{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:41:10Z","timestamp":1787330470248,"version":"build-2736575974"},"reference-count":67,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-12-1-0484"],"award-info":[{"award-number":["FA9550-12-1-0484"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CBET-1508713"],"award-info":[{"award-number":["CBET-1508713"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0010518"],"award-info":[{"award-number":["DE-SC0010518"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0009286"],"award-info":[{"award-number":["DE-SC0009286"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>Current state-of-the-art preconditioners for the reduced Hessian and the Karush--Kuhn--Tucker (KKT) operator for large-scale inverse problems are typically based on approximating the reduced Hessian with the regularization operator. However, the quality of this approximation degrades with increasingly informative observations or data. Thus the best case scenario from a scientific standpoint (fully informative data) is the worse case scenario from a computational perspective. In this paper we present an augmented Lagrangian-type preconditioner based on a block diagonal approximation of the augmented upper left block of the KKT operator. The preconditioner requires solvers for two linear subproblems that arise in the augmented KKT operator, which we expect to be much easier to precondition than the reduced Hessian. Analysis of the spectrum of the preconditioned KKT operator indicates that the preconditioner is effective when the regularization is chosen appropriately. In particular, it is effective when the regularization does not overpenalize highly informed parameter modes and does not underpenalize uninformed modes. Finally, we present a numerical study for a large data\/low noise Poisson source inversion problem, demonstrating the effectiveness of the preconditioner. In this example, three MINRES iterations on the KKT system with our preconditioner results in a reconstruction with better accuracy than 50 iterations of CG on the reduced Hessian system with regularization preconditioning.<\/jats:p>","DOI":"10.1137\/16m1084365","type":"journal-article","created":{"date-parts":[[2017,10,24]],"date-time":"2017-10-24T11:13:20Z","timestamp":1508843600000},"page":"A2365-A2393","source":"Crossref","is-referenced-by-count":11,"title":["A Data Scalable Augmented Lagrangian KKT Preconditioner for Large-Scale Inverse Problems"],"prefix":"10.1137","volume":"39","author":[{"given":"Nick","family":"Alger","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Umberto","family":"Villa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tan","family":"Bui-Thanh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Omar","family":"Ghattas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,10,24]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/070687426"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/080738064"},{"key":"atypb3","volume-title":"Proceedings of the 2005 ACM\/IEEE Conference on Supercomputing","author":"Ak\u00e7elik V.","year":"2005"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2008.09.021"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7930(98)00060-7"},{"key":"atypb6","volume-title":"Tenth Copper Mountain Conference on Multigrid Methods","author":"Ascher U. M.","year":"2001"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/06066391X"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01405194"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.4208\/cicp.190914.080415a"},{"key":"atypb10","volume-title":"Version 2.1","author":"Bell W. N.","year":"2013"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000212"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-011-9173-8"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/S106482750241565X"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827502415661"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(03)00417-5"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1002\/fld.904"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/060671590"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1988-0917816-8"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/28\/5\/055001"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/28\/5\/055002"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.3934\/ipi.2013.7.1139"},{"key":"atypb23","unstructured":"Y. Choi,\n                      Simultaneous Analysis and Design in PDE-Constrained Optimization\n                      , Ph.D. thesis, Stanford University, Stanford, CA, 2012."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2011.03.006"},{"key":"atypb25","volume-title":"Report 06-08","author":"Demkowicz L.","year":"2006"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-08-02100-5"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2012.11.070"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199678792.001.0001"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-1740-8"},{"key":"atypb30","first-page":"127647","volume":"2012","author":"Ferronato M.","year":"2012","journal-title":"IRSN Appl. Math."},{"key":"atypb31","unstructured":"P. H. Flath,\n                      Hessian-Based Response Surface Approximations for Uncertainty Quantification in Large-Scale Statistical Inverse Problems, with Applications to Groundwater Flow\n                      , Ph.D. thesis, The University of Texas at Austin, Austin, TX, 2013."},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1137\/090780717"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1137\/0716031"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/s00285-015-0888-x"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/04060679X"},{"key":"atypb36","volume-title":"Perspectives in Flow Control and Optimization","author":"Gunzburger M. D.","year":"2003"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/17\/6\/319"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2007.06.007"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1137\/090779127"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1137\/140973050"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.21582"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-012-0507-3"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-23099-8"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-016-0635-8"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1002\/nla.716"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827599355153"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1137\/080719960"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1137\/120871547"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1007\/BF02995904"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.1137\/100803031"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1137\/0712047"},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1137\/110847949"},{"key":"atypb53","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1863"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1002\/nla.814"},{"key":"atypb55","doi-asserted-by":"publisher","DOI":"10.1137\/080727154"},{"key":"atypb56","first-page":"341","volume":"46","author":"Rees T.","year":"2010","journal-title":"Kybernetika"},{"key":"atypb57","doi-asserted-by":"publisher","DOI":"10.1137\/100798491"},{"key":"atypb58","doi-asserted-by":"publisher","DOI":"10.1137\/120877532"},{"key":"atypb59","doi-asserted-by":"publisher","DOI":"10.1137\/060660977"},{"key":"atypb60","doi-asserted-by":"publisher","DOI":"10.1137\/070688961"},{"key":"atypb61","volume-title":"Tech. report","author":"Stoll M.","year":"2010"},{"key":"atypb62","doi-asserted-by":"publisher","DOI":"10.1007\/s00791-011-0168-2"},{"key":"atypb63","doi-asserted-by":"publisher","DOI":"10.1007\/BF02238511"},{"key":"atypb64","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717570"},{"key":"atypb65","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050138"},{"key":"atypb66","unstructured":"J. Xu and L. Zikatanov,\n                      Some Observations on Babu\u0161ka and Brezzi Theories\n                      , Tech. Report AM222, Penn State University, University Park, PA, 2000,http:\/\/www.math.psu.edu\/ccma\/reports.html."},{"key":"atypb67","doi-asserted-by":"publisher","DOI":"10.1137\/100814767"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1084365","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:07:28Z","timestamp":1787328448000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1084365"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":67,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1084365"],"URL":"https:\/\/doi.org\/10.1137\/16m1084365","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}