[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84876-en":3,"doc-seo-84876-105":30,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84876,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Metagraph-Based Domain-Decomposed Galerkin Reduced-Order Model","This study proposes a metagraph-based domain-decomposed Galerkin reduced-order model (MBDD-G-ROM) for distributed-memory parallel reduced-order analysis of large-scale problems. The approach uses calculation-point graphs for interaction modeling and metagraphs for connectivity among local approximation-space subdomains. In the POD implementation, local subdomains form POD computation subdomains with local POD bases, and the metagraph encodes block-sparsity via overlaps of local basis supports. A two-level graph-based partitioning decouples POD computation from parallel computation, enabling offline/online distributed-memory parallelization without forcing one-to-one subdomain correspondence. Static load balancing is integrated through metanode weights. Verification uses unsteady diffusion and incompressible Navier–Stokes flow around a 3D cylinder, showing maintained accuracy and high online parallel efficiency, with improved efficiency under estimated-cost weighting.","arXiv :2607 .06011v1 [math .NA] 7 Jul 2026  \nMetagraph-Based Domain-Decomposed  \nGalerkin Reduced-Order Model  \nKyohei Shintatea , Naoki Moritab , Shigeki Kanekoc , Nozomi Magomea , Naoto Mitsumeb,∗  \na University of Tsukuba, Degree Programs in Systems and Information Engineering, Tennodai 1 -1-1, Tsukuba, 3058573, Ibaraki, Japan b University of Tsukuba, Institute of Systems and Information Engineering, Tennodai 1 -1-1, Tsukuba, 3058573, Ibaraki, Japan c Graduate School of Engineering, Nagoya Institute of Technology, Gokishocho, Showa-ku, Nagoya, 4660061, Aichi, Japan  \nAbstract  \nThis study proposes a metagraph-based domain-decomposed Galerkin reduced-order model (MBDD-G-ROM) for distributed-memory parallel reduced-order analysis of large-scale problems. The method provides a graph-based representation of domain-decomposed Galerkin reduced-order models defined over arbitrary domain decompositions. Calculation-point graphs describe interactions among calculation points, whereas metagraphs describe connectivity among local approximation-space subdomains. In the proper orthogonal decomposition (POD)-based implementation considered in this study, these local subdomains correspond to POD computation subdomains, where local POD bases are constructed. In the resulting metagraph, POD computation subdomains are treated as metanodes, and metaedges encode the block-sparsity pattern induced by overlaps between the supports of local POD basis functions. Based on this representation, a two-level graph-based domain-partitioning strategy decouples the POD computation subdomains from the parallel computation subdomains. This decoupling enables distributed-memory parallelization of the offline and online phases, including reduced-system assembly and the linear solver, without imposing a one-toone correspondence between the two subdomain types in number or geometry. The metagraph structure also provides a natural way to incorporate static load balancing through metanode weights representing estimated computational costs. The method is verified using an unsteady diffusion equation and the incompressible Navier–Stokes equations for flow around a three-dimensional cylinder; these tests assess accuracy degradation caused by model reduction and strong-scaling behavior. Numerical results show that the method maintains solution accuracy while achieving high parallel efficiency in the online phase. The static load-balancing test shows that metanode weights based on estimated computational costs improve computational efficiency in the tested case.  \nKeywords:  \nParallel Computing, Domain Decomposition, Graph Structure, Reduced-Order Modeling, Proper Orthogonal Decomposition  \n1. Introduction  \nParametric studies play an important role in many engineering applications, where analyses are repeatedly performed for different values of the parameters defining the governing partial differential equation model. However, repeated simulations using full-order models (FOMs) entail substantial computational costs and are therefore often impractical. At the same time, because the target systems in such studies often share similar geometries and physical conditions across parameter instances, previously obtained simulation data can potentially be exploited to improve computational efficiency. Recently, reduced-order models (ROMs) [1–3] have been studied extensively as fast surrogate models for FOMs. Among them, Galerkin reduced-order models (G-ROMs), which compute approximate solutions by projecting the governing equations onto a low-dimensional approximation space, either a linear subspace or a nonlinear manifold, have attracted considerable attention owing to their favorable balance between computational  \n∗[Corresponding author. Email: mitsume@kz.tsukuba.ac.jp](Corresponding author. Email: mitsume@kz.tsukuba.ac.jp); Telephone: +81-29-853-5268  \naccuracy and efficiency. Approaches to defining the low-dimensional approximation spaces used in G-ROMs can be broadly d","cbCaibCtQ57fAMoe","https://ap.wps.com/l/cbCaibCtQ57fAMoe","pdf",4243559,2,1,41,"English","en",105,"# Introduction\n## Reduced-order modeling motivation and background\n## Offline/online workflow for data-driven ROMs\n## Approaches to constructing approximation spaces","[{\"question\":\"What does the metagraph represent in MBDD-G-ROM?\",\"answer\":\"The metagraph encodes connectivity among local approximation-space subdomains. In the POD-based implementation, POD computation subdomains become metanodes, and metaedges capture block-sparsity patterns induced by overlaps between local POD basis supports.\"},{\"question\":\"How does the two-level graph-based partitioning affect parallelization?\",\"answer\":\"It decouples POD computation subdomains from parallel computation subdomains, enabling distributed-memory parallelization of both offline and online phases (including reduced-system assembly and the linear solver) without requiring a one-to-one match in subdomain counts or geometry.\"},{\"question\":\"How is static load balancing incorporated and evaluated?\",\"answer\":\"Static load balancing is incorporated via metanode weights representing estimated computational costs. The method is tested on diffusion and incompressible Navier–Stokes problems, where cost-based metanode weights improve computational efficiency in the reported case.\"}]",1784198961,103,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"metagraph-based-domain-decomposed-galerkin-reduced-order-model","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/metagraph-based-domain-decomposed-galerkin-reduced-order-model/84876/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the metagraph represent in MBDD-G-ROM?","Question",{"text":75,"@type":76},"The metagraph encodes connectivity among local approximation-space subdomains. In the POD-based implementation, POD computation subdomains become metanodes, and metaedges capture block-sparsity patterns induced by overlaps between local POD basis supports.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the two-level graph-based partitioning affect parallelization?",{"text":80,"@type":76},"It decouples POD computation subdomains from parallel computation subdomains, enabling distributed-memory parallelization of both offline and online phases (including reduced-system assembly and the linear solver) without requiring a one-to-one match in subdomain counts or geometry.",{"name":82,"@type":73,"acceptedAnswer":83},"How is static load balancing incorporated and evaluated?",{"text":84,"@type":76},"Static load balancing is incorporated via metanode weights representing estimated computational costs. The method is tested on diffusion and incompressible Navier–Stokes problems, where cost-based metanode weights improve computational efficiency in the reported case.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]