Abstract
Parallel Block Jacobi (PBJ) [1] is an asynchronous spatial domain decomposition with application in solving the neutron transport equation due to its extendibility to massively parallel solution in unstructured spatial meshes (grids) without the use of the computationally complex and expensive sweeps required by the Source Iteration (SI) method in these applications. [2] However, PBJ iterative methods suffer a lack of iterative robustness in problems with optically thin cells, [1] which we have previously demonstrated to be a consequence of PBJ’s asynchronicity. To mitigate this effect, we have developed multiple PBJ / SI hybrid methods which employ a PBJ method (Parallel Block Jacobi - Integral Transport Matrix Method (PBJ-ITMM) or Inexact Parallel Block Jacobi (IPBJ)) along with SI. [3,4] In this work, we perform a parametric study to determine performance of numerous PBJ / SI hybrid methods as a function of multiple problem parameters. This parametric study reached 5 main conclusions: 1) our hybrid approach is more effective with PBJ-ITMM than with IPBJ, 2) for PBJ-ITMM, there is a hybrid method that mitigates the aforementioned iterative slowdown in optically thin cells without diminishing the method’s potential parallelism in unstructured grids, 3) this hybrid method is most effective in problems with large, continuous regions of very thin cells, 4) the best performing hybrid method consistently executes within a factor of ten slower than current state-of-the-art acceleration methods that are not efficiently extendable to the massively parallel regime, and 5) both PBJ-ITMM and IPBJ are observed to be viable approaches for our desired applications. In the pursuit of implementing PBJ-ITMM in unstructured grids, we conclude with a description of the Green’s Function ITMM Construction (GFIC) algorithm, which allows for the ITMM matrices to be constructed using the pre-existing SI sweep algorithm already present in unstructured grid SN transport codes.
Highlights
Parallel Block Jacobi (PBJ) [1] is a spatial domain decomposition that partitions a spatial mesh into multiple asynchronous sub-domains; asynchronous meaning that the incoming angular fluxes on subdomain interfaces are lagged by an iteration
This parametric study reached 5 main conclusions: 1) our hybrid approach is more effective with PBJ-ITMM than with IPBJ, 2) for PBJ-ITMM, there is a hybrid method that mitigates the aforementioned iterative slowdown in optically thin cells without diminishing the method’s potential parallelism in unstructured grids, 3) this hybrid method is most effective in problems with large, continuous regions of very thin cells, 4) the best performing hybrid method consistently executes within a factor of ten slower than current state-of-the-art acceleration methods that are not efficiently extendable to the massively parallel regime, and 5) both PBJ-ITMM and IPBJ are observed to be viable approaches for our desired applications
This work was performed in pursuit of obtaining algorithms for solving the transport equation on unstructured grids that are suitable for massively parallel computers
Summary
Parallel Block Jacobi (PBJ) [1] is a spatial domain decomposition that partitions a spatial mesh into multiple asynchronous sub-domains; asynchronous meaning that the incoming angular fluxes on subdomain interfaces are lagged by an iteration. [1] the number of required iterations grows without bound as the optical thickness of cells decreases, implying lack of iterative robustness This lack of robustness is a consequence of PBJ’s asynchronicity, as sub-domains only exchange information with adjacent sub-domains after each iteration, while distant sub-domains become increasingly coupled with thinning cells. To mitigate this issue, we have developed numerous PBJ / SI hybrid methods [3,4] which utilize the synchronicity of SI to improve the convergence rate of a PBJ method. The AH method was motivated by the observation that in almost all homogeneous problems, in the preconditioning approach, one method was primarily responsible for the convergence of the problem, with little contribution from the other
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