{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T17:02:30Z","timestamp":1788368550142,"version":"build-2803163510"},"reference-count":35,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2208456"],"award-info":[{"award-number":["DMS-2208456"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2208412"],"award-info":[{"award-number":["DMS-2208412"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["RTG DMS-2038118"],"award-info":[{"award-number":["RTG DMS-2038118"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2024,12,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Anderson acceleration (AA) is a popular algorithm designed to enhance the convergence of fixed-point iterations. In this paper, we introduce a variant of AA based on a truncated Gram\u2013Schmidt process (AATGS) which has a few advantages over the classical AA. In particular, an attractive feature of AATGS is that its iterates obey a three-term recurrence in the situation when it is applied to solving symmetric linear problems and this can lead to a considerable reduction of memory and computational costs. We analyze the convergence of AATGS in both full-depth and limited-depth scenarios and establish its equivalence to the classical AA in the linear case. We also report on the effectiveness of AATGS through a set of numerical experiments, ranging from solving nonlinear partial differential equations to tackling nonlinear optimization problems. In particular, the performance of the method is compared with that of the classical AA algorithms.<\/jats:p>","DOI":"10.1137\/24m1648600","type":"journal-article","created":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T04:57:17Z","timestamp":1728363437000},"page":"1850-1872","source":"Crossref","is-referenced-by-count":4,"title":["Anderson Acceleration with Truncated Gram\u2013Schmidt"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9594-6840","authenticated-orcid":true,"given":"Ziyuan","family":"Tang","sequence":"first","affiliation":[{"name":"Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tianshi","family":"Xu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Emory University, Atlanta, GA 30322 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huan","family":"He","sequence":"additional","affiliation":[{"name":"Work done in Department of Computer Science, Emory University, Atlanta, GA 30322 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yousef","family":"Saad","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0361-0931","authenticated-orcid":true,"given":"Yuanzhe","family":"Xi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Emory University, Atlanta, GA 30322 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,10,8]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1145\/321296.321305"},{"key":"ref2","doi-asserted-by":"crossref","unstructured":"W. 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Lacoste-Julien, A variational inequality perspective on generative adversarial networks, in Proceedings of the 7th International Conference on Learning Representations, ICLR, 2019."},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-017-0455-1"},{"key":"ref12","doi-asserted-by":"crossref","unstructured":"H. He, Z. Tang, S. Zhao, Y. Saad, and Y. Xi, nlTGCR: A class of nonlinear acceleration procedures based on conjugate residuals, SIAM J. Matrix Anal. Appl., 45 (2024), pp. 712\u2013743, https:\/\/doi.org\/10.1137\/23M1576360.","DOI":"10.1137\/23M1576360"},{"key":"ref13","unstructured":"H. He, S. Zhao, Y. Xi, J. C. Ho, and Y. Saad, GDA-AM: On the effectiveness of solving min-imax optimization via anderson mixing, in Proceedings of the Tenth International Conference on Learning Representations, ICLR 2022, OpenReview.net, 2022, https:\/\/openreview.net\/forum?id=3YqeuCVwy1d."},{"key":"ref14","doi-asserted-by":"crossref","unstructured":"C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, 1995, https:\/\/doi.org\/10.1137\/1.9781611970944.","DOI":"10.1137\/1.9781611970944"},{"key":"ref15","unstructured":"D. Lin, H. Ye, and Z. Zhang, Explicit Superlinear Convergence Rates of Broyden\u2019s Methods in Nonlinear Equations, preprint, arXiv:2109.01974, 2021."},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2562"},{"key":"ref17","unstructured":"V. V. Mai and M. Johansson, Anderson acceleration of proximal gradient methods, in Proceedings of the 37th International Conference on Machine Learning, ICML\u201920, 2020."},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1063\/1.1725600"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1145\/3197517.3201290"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1137\/22M1536741"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1137\/18M1206151"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1016\/0009-2614(80)80396-4"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1002\/jcc.540030413"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-023-02262-x"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1002\/nla.617"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718003"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2343"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1137\/130919398"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1137\/10078356X"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-021-01548-2"},{"key":"ref32","volume-title":"Advances in Neural Information Processing Systems","author":"Wei F.","year":"2021"},{"key":"ref33","unstructured":"F. 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