Alternating Anderson Acceleration-like Algorithms for Solving Large Sparse Linear Systems

Alternating Anderson Acceleration-like Algorithms for Solving Large Sparse Linear Systems | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Alternating Anderson Acceleration-like Algorithms for Solving Large Sparse Linear Systems Xue-Chen Zhai, Long-Ze Tan, Xue-Ping Guo This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7053861/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The increasing scale of scientific and engineering computations presents significant challenges in efficiently solving large sparse linear systems. In this work, we develop a class of enhanced iterative solvers by integrating Anderson Acceleration with the Hermitian and skew-Hermitian splitting (HSS) framework. Specifically, we propose the alternating Anderson-accelerated HSS (AAHSS) method, which incorporates the weighted HSS iterations and periodically applies Anderson Acceleration to improve convergence. A spectral radius-based convergence analysis is conducted to theoretically support the proposed strategy. Building upon the AAHSS method, we further design its four algorithmic variants: the single-step Anderson-accelerated HSS (AASHSS) method, its preconditioned version (AASPHSS), a parameterized single-step variant (AAPSHSS), and the parameterized preconditioned single-step (AAPSPHSS) methods. Comprehensive numerical experiments on a range of benchmark problems demonstrate that all proposed methods significantly reduce iteration counts and improve computational efficiency compared to classical Krylov subspace methods, such as the alternating Anderson-accelerated Richardson (AAR), GMRES, BiCGStab, Quasi-Minimal Residual (QMR) and CG methods. Among them, the AAPSHSS method performs particularly well on nonsymmetric systems, while the AAPSPHSS method is especially effective for symmetric problems. Anderson Acceleration HSS method Sparse linear systems Iterative methods Preconditioning Spectral analysis Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7053861","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":484683843,"identity":"f300a759-bdab-47b4-80aa-f410765ad8eb","order_by":0,"name":"Xue-Chen Zhai","email":"","orcid":"","institution":"East China Normal University","correspondingAuthor":false,"prefix":"","firstName":"Xue-Chen","middleName":"","lastName":"Zhai","suffix":""},{"id":484683844,"identity":"e15cc459-5f7d-457e-a727-983939784195","order_by":1,"name":"Long-Ze Tan","email":"","orcid":"","institution":"University of Science and Technology of China","correspondingAuthor":false,"prefix":"","firstName":"Long-Ze","middleName":"","lastName":"Tan","suffix":""},{"id":484683845,"identity":"9cfa7725-924c-4a45-aee5-b99dcda0a725","order_by":2,"name":"Xue-Ping Guo","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7UlEQVRIiWNgGAWjYNCCCiCWgPMSiNFyhoGBhzQtjG2kaDG43X5N4uO8bfL20g2MD37uOMzAz55jwPBzB24tknPOlEnO3HbbsEfmALNh75nDDJI9bwwYe8/g1sIvkZMmzbvtNmOPRAKbNGPbYQaDGzkGzCCn4gJsYC1zbtsDtbD/BmmxJ6SFXyL9mDRvw+1EkC3MYFskCGgB+oXZcsax28k9NxKbJXvb0nkkzjwrONiLRwswxB7e+FBz27Z9RvLBDz/brOX425M3PviJRwuDBI8BlMXYACJ5QMQBPBqAWtgf4JUfBaNgFIyCUcAAABfCTnlSs/E+AAAAAElFTkSuQmCC","orcid":"","institution":"East China Normal University","correspondingAuthor":true,"prefix":"","firstName":"Xue-Ping","middleName":"","lastName":"Guo","suffix":""}],"badges":[],"createdAt":"2025-07-05 15:08:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7053861/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7053861/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":91782505,"identity":"2f46ac4f-1db5-4c1b-be2e-59c290a14fe8","added_by":"auto","created_at":"2025-09-21 05:01:19","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":794256,"visible":true,"origin":"","legend":"","description":"","filename":"zxcpaperv10.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7053861/v1_covered_2dd97cdd-9c9f-451e-812d-c9a58adc21f4.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Alternating Anderson Acceleration-like Algorithms for Solving Large Sparse Linear Systems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Anderson Acceleration, HSS method, Sparse linear systems, Iterative methods, Preconditioning, Spectral analysis","lastPublishedDoi":"10.21203/rs.3.rs-7053861/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7053861/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The increasing scale of scientific and engineering computations presents significant challenges in efficiently solving large sparse linear systems. 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Comprehensive numerical experiments on a range of benchmark problems demonstrate that all proposed methods significantly reduce iteration counts and improve computational efficiency compared to classical Krylov subspace methods, such as the alternating Anderson-accelerated Richardson (AAR), GMRES, BiCGStab, Quasi-Minimal Residual (QMR) and CG methods. 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