arXiv subjects

Jose Jimenez-Merchan

Publications and source records attributed to Jose Jimenez-Merchan.

4 recordsLinked to original sources

arXiv2026

Variance reduction with probing and Multilevel Monte Carlo in Lattice QCD

Trace estimation is central in many lattice QCD computations, but the accuracy of the standard, stochastic Hutchinson method improves only with the square root of the sample size, making precise results expensive. We investigate two complementary variance reduction strategies. First, multigrid multilevel Monte Carlo uses a multigrid hierarchy to construct an unbiased multilevel estimator via recursive coarse grid corrections available from the multigrid hierarchy of the solver. Second, stochastic probing uses distance-$d$ graph colorings; we propose a torus based coloring that requires substantially fewer colors than hierarchical probing at the same distance. We test these approaches on two representative problems: the connected pseudoscalar correlator and disconnected fermion loops. For the connected pseudoscalar two-point function, the multilevel decomposition yields a variance reduction of up to $\mathcal{O}(10^5)$ at large time separations and translates into a clear cost reduction at fixed accuracy, thus confirming earlier results of arXiv:2412.06347. For the disconnected loops, in contrast, the multilevel decomposition provides only moderate gains, whereas probing combined with dilution delivers a substantial cost reduction that improves as the number of probing vectors is increased. Overall, the results highlight a pronounced complementarity: deflation schemes are most effective for observables dominated by long distance propagation, while probing is most effective for localized quantities.

hep-lat↗

arXiv2026

Probing and graph coloring techniques for trace estimation in Lattice QCD

The computation of $\mathrm{Tr}[D^{-1}]$, where $D$ is the Wilson-Dirac matrix of Lattice QCD, is a fundamental and computationally demanding task with applications to disconnected hadronic correlation functions. Since $D^{-1}$ is a dense matrix of prohibitive size, its trace cannot be computed exactly, and one must resort to stochastic estimation via the Hutchinson estimator. The variance of the resulting estimation, however, can be large, as it is dominated by the off-diagonal entries of $D^{-1}$. We review the stochastic probing technique, which reduces the variance by constructing structured sampling vectors from distance-$d$ colorings of the graph associated with $D$, exploiting the exponential off-diagonal decay of $D^{-1}$ to eliminate dominant short-range contributions to the variance. We then present a novel multiplier-based coloring scheme, which achieves valid distance-$d$ colorings at arbitrary distances with significantly fewer colors than the established hierarchical probing construction. We prove that at any intermediate coloring falling between two consecutive hierarchical levels, the multiplier-based estimator achieves strictly lower variance than the partial hierarchical estimator, for large enough $d$. This is confirmed by numerical experiments showing that the multiplier-based variance decreases smoothly and monotonically with the number of colors, avoiding the irregular behavior affecting hierarchical probing at intermediate colorings, and achieving a substantial improvement in relative accuracy.

hep-lat↗

arXiv2026

Performance-Portable Optimization and Analysis of Multiple Right-Hand Sides in a Lattice QCD Solver

Managing the high computational cost of iterative solvers for sparse linear systems is a known challenge in scientific computing. Moreover, scientific applications often face memory bandwidth constraints, making it critical to optimize data locality and enhance the efficiency of data transport. We extend the lattice QCD solver DD-$\alpha$AMG to incorporate multiple right-hand sides (rhs) for both the Wilson-Dirac operator evaluation and the GMRES solver, with and without odd-even preconditioning. To optimize auto-vectorization, we introduce a flexible interface that supports various data layouts and implement a new data layout for better SIMD utilization. We evaluate our optimizations on both x86 and Arm clusters, demonstrating performance portability with similar speedups. A key contribution of this work is the performance analysis of our optimizations, which reveals the complexity introduced by architectural constraints and compiler behavior. Additionally, we explore different implementations leveraging a new matrix instruction set for Arm called SME and provide an early assessment of its potential benefits.

cs.DC↗

arXiv2025

Using orthogonal projectors in multigrid multilevel Monte Carlo for trace estimation in lattice QCD

We introduce a multigrid multilevel Monte Carlo method for stochastic trace estimation in lattice QCD based on orthogonal projections. This formulation extends the previously proposed oblique decomposition and it is assessed on three representative problems: the connected pseudoscalar correlator, the trace of the full Dirac operator's inverse $\mathrm{tr}(D^{-1})$, and disconnected fermion loops. For the connected correlator, variance reductions grow systematically with the time separation and lead to cost savings of up to a factor of 30 at large separations, outperforming both the plain Hutchinson's estimator and the oblique formulation. For $\mathrm{tr}(D^{-1})$, reductions are more modest but remain systematic, with stronger effects on more ill-conditioned systems. Disconnected loops show no improvement, since their variance is dominated by local same-slice contributions not targeted by the decomposition.

hep-lat↗