Delayed approximate matrix assembly in multigrid with dynamic precisions. (22nd July 2020)

• Record Type: Journal Article
• Authors: Murray, Charles D.; Weinzierl, Tobias
• Other Names: Wang Zhibo guestEditor.; Jiang Lin guestEditor.; Suman Bilial guestEditor.; Wyrzykowski Roman guestEditor.; Szymanski Boleslaw K. guestEditor.; Bentes Cristiana Barbosa guestEditor.; França Felipe M.G. guestEditor.; Marzulo Leandro Augusto Justen guestEditor.; Mencagli Gabriele guestEditor.; Pilla Mauricio Lima guestEditor.
• Is Part Of: Concurrency and computation. Volume 33:Number 11(2021)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2020-07-22
• Subjects: algebraic‐geometric multigrid -- asynchronous multigrid -- delayed operator computation -- dynamically adaptive Cartesian grids -- finite element assembly -- mixed precision computing; Parallel processing (Electronic computers) -- Periodicals; Parallel computers -- Periodicals; 004.35
• DOI: 10.1002/cpe.5941
• Languages: English
• ISSNs: 1532-0626

Abstract

Summary: The accurate assembly of the system matrix is an important step in any code that solves partial differential equations on a mesh. We either explicitly set up a matrix, or we work in a matrix‐free environment where we have to be able to quickly return matrix entries upon demand. Either way, the construction can become costly due to nontrivial material parameters entering the equations, multigrid codes requiring cascades of matrices that depend upon each other, or dynamic adaptive mesh refinement that necessitates the recomputation of matrix entries or the whole equation system throughout the solve. We propose that these constructions can be performed concurrently with the multigrid cycles. Initial geometric matrices and low accuracy integrations kickstart the multigrid iterations, while improved assembly data is fed to the solver as and when it becomes available. The time to solution is improved as we eliminate an expensive preparation phase traditionally delaying the actual computation. We eliminate algorithmic latency. Furthermore, we desynchronize the assembly from the solution process. This anarchic increase in the concurrency level improves the scalability. Assembly routines are notoriously memory‐ and bandwidth‐demanding. As we work with iteratively improving operator accuracies, we finally propose the use of a hierarchical, lossy compression scheme such that the memory footprint is brought down aggressively where the system matrix entries carry little information or are not yet available with high accuracy.