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Delayed approximate matrix assembly in multigrid with dynamic precisions

Murray, Charles D.; Weinzierl, Tobias

Delayed approximate matrix assembly in multigrid with dynamic precisions Thumbnail


Charles D. Murray


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.


Murray, C. D., & Weinzierl, T. (2021). Delayed approximate matrix assembly in multigrid with dynamic precisions. Concurrency and Computation: Practice and Experience, 33(11), Article e5941.

Journal Article Type Article
Acceptance Date Jun 9, 2020
Online Publication Date Jul 22, 2020
Publication Date Jun 10, 2021
Deposit Date May 25, 2020
Publicly Available Date Jul 24, 2020
Journal Concurrency and Computation: Practice and Experience
Print ISSN 1532-0626
Electronic ISSN 1532-0634
Publisher Wiley
Peer Reviewed Peer Reviewed
Volume 33
Issue 11
Article Number e5941
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Published Journal Article (Advance online version) (1.6 Mb)

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Copyright Statement
Advance online version © 2020 The Authors. Concurrency and Computation: Practice and Experience published by John Wiley & Sons, Ltd. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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