Scaling the memory wall using mixed-precision -- HPG-MxP on an exascale machine

Aditya Kashi; Feiyi Wang; Hao Lu; Michael Matheson; Nicholson Koukpaizan; Sarp Oral

REVIEW 2 major objections 5 minor 33 references

Scaling the memory wall using mixed-precision -- HPG-MxP on an exascale machine

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A highly optimized HPG-MxP implementation on a 9,408-node GPU exascale system shows a penalized 1.6x speedup for mixed double-single precision GMRES-IR over double-precision GMRES.

desk verdict A solid engineering benchmark paper: real 1.6x mixed-precision speedup on Frontier, but the full-scale validation penalty is measured at a looser residual than the official 1e-9, so the "no convergence loss at scale" claim is weaker than stated. read the letter →

arxiv 2507.11512 v1 pith:NWXL55LH submitted 2025-07-15 cs.DC cs.NAcs.PFmath.NA

Aditya Kashi , Nicholson Koukpaizan , Hao Lu , Michael Matheson , Sarp Oral , Feiyi Wang This is my paper

classification cs.DCcs.NAcs.PFmath.NA

keywords mixedprecisionGMRESiterativerefinementHPG-MxPbenchmarkmemorybandwidthGauss-SeidelGPUexascalecomputing

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper builds a highly optimized implementation of the HPG-MxP benchmark, which measures how fast a machine solves a memory-bandwidth-limited sparse linear system with GMRES, and uses it to ask whether mixed double-single precision can beat pure double precision on an exascale GPU machine. It reports a 1.6x speedup from running most of GMRES in single precision while keeping residual and solution updates in double precision, on a full-system run of 9,408 nodes reaching 17.23 PFLOPS with 78% weak-scaling efficiency. The result matters because most PDE-based simulation workloads are memory-bandwidth limited and have not benefited from the low-precision arithmetic gains seen in AI; the paper provides a yardstick for what careful mixed-precision optimization can deliver on real sparse solvers.

What carries the argument

The load-bearing object is the GMRES-IR solver with classical Gram-Schmidt with reorthogonalization (CGS2), where all steps except the double-precision residual update and solution update may run in single precision. The preconditioner is one geometric multigrid cycle with multicolor Gauss-Seidel smoothing; the implementation reorders each subdomain into independent sets (eight colors for the 27-point stencil), stores the matrix in ELLPACK format, a padded sparse layout that keeps GPU warps busy, fuses residual calculation with restriction, and overlaps halo communication with interior computation using GPU streams and events. The benchmark's final score is penalized by the validation-phase iteration ratio, which is what converts raw speed into a claim about useful performance.

What would settle it

Run double-precision GMRES and mixed-precision GMRES-IR on the full 9,408-node system without the 10,000-iteration cap, both to a relative residual of 1e-9, and compare the resulting iteration ratio to the 0.968 ratio measured at one node; if the ratio rises materially, the penalized 1.6x speedup overestimates the useful gain.

Watch

Extended reading notes

Core claim

The central claim is that a mixed double-single precision GMRES with iterative refinement (GMRES-IR) can run the HPG-MxP benchmark about 1.6 times faster than an equally optimized double-precision GMRES, while still giving a solution that is close to double-precision quality. The benchmark's penalty mechanism counts floating-point operations of both precisions equally and scales the mixed-precision throughput by the ratio $n_d/n_{ir}$ of iterations needed by the double and mixed solvers to converge nine orders of magnitude, so the reported speedup is a penalized, useful-performance figure. The paper further claims this is the first full-system HPG-MxP run, at 9,408 nodes or 75,264 GPUs, and that validation on a single node's small problem is as stringent as a new full-scale validation in capturing any loss of convergence from mixed precision.

Load-bearing premise

The result rests on the assumption that the ratio of double-precision to mixed-precision iteration counts measured on a single small node carries over to full-system scale, so the penalty applied to the mixed-precision score stays accurate.

Editorial extensions

If this is right

Production sparse solvers that use GMRES and multigrid preconditioning can expect a realizable speedup of about 1.6x from double-single precision without sacrificing final double-precision accuracy, according to these benchmarks.
The optimized kernels sit at the HBM bandwidth roofline, so the speedup comes from halving the bytes moved per operation in the single-precision parts rather than from hiding implementation inefficiencies.
At full-system scale the orthogonalization phase takes a growing share of time because of all-reduce latency, making communication efficiency the main limiter of mixed-precision scaling above thousands of nodes.
Validation on one node gives essentially the same penalty ratio as full-scale validation, so benchmark runs can save substantial time by validating on a small subset of processors.
The benchmark matrix is artificial, so speedups in real applications will depend on the condition numbers and pseudo-spectra of the actual matrices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the paper is right, the 1.6x speedup being close to the 2x bandwidth-only ceiling implies that most remaining time is memory traffic, so the next big gain for production solvers would come from reducing bytes moved, for example by storing auxiliary vectors in lower precision, rather than from faster arithmetic.
The full-scale validation shows that above 64 nodes the double-precision reference no longer reaches a relative residual of 1e-9, so the penalty ratio is measured at a looser residual; a direct test pushing both solvers to 1e-9 at thousands of nodes would reveal whether single-node validation misses any scale-dependent convergence loss.
For matrix-free or nonlinear GMRES variants, the mixed solver's extra low-precision matrix copy would disappear, making the memory-footprint objection smaller and potentially making the speedup easier to realize in production PDE codes.
The benchmark counts single- and double-precision flops equally, so its PFLOPS are mixed-precision and are not directly comparable to HPCG's double-precision PFLOPS despite both being memory-bound.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper delivers what it claims: a heavily optimized HPG-MxP implementation on Frontier, a first full-system run at 9408 nodes (17.23 PFLOPS, 78% weak scaling), and a reproducible 1.6x speedup for double-single GMRES-IR over double GMRES while keeping the validation penalty. The optimizations are sensible and well-described: multicolored Gauss-Seidel, ELLPACK format, compute-communication overlap, fused SpMV-restriction. The roofline and traces substantiate the memory-bandwidth-limited story. The comparison to the reference implementation is honest, and the authors note the xsdk points are read off a graph. Credit where due: the full-scale validation mode is a genuinely useful addition to the benchmark, and the open-source code commitment is good practice. The plot twist is that the full-scale validation doesn't fully validate what it claims. In the "fullscale" mode, at 64 nodes and above, the double-precision GMRES hits the 10,000-iteration cap at relative residual ~1e-5 to 1e-6, not 1e-9. The iteration ratios n_d/n_ir at those scales (1.05, 1.02, 1.07, 0.96) are therefore measured against a much looser target than the standard validation ratio 0.968 at 1e-9. The conclusion that the standard validation "sufficiently captures" convergence loss is not directly established: the comparison at scale happens at 1e-5, not at the 1e-9 tolerance that defines the official penalty. If mixed-precision convergence degrades as the target tightens (single-precision Krylov basis, loss of orthogonality), the penalized 1.6x could overstate useful speedup for applications needing double-precision-accurate solutions. The stress-test concern lands, though it is a moderate rather than fatal issue: the ratios at scale are all close to 1, so the practical risk is bounded. Still, the paper should either run the double solver to 1e-9 at larger scales (if feasible), or explicitly hedge that the penalty ratio is only verified to ~1e-5 at scale. Minor but real: no error bars or repeated-run statistics, and the code/commit details are deferred to the reproducibility appendix without giving specifics in the paper. Those are fixable. The paper is a strong engineering contribution. I'd send it to a serious referee, mostly to pressure-test the validation interpretation. The 1.6x number will get cited regardless; the question is whether the citation is "mixed precision gives 1.6x on Frontier for this benchmark" or "...and this extends unchanged to production tolerances." The former is solid; the latter needs the caveat.

Referee Report

2 major / 5 minor

Summary. The paper presents an optimized implementation of the HPG-MxP benchmark for GPU-based exascale systems, introducing several algorithmic improvements (multicolor Gauss-Seidel, ELLPACK storage, compute-communication overlap, fused SpMV-restriction) that apply to both the mixed- and double-precision GMRES solvers. On Frontier, the authors report a first full-system run at 9408 nodes reaching 17.23 PFLOPS with 78% weak scaling efficiency, and a penalized 1.6x speedup of double-single precision GMRES-IR over double-precision GMRES. They also introduce a 'full-scale validation' mode intended to test whether the benchmark's standard small-scale validation penalty is representative at larger scales.

Significance. The paper's strongest contribution is a carefully engineered, open-source implementation of HPG-MxP with a controlled comparison: the mixed- and double-precision solvers share all optimizations, so the reported 1.6x speedup is not an artifact of comparing an optimized mixed-precision code against an unoptimized double-precision baseline. The roofline plot (Figure 8) substantiates the bandwidth-limited behavior of the kernels, and the traces (Figure 9) support the claimed compute-communication overlap. If the penalized speedup holds at the official 1e-9 tolerance at scale, this would be an important data point showing that mixed precision can deliver practical gains for memory-bound sparse iterative solvers on exascale GPU systems.

major comments (2)

[§4.2, Table 2; §5] The claim in §5 that 'the original benchmark's validation method sufficiently captures any loss of convergence rate' is not directly supported by the full-scale validation data in Table 2. At 64, 128, 1024, and 4096 nodes the double-precision GMRES run stops at the 10,000-iteration cap with an achieved relative residual of about 1e-5 to 1e-6, so the ratios n_d/n_ir (1.050, 1.023, 1.067, 0.958) are measured against a much looser target than the 1e-9 tolerance used in the standard validation (ratio 0.968). The data do not rule out that the iteration ratio between mixed-precision GMRES-IR and double-precision GMRES deteriorates as the target residual is tightened from 1e-5 to 1e-9 at large node counts; if this occurred, the penalized speedup in Figure 5 would overestimate the useful gain for applications requiring double-precision-accurate solutions. The authors should either run the full-scale validation to the official 1e-9 tolerance at additional scales, or provide residual-history evidence that the iteration ratio is tolerance-independent.
[§4, Figure 4] The quantitative comparison to the previous reference implementation (Anzt et al., labelled 'xsdk') in Figure 4 relies on values read off a graph, as the authors themselves acknowledge in §4. Because one of the stated contributions is that the optimized implementation 'achieves much higher performance than the reference implementation,' the paper should report the xsdk numbers directly (or with explicit uncertainty bounds) so the comparison is reproducible and not subject to graph-reading error.

minor comments (5)

[Algorithm 2] The text contains a typographical error: 'Kyrlov' should be 'Krylov', and 'Given's rotations' should be 'Givens rotations'.
[§4.1] The phrase 'owing to their the need to fetch index arrays' is grammatically incorrect; it should be 'owing to the need to fetch index arrays.'
[Figure 5] The text says SpMV and multigrid 'drag the speedup down somewhat,' but the figure does not clearly indicate whether these motifs have speedup above or below 1.0; annotating the figure or explaining the values would improve clarity.
[§4.2] The sentence beginning 'It is clear from the full-scale residual norms...' should specify that at 2 and 8 nodes the achieved residuals (9.98e-10 and 9.99e-10) are just below the 1e-9 threshold, while at larger scales the residuals are three to four orders of magnitude larger.
[§4 (general)] The paper reports no repeated runs or error bars for any performance measurement; at least for a few representative node counts, a repeated-run or variability analysis would strengthen the reported speedup and scaling efficiency.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 1.6x speedup is a measured benchmark outcome with transparent, non-fitted penalty factors.

full rationale

The paper is an empirical performance study rather than a derivation. The central quantity, the 1.6x speedup, is the ratio of measured penalized throughputs of the mixed-precision GMRES-IR solver and the double-precision GMRES reference, where the penalty is fixed by the HPG-MxP benchmark definition (Yamazaki et al.) as the validation-phase iteration ratio n_d/n_ir. These iteration counts are measurements, not fitted parameters, and no quantity is renamed as a prediction after being used to construct it. The full-scale validation in Section 3.3 compares n_d/n_ir at the double-precision solver's achieved residual tau, which at 64 nodes and above is around 1e-5 rather than 1e-9; this is a genuine evidentiary limitation for the conclusion that the standard validation 'sufficiently captures any loss of convergence rate', but it is not a circular step: the full-scale ratios are new measurements made under a different tolerance, not inputs to the speedup calculation. Self-citations (Kashi's 2020 thesis on Gauss-Seidel variants and the 2025 mixed-precision survey) support implementation context and background, not the speedup claim, so they are not load-bearing. No quoted equation reduces to its own input, no fitted value is presented as an independent prediction, and no uniqueness or correctness claim is imported from the authors' prior work. The result is benchmark data, not a derivation circular with its inputs.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted to data; the benchmark's configuration (restart length 30, 320^3 mesh) is standard. The central claim rests on standard numerical linear algebra assumptions and implementation correctness assumptions, the most important being that multicolor Gauss-Seidel and computed communication overlap do not alter convergence behavior.

assumptions (3)

domain assumption Multicolor Gauss-Seidel with 8 colors preserves the multigrid preconditioner's effectiveness so that mixed-precision GMRES-IR requires a comparable number of iterations to double-precision GMRES.
The paper relies on this in Section 4 (validation ratios near 1) but does not prove it for all scales; it is tested empirically only up to 4096 nodes in full-scale validation.
domain assumption The compute-communication overlap using two streams and events produces results identical to non-overlapped SpMV/Gauss-Seidel, so the measured performance is not obtained by changing the numerical algorithm.
Section 3.2.3 describes stream/event synchronization, but no numerical equivalence proof is given; this is a standard implementation assumption.
domain assumption The HPG-MxP flop-count model, updated for the fused SpMV-restriction, counts operations consistently between the mixed and double runs.
Section 3.2.4 says the accounting was updated, but the exact formula is not fully detailed; a biased flop count would directly affect the GFLOP/s speedup ratio.

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Cite this review

Pith. "Pith review of Scaling the memory wall using mixed-precision -- HPG-MxP on an exascale machine." pith.science (2026). https://pith.science/paper/NWXL55LH

@misc{pith2026250711512,
  author       = {Pith},
  title        = {Pith review of: Scaling the memory wall using mixed-precision -- HPG-MxP on an exascale machine},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NWXL55LH}},
  note         = {Machine review of arXiv:2507.11512}
}

read the original abstract

Mixed-precision algorithms have been proposed as a way for scientific computing to benefit from some of the gains seen for artificial intelligence (AI) on recent high performance computing (HPC) platforms. A few applications dominated by dense matrix operations have seen substantial speedups by utilizing low precision formats such as FP16. However, a majority of scientific simulation applications are memory bandwidth limited. Beyond preliminary studies, the practical gain from using mixed-precision algorithms on a given HPC system is largely unclear. The High Performance GMRES Mixed Precision (HPG-MxP) benchmark has been proposed to measure the useful performance of a HPC system on sparse matrix-based mixed-precision applications. In this work, we present a highly optimized implementation of the HPG-MxP benchmark for an exascale system and describe our algorithm enhancements. We show for the first time a speedup of 1.6x using a combination of double- and single-precision on modern GPU-based supercomputers.

Figures

Figures reproduced from arXiv: 2507.11512 by the authors.

**Figure 2.** An example of independent sets for a 2D 9-point [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 4.** ‘Weak’ scaling of the overall benchmark on Frontier. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: Speedups obtained on Frontier for different compu [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 7.** Figure 7: Breakdown of time spent in the multigrid smoother [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: Roofline of the benchmark on a single GCD of an [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: Traces of our GMRES-IR implementation in an 8-node benchmark run on Frontier. Purple bars in the ‘GPU’ section [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

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