[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-144473-en":3,"doc-seo-144473-105":30,"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":27,"seo_description":14,"update_tm":28,"read_time":29},144473,687207017582,"Ethan Miller","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",6,"Technology","Large-scale op-miza-on-based non-nega-ve computa-onal framework for diffusion equa-ons - Parallel implementa-on and performance studies","Large-scale optimization-based, non-negative computational framework for anisotropic diffusion equations is presented, motivated by the fact that the Galerkin finite element method fails discrete maximum principles under anisotropic diffusion. Convex optimization techniques are reviewed as a remedy, while the work targets larger and more realistic problems beyond small 2D tests. The approach leverages PETSc TAO and DMPlex features, reports solver iteration and wall-clock time, and evaluates efficiency and speedup under varying processor counts.","Large-­‐scale op-miza-on-­‐based non-­‐nega-ve computa-onal framework for diffusion equa-ons: parallel implementa-on and performance studies  \nJus-n Chang and Kalyana Nakshatrala (University of Houston)  \nSa9sh Karra (Los Alamos Na9onal Laboratory)  \nPETSc Conference and Workshop  \nJune 15th – 18th, 2015  \nIntroduc9on and Mo9va9on  \n• The Galerkin Finite Element Method does not sa9sfy the discrete maximum principles for anisotropic diffusion  \n• Recent studies have proposed convex op-miza-on techniques to overcome this setback  \n• So far these have only been tested on small and 2D academic problems  \nMesh representa9on has a Hasse Diagram  \nGalerkin solu9on (with nega9ve concentra9ons)  \nAim of this work: Non-­‐nega9ve methodology  \n1. Solve anisotropic diffusion leveraging PETSc’s TAO and DMPlex features  \n2. Ensures non-­‐nega9ve solu9ons for larger and more realis9c problems  \n3. Document the performance metrics  \nPerformance studies  \nSolver Iterations  \n2000  \n1750  \n1500  \n1250  \n1000  \n750  \n500  \n250  \n0  \nA1 B1 C1 A2 B2 C2 A3 B3 C3 Cases  \nAr ithmet ic Intensity  \n0.16  \n0.14  \n0.12  \n0.1  \n0.08  \n0.06  \n0.04  \n0.02  \n0  \nA1 B1 C1 A2 B2 C2 A3 B3 C3 Cases  \nEff ic iency (%)  \n100  \n80  \n60  \n40  \n20  \n0  \nWall-­‐clock 9me (s) Solver itera9ons  \n\n|  |  |  |  Mustang-Original\u003Cbr> Mustang-Nonnegative |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n|  ~~ ~~  Wolf-Original\u003Cbr>  Wolf-Nonnegative\u003Cbr>\u003Cbr>|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n\nA1 B1 C1 A2 B2 C2 A3 B3 C3  \nCases  \nEfficiency based on the Roofline model  \nSpeedup  \nArithme9c Intensity  \n64  \n48  \n32  \n16  \n1  \n64  \n48  \n32  \n16  \n1  \n64  \n48  \n32  \n16  \n1  \n64  \n48  \n32  \n16  \n1  \n64  \n48  \n32  \n16  \n1  \n64  \n48  \n32  \n16  \n1  \n\\# of processors  \nChromium contamina9on  \nWa l l-c lock time (s)  \n105  \n104  \n103  \n102  \n16 32 64 128 256 512 1024 2048  \n\\# of processors","cbCaipwC39hhj4sS","https://ap.wps.com/l/cbCaipwC39hhj4sS","pdf",814294,1,4,"English","en",105,"# Introduc9on and Mo9va9on\n## Non-nega9ve methodology\n## Performance studies","[{\"question\":\"Why does the Galerkin finite element method have limitations for anisotropic diffusion?\",\"answer\":\"It does not satisfy discrete maximum principles for anisotropic diffusion, which motivates the search for alternative methods.\"},{\"question\":\"What is the core aim of the presented work?\",\"answer\":\"Develop a non-negative methodology to compute anisotropic diffusion solutions that remain non-negative for larger, more realistic problems.\"},{\"question\":\"How does the framework implement the solver and evaluate performance?\",\"answer\":\"It solves anisotropic diffusion using PETSc TAO and DMPlex features, then documents performance metrics such as solver iterations, wall-clock time, efficiency, and speedup across processor counts.\"}]","Large-scale op-miza-on-based non-nega-ve computa-onal framework for diffusion equa-ons - Parallel implementa-on and performance studies | PDF",1787702982,10,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":28},"large-scale-op-miza-on-based-non-nega-ve-computa-onal-framework-for-diusion-equa-ons-parallel-implementa-on-and-performance-studies","",{"@graph":36,"@context":84},[37,53,67],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/technology/",3,{"item":52,"name":13,"@type":43,"position":21},"https://docshare.wps.com/document/large-scale-op-miza-on-based-non-nega-ve-computa-onal-framework-for-diusion-equa-ons-parallel-implementa-on-and-performance-studies/144473/",{"url":52,"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":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-08-26",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 does the Galerkin finite element method have limitations for anisotropic diffusion?","Question",{"text":74,"@type":75},"It does not satisfy discrete maximum principles for anisotropic diffusion, which motivates the search for alternative methods.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What is the core aim of the presented work?",{"text":79,"@type":75},"Develop a non-negative methodology to compute anisotropic diffusion solutions that remain non-negative for larger, more realistic problems.",{"name":81,"@type":72,"acceptedAnswer":82},"How does the framework implement the solver and evaluate performance?",{"text":83,"@type":75},"It solves anisotropic diffusion using PETSc TAO and DMPlex features, then documents performance metrics such as solver iterations, wall-clock time, efficiency, and speedup across processor counts.","https://schema.org",{"og:url":52,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,112,117,122,127,130,133],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":110,"slug":111},50,"technology",{"id":113,"doc_module":4,"doc_module_name":46,"category_name":114,"show_sort_weight":115,"slug":116},7,"Healthcare",40,"healthcare",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":119,"show_sort_weight":120,"slug":121},8,"Research & Report",30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":29,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":29,"slug":132},"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]