[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84791-en":3,"doc-seo-84791-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84791,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Variance Reduction with Probing and Multilevel Monte Carlo in Lattice QCD","Stochastic Hutchinson trace estimation in lattice QCD becomes expensive because its accuracy improves only with the square root of the sample size. The work develops two complementary variance-reduction strategies: multigrid multilevel Monte Carlo, which builds an unbiased multilevel estimator from multigrid hierarchy coarse-grid corrections, and stochastic probing using distance-3 graph colorings, including a torus-based coloring that reduces the required number of colors. Tests cover connected pseudoscalar correlators and disconnected fermion loops, showing strong gains for long-distance observables from multilevel/deflation and for localized fluctuations from probing combined with dilution.","arXiv :2607 .05 157v 1 [hep-lat] 6 Jul 2026  \nVariance reduction with probing and Multilevel Monte Carlo in Lattice QCD  \nAndreas Frommer, 􀀰 Jose Jimenez-Merchan,􀀰,∗ Bruno Lang, 􀀰 Mario Papace􀀰 and Gustavo Ramirez-Hidalgo􀀱  \n􀀰 Bergische Universität Wuppertal  \nFakultät für Mathematik und Naturwissenschaften Gaußstraße 20, 42119 Wuppertal, Germany  \n􀀱 Jülich Supercomputing Centre,  \nForschungszentrum Jülich GmbH, Wilhelm-Johnen-Straße 52428 Jülich, Germany E-mail: {frommer,jimenezmerchan,lang,[papace}@uni-wuppertal.de](papace}@uni-wuppertal.de) , [g.ramirez.hidalgo@fz-juelich.de](g.ramirez.hidalgo@fz-juelich.de)  \nTrace 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.  \nWe 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-􀀳 graph colorings; we propose a torus based coloring that requires substantially fewer colors than hierarchical probing at the same distance.  \nWe test these approaches on two representative problems: the connected pseudoscalar correlatorand disconnected fermion loops. For the connected pseudoscalar two-point function, the multilevel decomposition yields a variance reduction of up to O (105) at large time separations and translates into a clear cost reduction at fixed accuracy, thus confirming earlier results of [1] . 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 asthe 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.  \n42nd International Symposium on Lattice Field Theory (Lattice 2025) Tata Institute of Fundamental Research (TIFR), Mumbai, India November 2-8 2025  \n∗ Speaker  \n© Copyright owned by the author(s) under the terms of the Creative Commons  \nAttribution-NonCommercial-NoDerivatives 4 .0 International License (CC BY-NC-ND 4 .0) .  \nAll rights for text and data mining, AI training, and similar technologies for commercial purposes, are reserved.  \nISSN 1824-8039. Published by SISSA Medialab. [https://pos.sissa.it/](https://pos.sissa.it/)  \n1. Introduction  \nWe consider the problem of computing tr 􀀂􀀗(􀁃)􀀙 −1(􀁃, 􀁃′) 􀀃, where 􀀙 ∈ C􀀽×􀀽 is the discretized Wilson-Dirac operator in Lattice Quantum Chromodynamics (QCD) . This problem arises when computing connected and disconnected contributions to hadronic correlation functions. The Wilson-Dirac matrix 􀀙 is a block-structured matrix with each lattice site represented by a 12 × 12 block corresponding to the internal degrees of freedom of spin (4 components) and color (3 components) . 􀀙 −1(􀁃, 􀁃′) is a block of the inverse Wilson-Dirac matrix correlating the time slices 􀁃 and 􀁃′, while 􀀗(􀁃) is an operator possibly acting on spin, color, and space indices at a fixed time slice 􀁃 . The dimension of such operators is usually so large that explicitly computing the trace becomes impractical, which necessitates the use of stochastic methods.  \nConsider a stochastic vector 􀁛 ∈ C􀀽 with components 􀁛 􀀹 that satisfy  \nE [􀁛 􀀹] = 0, E [􀀹􀁛 􀀺] = 􀁘 􀀹 􀀺 , for 􀀸, 􀀺 = 1, ..., 􀀽 . 􀀹 ≠ 􀀺 . (1)  \nThen we have the following relation:  \ntr ( 􀀵 (􀀙)) = E [􀁛†􀀵 (􀀙)􀁛], (2)  \nwhere 􀀵 denotes any function that can be applied to 􀀙 . Since we are interested in the inverse, we take 􀀵 to be the inverse. Using eq. (2), this yields the Hutchinson estimator [2] for tr(􀀙 −1) :  \n􀀣  \ntr (􀀙 −1) ≈ (􀀙 −1) = 1􀀣 ∑︁􀀸 = 1 (􀁛 (􀀸))†􀀙 −1􀁛 (􀀸) , (3)  \nwhere 􀁛(􀀸) ar","cbCaigU6Drb9Mjf8","https://ap.wps.com/l/cbCaigU6Drb9Mjf8","pdf",452282,1,10,"English","en",105,"# Introduction\n## Variance reduction: multigrid multilevel Monte Carlo\n## Variance reduction: stochastic probing with graph coloring\n## Numerical results: connected correlators and disconnected loops","[{\"question\":\"Why is standard Hutchinson trace estimation costly in lattice QCD?\",\"answer\":\"Its estimator accuracy improves only proportionally to the square root of the sample size, so achieving higher precision requires many more stochastic samples.\"},{\"question\":\"How does multigrid multilevel Monte Carlo reduce variance?\",\"answer\":\"It uses a multigrid hierarchy to form an unbiased multilevel estimator, using recursive coarse-grid corrections derived from the solver’s multigrid structure.\"},{\"question\":\"When is stochastic probing (graph coloring with dilution) most effective?\",\"answer\":\"It provides substantial cost reduction for disconnected fermion loops and for localized, short-range fluctuations, especially as the number of probing vectors increases.\"}]",1784198264,25,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"variance-reduction-with-probing-and-multilevel-monte-carlo-in-lattice-qcd","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/variance-reduction-with-probing-and-multilevel-monte-carlo-in-lattice-qcd/84791/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"Why is standard Hutchinson trace estimation costly in lattice QCD?","Question",{"text":74,"@type":75},"Its estimator accuracy improves only proportionally to the square root of the sample size, so achieving higher precision requires many more stochastic samples.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does multigrid multilevel Monte Carlo reduce variance?",{"text":79,"@type":75},"It uses a multigrid hierarchy to form an unbiased multilevel estimator, using recursive coarse-grid corrections derived from the solver’s multigrid structure.",{"name":81,"@type":72,"acceptedAnswer":82},"When is stochastic probing (graph coloring with dilution) most effective?",{"text":83,"@type":75},"It provides substantial cost reduction for disconnected fermion loops and for localized, short-range fluctuations, especially as the number of probing vectors 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