University:
Boston University
Course:
GRS CS 910 | Directed Study: Systems
Academic year:

2024
Views:

436

Pages:

25
Author:

forrastaten54

Mixed precision with reliable updates As discussed yesterday, in the usual method of iterative refinement (or "defect correction"), the Krylov subspace is thrown away at every restart: ro = b - Axo; k = 0; while ||rk||> e do Solve Apk+1 rk to precision e"; Ik+1Tk+Pk+1+ k = k + 1; end An alternative is "reliable updates," originally introduced to combat residual drift caused by the erratic convergence of BiCGstab Mixed precision with reliable updates New (?) idea is to apply this approach to mixed precision. (Clark et al., arXiv:0911.3191) rob-Axo: for: 10 = 0; k=0; while e do Low precision solver iteration: ずんずんまん→+1 if ||+|| SM(7) then 21+1=21++ +1=0: 1=1+1; end δ M(4) Adenotes reduced precision. is a parameter determining the frequency of updates. denotes the maximum iterated residual since the last update. k=k+1: end Reliable updates seems to win handily at light quark masses (and is no worse than iterative refinement at heavy masses). Performance model For the Wilson matrix-vector product, we have: (1320 flops/site) x (L¹/2 flops) = 660L4 flops (24/2 x 4 bytes/boundary site) × (8L³/2 sites) = 192L3 bytes 660L4 Perf 192L3 Bandwidth Bandwidth [MB/s] = 0.29(Perf [Mflop/s]) L MessageSize [Bytes] = 24L3 Performance model This model is pessimistic in the sense that it assumes we are going to parallelize in all 4 dimensions. For small numbers of nodes, this is never optimal. It is optimistic in all other respects (assuming perfect overlapping of communication and computation, for example), telling us the best we can possibly do. For this exercise, we're interested in the strong scaling regime (smallest possible sub-volumes). How small can we go before the surface/volume ratio limits us?