[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84769-en":3,"doc-seo-84769-105":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},84769,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","An Equivalence Result for Sketched Anderson Acceleration and Sketched GMRES","An equivalence result is developed between randomized Anderson Acceleration and randomized GMRES for linear fixed-point problems. The classical connection is extended to the sketched setting where the Anderson least-squares step is carried out in a subspace defined by a random projection. The analysis relies on sketched norms/inner products via ε-embeddings and embedding dimensions s much smaller than n. The outcome motivates further study of randomized acceleration design and theoretical guarantees for sketched Krylov-type methods.","arXiv :2607 .04904v1 [math .NA] 6 Jul 2026  \nAn Equivalence result for sketched Anderson Acceleration and  \nsketched GMRES⋆  \nAlberto Bucci  \nSchool of Mathematics, University of Edinburgh, Peter Guthrie Tait Rd, Edinburgh, EH9 3FD, Scotland, UK  \nFabio Durastante  \nDipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, Pisa, 56127, PI, Italy  \nAbstract  \nIn this paper we present an equivalence result between a randomized version of Anderson Acceleration and of randomized GMRES for linear problems. Namely, we extend the classical result of Walker and Ni (2011) to the case in which the least-squares problem in Anderson Acceleration is solved in a sketched space defined by a random projection. This equivalence suggests potential avenues for further research in the design and analysis of randomized acceleration methods.  \nKeywords: Sketching, GMRES, Anderson Acceleration  \n2020 MSC: 65F10, 65B99, 60B20  \n1. Introduction  \nAnderson acceleration (AA), originally introduced in [2], is a widely used extrapolation strategy for fixed-point iterations and nonlinear equations; see, for instance, [3] . For linear problems, i.e., when applied to a linear fixed-point iteration for solving the linear system  \nAx = b, A ∈ Rn×n , b ∈ Rn , (1)  \nAA is known to be closely related to Krylov subspace methods. Several convergence and equivalence properties are now well understood [4, 5] . On the Krylov side, the Generalized Minimal Residual (GMRES) method is the reference method for nonsymmetric systems. Following a recent trend of interest in randomized numerical linear algebra [6], both randomized and sketched variants of GMRES [7, 8, 9] and randomized alternating variants of AA [10] have been introduced. This motivates the present investigation, which extends the classical results of [1] to establish equivalence between appropriately defined variants of sketched AA and sketched GMRES.  \n⋆ AB is supported by the UK’s Engineering and Physical Sciences Research Council (EPSRC grant EP/Z533786/1) . ABand FD acknowledge the INdAM GNCS project with CUP E53C25002010001 . FD acknowledges the MUR Excellence Department Project awarded to the Department of Mathematics, University of Pisa, CUP I57G22000700001 .  \n2. Sketched GMRES and Sketched Anderson Acceleration  \nWe start by recalling a randomized construction of GMRES based on sketched Gram–Schmidt orthogonalization and randomized Arnoldi iteration, as described in [7] . For this construction, we need the notion of an ε-embedding for a subspace [6, §8.1] .  \nDefinition 1. A matrix Θ ∈ Rs×n is an ε-embedding for a subspace V ⊂ Rn if  \n(1 − ε)∥v∥22 ≤ ∥Θv∥22 ≤ (1 + ε)∥v∥22 , ∀v ∈ V.  \nWe define ∥ · ∥Θ ≜ ∥Θ · ∥2 and ⟨·, ·⟩Θ ≜ ⟨Θ·, Θ·⟩ as the sketched norm and inner product.  \nChoose an embedding dimension s ≪ n and a sketching matrix Θ ∈ Rs×n, so that inner products in the Krylov space are approximated by the sketched inner product ⟨u, v⟩ ≈ ⟨u, v⟩Θ . The method can be used to construct a basis that is orthonormal in the sketched inner product, or exploiting the solution of the sketched least-squares problem to reduce the depth of the orthogonalization process, see, e.g. [8] . In the following, we recall the construction with full orthogonalization, and then comment on the equivalence result with a truncated orthogonalization process. For a target iteration count m, and an initial guess x0 , define q 1 = (b − Ax0)/∥b − Ax0 ∥Θ and, for i = 1, . . . , m − 1, compute wi+1 = Aqi , and then apply one sketched Gram–Schmidt step:  \npi+1 = Θwi+1 , S i = ΘQi , [H](1:i,i) = arg min ∥S iy − pi+1∥2 ,  \ny∈Ri  \nq′i+1 = wi+1 − Qi [H](1:i,i) , hi+1,i = ∥q′i+1∥Θ , qi+1 = q′i+1/hi+1,i.  \nCollecting coefficients gives an upper Hessenberg matrix Hm ∈ Rm×(m−1) and a basis matrix Qm = [q 1 , . . . , qm ] satisfying the Arnoldi relation AQm−1 = QmHm.  \nThe randomized GMRES iterate is then obtained by solving the small least-squares problem ym−1 = arg min ∥Hmz − h1,0e 1 ∥2 , and setting xm−1 = Qm−1ym−1, where h","cbCaiaZXq2Aql2gG","https://ap.wps.com/l/cbCaiaZXq2Aql2gG","pdf",222650,3,1,6,"English","en",105,"# Introduction\n# Sketched GMRES and Sketched Anderson Acceleration","[{\"question\":\"What equivalence does the paper establish between Anderson Acceleration and GMRES?\",\"answer\":\"The paper proves an equivalence between a randomized version of Anderson Acceleration and a randomized GMRES method for linear problems when Anderson’s least-squares step is solved in a sketched space defined by random projection.\"},{\"question\":\"How is the sketched GMRES construction defined in the paper?\",\"answer\":\"It uses a randomized construction based on sketched Gram–Schmidt orthogonalization and randomized Arnoldi iteration, where inner products and norms are approximated using a sketching matrix Θ.\"},{\"question\":\"What is an ε-embedding, and why is it important here?\",\"answer\":\"An ε-embedding is a matrix Θ that approximately preserves norms of all vectors in a subspace V up to factors (1−ε) and (1+ε). It ensures the sketched residual minimization behaves like residual minimization in the original norm, up to distortion from the embedding.\"}]",1784198126,15,{"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},"an-equivalence-result-for-sketched-anderson-acceleration-and-sketched-gmres","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/an-equivalence-result-for-sketched-anderson-acceleration-and-sketched-gmres/84769/",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","2026-07-22","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 equivalence does the paper establish between Anderson Acceleration and GMRES?","Question",{"text":75,"@type":76},"The paper proves an equivalence between a randomized version of Anderson Acceleration and a randomized GMRES method for linear problems when Anderson’s least-squares step is solved in a sketched space defined by random projection.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the sketched GMRES construction defined in the paper?",{"text":80,"@type":76},"It uses a randomized construction based on sketched Gram–Schmidt orthogonalization and randomized Arnoldi iteration, where inner products and norms are approximated using a sketching matrix Θ.",{"name":82,"@type":73,"acceptedAnswer":83},"What is an ε-embedding, and why is it important here?",{"text":84,"@type":76},"An ε-embedding is a matrix Θ that approximately preserves norms of all vectors in a subspace V up to factors (1−ε) and (1+ε). It ensures the sketched residual minimization behaves like residual minimization in the original norm, up to distortion from the embedding.","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},[93,97,101,105,110,114,119,122,127,130,134],{"id":21,"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":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":22,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},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":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]