probabilistic method in FGaBP, where computations are carried out using distributed message updates on a matrix-free data structure.…”

“…Most importantly, both the FGaBP method and its multigrid-adapted (FMGaBP) solver [2] eliminate the need to perform global algebraic operations, such as sparse matrix vector multiplies (SMVMs). Nonetheless, to help extend the applicability of the FGaBP solver to a wider range of applications, we here explore combining it using residual minimization with variants of Krylov subspace methods.…”

Parallel finite element technique using Gaussian belief propagation

“…The basic FGaBP formulation was previously demonstrated in [28,30] for 2D Laplace problems using both triangular and quadrilateral FEM elements as provided by the libraries GetFEM++ and deal.II. In this work, we verify the numerical results of the new AU-FGaBP formulation using the definite Helmholtz problem with a known solution for 2D and 3D domains as well as higher order FEM elements.…”