[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117482-en":3,"doc-seo-117482-105":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":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},117482,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Accelerating Convergence of Linear Iterative Solvers using Machine Learning - Thesis Abstract","This thesis investigates how Machine Learning can accelerate iterative numerical solvers, with emphasis on the Generalized Minimal RESidual (GMRES) method for solving arbitrary invertible linear systems. A sequence of previous linear problems is used to learn GMRES convergence behavior and to predict a suitable initial guess for each new system, aiming to reduce the number of matrix-vector products needed to reach stopping criteria. The study first targets simpler PDE settings such as Laplace and advection-diffusion with time-dependent forcing, then extends to stiff Navier-Stokes test cases like flow around a cylinder, a NACA0012 airfoil, and the Taylor-Green vortex. Classical GMRES is compared against the ML-driven approach, including effects of coupling cost and parameter sensitivity, demonstrating improved efficiency and accuracy potential.","Accelerating Convergence of Linear Iterative Solvers using Machine Learning  \nTesi di Laurea Magistrale in  \nMathematical Engineering-Ingegneria Matematica  \nAuthor: Luca Saverio  \nStudent ID: 990172  \nAdvisor Politecnico di Milano: Prof. Nicola Parolini  \nAdvisor Sorbonne Université: Prof. Corrado Maurini  \nONERA Supervisors: Emeric Martin, Jorge Nuñez, Florent Renac  \nAcademic Year: 2022-2023  \ni  \nAbstract  \nThis thesis explores the application of Machine Learning techniques to accelerate iterative numerical methods, with a particular focus on the Generalized Minimal RESidual (GMRES) method, for solving arbitrary invertible linear systems. By training on the GMRES convergence behaviour obtained on previous linear systems in a sequence, the main goal of this work is to predict an adequate initial guess for each system of the sequence. This thesis is organized as follows: first, Machine Learning is applied to simple problems, such as the Laplace equation, where the right-hand side is modified at each iteration, and to the Advection-Diffusion problem with a time-dependent right-hand side. The matrix or operator does not change over the sequence. The model is trained using online learning techniques and the prediction of the initial guess results in a significant speed-up of the GMRES convergence on the considered linear systems. In the second part of the thesis, Neural Networks are applied to more complex and stiff systems, starting from the Navier-Stokes equations: the flow around a cylinder, the flow around a NACA0012 airfoil and the Taylor-Green Vortex test case are solved. The results of the classical GMRES algorithm are then compared to the ones of the Machine Learning scheme. The effectiveness of this approach is evaluated and compared to traditional methods, in terms of number of matrix-vector products to satisfy user parameters driving the stopping of the iterative solver. Considerations about gain in time taking into account the cost of the coupling and the sensitivity of certain parameters on the ML strategy performance are also addressed. Overall, this thesis demonstrates the potential of Machine Learning to improve the efficiency and accuracy of iterative numerical methods, particularly in the context of solving complex mathematical problems.  \nKeywords: GMRES, Neural Networks, Graph Neural Networks, CPU, GPU, Machine Learning, CFD, Performance, Pytorch  \niii  \nAbstract in lingua italiana  \nQuesta tesi esplora l’applicazione delle tecniche di Machine Learning per accelerare imetodi numerici iterativi, con particolare attenzione al metodo del residuo minimo generalizzato, per la risoluzione di sistemi lineari arbitrariamente invertibili. L’obiettivo principale è quello di prevedere un’ipotesi iniziale adeguata all’algoritmo GMRES addestrando su precedenti sistemi lineari della sequenza. Questa tesi è organizzata come segue: in primo luogo, l’apprendimento automatico viene applicato a problemi semplici, come l’equazione di Laplace, in cui il lato destro viene modificato ad ogni iterazione per le stessematrici, e al problema di Avvezione-Diffusione con un vettore lato destro dipendente dal tempo. Il modello viene addestrato utilizzando tecniche di apprendimento online mantenendo fissa la matrice o l’operatore. I risultati mostrano una significativa accelerazionenella convergenza. Nella seconda parte della tesi, le Reti Neurali vengono applicate asistemi più complessi e rigidi, partendo dalle equazioni di Navier-Stokes: vengono risolti il flusso attorno ad un cilindro, il flusso attorno ad un profilo alare NACA0012 e il casoddel vortice di Taylor-Green. I risultati del classico algoritmo GMRES vengono quindi confrontati con quelli dello schema di Machine Learning. L’efficacia di questo approccio viene valutata e confrontata con i metodi tradizionali, in termini di numero di prodottimatrice-vettore per raggiungere la convergenza del sistema lineare. Vengono inoltre affrontate considerazioni sul guadagno nel tempo con l’","cbCaihlgcKSkncx0","https://ap.wps.com/l/cbCaihlgcKSkncx0","pdf",5144872,1,141,"English","en",105,"# Abstract\n## Learning-based acceleration of GMRES\n## Experiments on PDE benchmarks\n## Neural networks for Navier-Stokes test cases\n## Performance evaluation and comparison","[{\"question\":\"How does the thesis use Machine Learning to improve GMRES performance?\",\"answer\":\"It trains on the GMRES convergence behavior from a sequence of prior linear systems to predict an adequate initial guess for each new system, speeding up convergence.\"},{\"question\":\"Which mathematical test problems are used before the Navier-Stokes applications?\",\"answer\":\"The work starts with simpler problems such as Laplace and an advection-diffusion problem with a time-dependent right-hand side, keeping the operator matrix fixed across the sequence.\"},{\"question\":\"What Navier-Stokes scenarios are compared between classical GMRES and the ML scheme?\",\"answer\":\"The thesis considers flow around a cylinder, flow around a NACA0012 airfoil, and the Taylor-Green vortex test case, comparing classical GMRES results to ML-accelerated outputs.\"}]","Accelerating Convergence of Linear Iterative Solvers using Machine Learning - Thesis Abstract | PDF",1785676109,355,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"accelerating-convergence-of-linear-iterative-solvers-using-machine-learning-thesis-abstract","",{"@graph":36,"@context":85},[37,54,68],{"@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/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/accelerating-convergence-of-linear-iterative-solvers-using-machine-learning-thesis-abstract/117482/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How does the thesis use Machine Learning to improve GMRES performance?","Question",{"text":75,"@type":76},"It trains on the GMRES convergence behavior from a sequence of prior linear systems to predict an adequate initial guess for each new system, speeding up convergence.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which mathematical test problems are used before the Navier-Stokes applications?",{"text":80,"@type":76},"The work starts with simpler problems such as Laplace and an advection-diffusion problem with a time-dependent right-hand side, keeping the operator matrix fixed across the sequence.",{"name":82,"@type":73,"acceptedAnswer":83},"What Navier-Stokes scenarios are compared between classical GMRES and the ML scheme?",{"text":84,"@type":76},"The thesis considers flow around a cylinder, flow around a NACA0012 airfoil, and the Taylor-Green vortex test case, comparing classical GMRES results to ML-accelerated outputs.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"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"]