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multigrid-design-notes Design, development and tuning of LQCD multigrid solvers for physical-mass Möbius DWF on Frontier (AMD MI250X); covers HDCG, PVdagM two-level solver, coarse operator performance, and Lüscher deflation of the coarse solve
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LQCD Multigrid: Design, Development and Tuning

Physical problem

Physical-mass Möbius DWF: Ls=24, b=1.5, c=0.5, M5=1.8, mass=0.00078. 48³×96 lattice. MPI geometry 3.6.4.4 (288 ranks / GCDs on Frontier). Target: accelerate HMC fermion force and CG solves.

Two solver paths

Path 1: HDCG (TwoLevelADEF2 on MdagM)

  • File: examples/Example_mdagm.cc
  • Operator: MdagM (Hermitian positive definite); outer solver is ADEF2 CG.
  • Subspace: 60 near-null vectors via CG inverse iteration (CreateSubspace).
  • Coarse geometry: block {4,4,3,4}, coarse lattice 12×12×16×24, Ls_coarse=1.
  • Coarse operator: GeneralCoarsenedMatrix (Petrov-Galerkin, npoint=33, NextToNearestStencil).
  • Smoother: fixed-iteration CG on shifted operator (M†M + lo), lo=hi/80, 20 iters.
  • Coarse solve: CG with DeflatedGuesser using 60 chi deflation vectors.
  • Chi vectors: extracted BEFORE block-GS by diagonalising W_ij=<ψ_i|M†M|ψ_j>, computing chi_k = Σ_i V[i,k] ψ_i. These are global near-null combinations; block-GS destroys this.
  • Deflation effect: 1089 coarse CG iters (undeflated) → 329 with 60 chi vectors.
  • Best result: 274 outer ADEF2 iters, ~400s on 288 ranks Frontier.
  • Coarse MVM performance: 541.7 GFlop/s kernel, 1119.6 GB/s (70% HBM), 97% roofline. MPI latency = 1188 μs = 37% of 3.2 ms per coarse MVM call. Single-RHS is bandwidth-bound.

Path 2: PVdagM two-level PGCR

  • File: examples/Example_pvdagm.cc and examples/Example_pvdagm_defl.cc
  • Operator: PVdagM = PV†M (non-Hermitian); outer solver is PGCR.
  • PV is the Pauli-Villars (mass=1) Möbius operator. PVdagM has exact zero modes.
  • Subspace: 60 near-null vectors via GCR inverse iteration (CreateSubspaceGCR). GCR setup is slow: each vector takes O(600) PGCR steps, total ~4100s setup on Frontier.
  • Coarse geometry: block 2^4 (lattice halved each dim), Ls_coarse=1.
  • Coarse operator: GeneralCoarsenedMatrix with non-Hermitian coarsening.
  • Preconditioner: MGPreconditioner V-cycle (pre-smooth, project, coarse solve, promote, post-smooth).
  • Smoother: PGCR on ShiftedPVdagM (shift=0.01).
  • Baseline (no deflation, 5e-2 coarse tol): 59 outer iters, 300s solve time. Coarse solve: 4.58s/call, 250 PGCR steps, NEVER converges ("did not converge" every call).
  • Outer iteration count vs coarse tolerance: 5e-2→59 iters, 1e-1→63 iters, 3e-2→34 iters. But at 3e-2 without deflation: 1000+ coarse PGCR steps per call (useless).

Key implementation work: GeneralCoarsenedMatrix performance

Grid/algorithms/multigrid/GeneralCoarsenedMatrix.h:

  1. accelerator_barrier fix: acceleratorBarrier() is not a Grid macro; correct call is accelerator_barrier(dummy) (takes a dummy argument). This caused SIGBUS on Frontier.

  2. Coalesced FT kernel: In CoarsenOperator, the loop filling A_vsss was serialised over j. Changed to:

    accelerator_for(sss, osites, nbasis, {
    int j = acceleratorSIMTlane(nbasis);
    A_v[sss](i,j) = FT_v[sss](j);
});

    This gives coalesced HBM access (nbasis consecutive elements per warp lane).

  3. Batched CoarsenOperator: Used MultiRHSBlockProject to batch all npoint=33 stencil directions in one GEMM call per basis vector, replacing serial blockProject calls. Reduced projection from dominant bottleneck to 12% of CoarsenOperator time. mat (linop applications) now dominates at 83%.

Lüscher deflation of the coarse solve (Example_pvdagm_defl.cc)

Theory (Lüscher arXiv:0706.2298, Section A.3)

For near-null vectors {ψ_s} of operator D, the Petrov-Galerkin initial guess is: guess = Ψ W⁻¹ Ψ† src where W_st = <ψ_s|D|ψ_t> and Ψ is the matrix of ψ columns.

Condition <ψ_s | src - D*guess> = 0 gives W c = b, b_t = <ψ_t|src>. No SVD needed — W is dense, invert directly (LU). The U,V from SVD are unitaries within the ψ-basis and cancel in W⁻¹; direct inverse is cleaner.

Diagnostic results (job 4948520, before deflation)

Fine projected matrix W (60×60):

  • ||W|| = 0.02399 — near-null vectors are genuinely small.
  • Singular values: range [0.00169, 0.00529], ratio ~3:1. Well-conditioned inverse.

Coarse null matrix C_kl = <P ψ_k | A_coarse | P ψ_l>:

  • ||C|| = 0.02399 — identical to ||W||. Galerkin property is exact.
  • ||C - C†|| / ||C|| = 2.59e-9 — C is Hermitian to machine precision. Despite PVdagM being non-Hermitian, the projected coarse matrix is numerically Hermitian.
  • C singular values match W singular values exactly (coarsening faithful).

This means: the coarse near-null vectors (projections of fine ψ_k) are exact near-null vectors of the coarse operator, AND the deflation can use a Hermitian eigensolver.

Implementation

CoarseDeflatedGuesser (in Example_pvdagm_defl.cc, before MGPreconditioner):

template<class Field>
class CoarseDeflatedGuesser : public LinearFunction<Field> {
    const std::vector<Field> &chi;   // coarse eigenvectors of C_sym
    const std::vector<RealD> &eval;  // eigenvalues
public:
    void operator()(const Field &src, Field &guess) {
        guess = Zero();
        for (int k = 0; k < chi.size(); k++)
            axpy(guess, TensorRemove(innerProduct(chi[k], src)) / eval[k], chi[k], guess);
    }
};

Chi vector construction (in runMG, after CoarsenOperator and C computation):

  1. Project pre-GS fine subspace to coarse grid: psi_coarse[k] = P ψ_k
  2. Compute C_sym = (C + C†)/2
  3. SelfAdjointEigenSolver(C_sym) → eigenvalues lambda[k], eigenvectors V
  4. chi_coarse[k] = Σ_i V[i,k] * psi_coarse[i] (satisfies <chi_j|A_c|chi_k> = lambda_k δ_jk)
  5. chi_eval[k] = lambda[k]

MGPreconditioner modified to hold CoarseSolver &_CoarseGuesser:

  • Added as constructor parameter and member reference.
  • In V-cycle: replaced Csol = Zero() with _CoarseGuesser(Csrc, Csol) before _CoarseSolve.

Note on pre-GS subspace: The subspace is copied at the top of runMG BEFORE CoarsenOperator modifies it via block-GS. This pre-GS copy is essential for computing W (fine projected matrix) and psi_coarse. After CoarsenOperator, AggregatesPD.subspace is the block-orthonormal basis.

Expected benefit

Without deflation at 3e-2 coarse tolerance: 1000+ PGCR steps per call, never converges. With deflation: near-null modes removed by guesser; PGCR sees well-conditioned complement. Target: O(100-200) coarse PGCR steps per call, 34 outer iterations, large total time reduction.

Memory considerations (Frontier, MI250X, 64 GB HBM per GCD)

Each fine fermion field: ~162 MB per rank (Ls=24, 48³×96/288 local vol, spincolour=12 complex). 60 fine fields (subspace): ~9.7 GB. The pre-GS subspace COPY in runMG adds another 9.7 GB. Chi/null vector construction: build one at a time into a temp field, project immediately. Do NOT allocate 2×nbasis additional fine fields (would add 19 GB → OOM).

Coarse vectors (~2 MB each): 120 coarse vectors (psi_coarse + chi_coarse) = ~240 MB. Fine.

Roofline analysis (coarse MVM, single-RHS)

Arithmetic intensity: ~0.5 flops/byte. Roofline crossover: 119.7 flops/byte. Deeply memory-bandwidth bound. Peak HBM: 1600 GB/s. Achieved: 1119.6 GB/s (70%). MPI latency: 1188 μs = 37% of 3.2 ms per call. Irreducible for single-RHS. Multi-RHS is the fundamental solution for throughput, but cannot be used in HMC (each trajectory has a new gauge field → new coarse operator).

Fine operator performance model (measured 2026-08-20, Frontier, 288 ranks)

48³×96, Ls=24, mpi 3.6.4.4 (36 nodes), --accelerator-threads 8 --shm-mpi 1 --comms-overlap.

Measured decomposition of one Möbius M (roctx trace, MG smoother)

ms
DhopInternalOverlappedComms (wall) 5.549
... of which self 4.746
hipStreamSynchronize x7 2.158
whole L1L2-Vcycle - Smoothers step = one M 7.128

Per M: one Meooe/DW (~4.7 ms, comms dominated), two M5D (~0.5 ms each), axpy (~0.4 ms). Dslash arithmetic is only 1.4 (interior) + 0.4 (exterior) ms. ~63% of the sequence has MPI in flight; comms and compute are essentially serialised despite --comms-overlap.

PVdagM.Op = M + Mdag = 2x7.128 = 14.3 ms. CoarsenOperator tmat measured 30.695 s / 1980 Ops = 15.5 ms/Op, i.e. within 9% of the fine operator's own in-production cost -- the coarsening is not the problem.

Benchmark_dwf, fp64 (mflop/s per node, avg of 4 runs)

accelerator-threads comms fp64 comms fp32 (SloppyComms)
16 1,769,808 2,842,248
8 1,803,904 2,862,619
4 1,802,937 2,731,611
  • Comms precision is the dominant knob: +59%. --shm-mpi 1 a further 2-4%.
  • Thread count is second-order (2-5% spread). A wavefront-fill model (block = Nsimd x nt, fp64 halves Nsimd, so nt=16 refills 64) predicts 16 and is WRONG -- the Dslash is bandwidth bound, not occupancy bound. The flatness is itself the disproof. 8 remains best, as tuned under fp32.
  • Trap: Benchmark_dwf defaults to Ls=16 and takes -Ls (single dash). Check the reported mflop/s against 1320*V5*ncall/t before comparing to Ls=24 work, or you will infer a 2x deficit that is not there.

Optimisation model (per M, from 7.1 ms)

Overlappable work = DslashInterior + DslashExterior + addQmu(noop) + axpby + M5D

  • gather = 1.4+0.4+0+0.4+0.5+0.4 ~ 3.1 ms, against 4.7 ms irreducible comms.
ms speedup
now 7.5 -
overlap only max(3.1,4.7) = 4.7 1.6x
overlap + sloppy comms 3.1 (compute bound) 2.3x

The two are independent. Overlap is 1.6x at zero precision cost -- do it unconditionally. Sloppy is the further 1.4x and is the only piece carrying a judgement.

Where sloppy comms may be applied

  • Whole V-cycle (fine smoother + coarse solves): safe. Only the outer PGCR operator application must be exact; the outer Krylov corrects the rest every iteration.
  • CoarsenOperator: different risk category. Error goes into A_c permanently and propagates to its inverse; it is not corrected by the outer solver. The dense bottom's certificate does NOT detect this. The import certificate compares Dense x against Op.M x; if A_c was built sloppily both sides carry the same error and it still reads ~1e-8. VERIFY is likewise the exact inverse of whatever matrix it was handed. Both certify the import and inversion, not the accuracy of A_c. The instruments that DO see it: coarsen twice and compare BLAS_A element by element against an exact run (the comparison Test_coarse_v2_coarsen already performs between V1 and V2), or the outer iteration count and time to solution.
  • Does not reach the coarse levels. SloppyComms sets a flag on Stencil/StencilEven/StencilOdd; V2's M uses PaddedCell::Face_exchange, which sends raw vobj bytes with no precision option. Coarse levels are less bandwidth sensitive, so this matters less than it sounds.
  • Object sharing caveat: the flag is per fermion operator. A single MobiusFermionD shared between PVdagM and ShiftedPVdagM cannot be sloppy for one and exact for the other -- the preconditioner needs its own pair on the same gauge field.

Pipelined mrhs (idea iii)

Ripple independent vectors so Dwbegin(n) overlaps Dwcomplete(n-1); needs double-buffered comms (two DWF operators alternating, or the stencil preserve_shm flag) and a larger SHM segment. CoarsenOperator is the natural first customer: its 1980 applications are already fully independent and the single-RHS driver is already the loop shape, so no algorithmic change is needed.

Three-level perspective

The chi deflation of the coarse solve can be viewed as a third level: coarsening all the way to 1⁴ × nbasis. The 60 near-null vectors of the coarse operator span this third level's null space. The Lüscher guesser is the exact inverse on this space.

References

  • HDCG paper (2014): arXiv:1409.xxxx (P. Boyle) — ADEF2 CG with coarse deflation.
  • Lüscher (2007): arXiv:0706.2298 — non-Hermitian deflation, Section A.3.
  • Physical DWF multigrid: arXiv:2409.03904.
  • Grid library: github.com/paboyle/Grid.
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