Mixed Precision Perturbation Analysis of Two-Level Domain Decomposition Preconditioners
Résumé
This article is concerned with domain decomposition preconditioners, and more particularly the additive Schwarz method (ASM) with a coarse space based on the GenEO (Generalized Eigenproblem in the Overlap) method, to solve large sparse symmetric positive definite problems. While such preconditioners can achieve scalability by being efficiently parallelized, they still require linear algebra operations in each local subdomain that can be quite expensive in time and memory. Motivated by the emergence of fast, low precision arithmetic in hardware, in this work we aim to reduce the costs of ASM-GenEO preconditioners using mixed precision. In order to determine which parts of these preconditioners can be switched to lower precision, we carry out a perturbation analysis that identifies the sensitivity of the different operations. We obtain theoretical error bounds that show that the precision selection should be guided by the local matrix condition numbers. This provides an actionable criterion for selecting the appropriate precision for each local subdomain while maintaining rigorous guarantees on the efficiency of the preconditioner. We perform extensive numerical experiments to validate our theoretical bounds and assess their sharpness. Overall, this study thus demonstrates that domain decomposition preconditioners can be constructed in mixed precision while reliably controlling their efficiency.
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