{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:38:52Z","timestamp":1787337532723,"version":"build-2736575974"},"reference-count":44,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100008905","name":"Alpen-Adria-Universit\u00e4t Klagenfurt","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100008905","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["I2271"],"award-info":[{"award-number":["I2271"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P30054"],"award-info":[{"award-number":["P30054"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006548","name":"K\u00e4rntner Wirtschaftsf\u00f6rderungsfonds","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100006548","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>The conventional way of formulating inverse problems such as identification of a (possibly infinite dimensional) parameter is via some forward operator, which is the concatenation of the observation operator with the parameter-to-state-map for the underlying model. Recently, all-at-once formulations have been considered as an alternative to this reduced formulation, avoiding the use of a parameter-to-state map, which would sometimes lead to overly restrictive conditions. Here the model and the observation are considered simultaneously as one large system, with the state and the parameter as unknowns. A still more general formulation of inverse problems, containing not only the reduced and all-at-once formulations but also the well-known and highly versatile variational approach (also called the Kohn--Vogelius functional approach) as special cases, is to formulate the inverse problem as a minimization problem---instead of an equation---for the state and parameter. Regularization can be incorporated via imposing constraints and\/or adding regularization terms to the objective. In this paper, after giving a motivation by formulating the electrical impedance tomography (EIT) problem by means of the classical Kohn--Vogelius functional, we explore the regularization aspects for such variational formulations in an abstract setting. Indeed, combination of regularization by constraints and by penalization leads to new methods that are applicable without solving forward problems. In particular, for the EIT problem we will consider a method employing box constraints in a very natural manner to incorporate the discrepancy principle for regularization parameter choice as well as a priori information on the searched for conductivity.<\/jats:p>","DOI":"10.1137\/17m1124036","type":"journal-article","created":{"date-parts":[[2018,3,8]],"date-time":"2018-03-08T09:18:56Z","timestamp":1520500736000},"page":"620-645","source":"Crossref","is-referenced-by-count":14,"title":["Minimization Based Formulations of Inverse Problems and Their Regularization"],"prefix":"10.1137","volume":"28","author":[{"given":"Barbara","family":"Kaltenbacher","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,3,8]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/22\/1\/007"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/12\/5\/002"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"A. Bakushinsky and M. Kokurin,\n                      Iterative Methods for Approximate Solution of Inverse Problems\n                      , Math. Appl. 577, Springer, Dordrecht, 2004.","DOI":"10.1007\/978-1-4020-3122-9"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/11083277X"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/27\/4\/045011"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/21\/6\/010"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/18\/4\/301"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901389980"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139994267444"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.3934\/ipi.2013.7.1183"},{"key":"atypb11","first-page":"499","volume":"6","author":"Dombrovskaja I.","year":"1965","journal-title":"Sibirsk. Mat. Z\u030c."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/5\/4\/007"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"H. W. Engl, M. Hanke, and A. Neubauer,\n                      Regularization of Inverse Problems\n                      , Kluwer Academic Publishers, Dordrecht, 1996.","DOI":"10.1007\/978-94-009-1740-8"},{"key":"atypb14","unstructured":"J. Flemming,\n                      Generalized Tikhonov Regularization: Basic Theory and Comprehensive Results on Convergence Rates\n                      , Shaker Verlag, Aachen, 2012; Ph.D. thesis, TU Chemnitz, 2011."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.1989.38.38025"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/17\/6\/319"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-012-0499-z"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/140984439"},{"key":"atypb19","first-page":"270","volume":"145","author":"Ivanov V. K.","year":"1962","journal-title":"Dokl. Akad. Nauk SSSR"},{"issue":"103","key":"atypb20","first-page":"211","volume":"61","author":"Ivanov V. K.","year":"1963","journal-title":"S.)"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"V. K. Ivanov, V. V. Vasin, and V. P. Tanana,\n                      Theory of Linear Ill-posed Problems and Its Applications\n                      , Inverse and Ill-Posed Problems Series, VSP, Utrecht, 2002.","DOI":"10.1515\/9783110944822"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/16M1060984"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1017\/S0956792502005089"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/30\/4\/045002"},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"B. Kaltenbacher, A. Neubauer, and O. Scherzer,\n                      Iterative regularization methods for nonlinear ill-posed problems\n                      , Radon Ser. Comput. Appl. Math. 6, Walter de Gruyter, Berlin, 2008.","DOI":"10.1515\/9783110208276"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1515\/jiip-2015-0087"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.3934\/ipi.2017033"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"A. Kirsch,\n                      An Introduction to the Mathematical Theory of Inverse Problems\n                      , Springer, New York, 1996.","DOI":"10.1007\/978-1-4612-5338-9"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/14\/6\/010"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/12\/6\/010"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/6\/3\/009"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160400605"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/29\/7\/075016"},{"key":"atypb35","first-page":"1225","volume":"175","author":"Morozov V.A.","year":"1967","journal-title":"Dokl. Akad. Nauk. SSSR"},{"key":"atypb36","first-page":"81","volume":"41","author":"Neubauer A.","year":"2014","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb37","unstructured":"C. Po\u0308schl,\n                      Tikhonov Regularization with General Residual Term\n                      , Ph.D. thesis, University of Innsbruck, 2008."},{"key":"atypb38","doi-asserted-by":"crossref","unstructured":"O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen,\n                      Variational Methods in Imaging\n                      , Springer, New York, 2008.","DOI":"10.1007\/978-0-387-69277-7"},{"key":"atypb39","doi-asserted-by":"crossref","unstructured":"T. Schuster, B. Kaltenbacher, B. Hofmann, and K. Kazimierski,\n                      Regularization Methods in Banach Spaces\n                      , Radon Ser. Comput. Appl. Math. 10, Walter de Gruyter, Berlin, New York, 2012.","DOI":"10.1515\/9783110255720"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/5\/2\/008"},{"key":"atypb41","first-page":"1035","volume":"4","author":"Tikhonov A. N.","year":"1963","journal-title":"Sov. Math. Dokl."},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/32\/1\/015007"},{"key":"atypb43","doi-asserted-by":"crossref","unstructured":"C. R. Vogel,\n                      Computational Methods for Inverse Problems\n                      , Frontiers Appl. Math. 23, SIAM, Philadelphia, 2002.","DOI":"10.1137\/1.9780898717570"},{"key":"atypb44","unstructured":"F. Werner,\n                      Inverse Problems with Poisson Data: Tikhonov-type Regularization and Iteratively Regularized Newton Methods\n                      , Der Andere Verlag, Uelvesbu\u0308ll, 2012. Ph.D. thesis, University of Go\u0308ttingen 2012."},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1515\/jiip-2013-0074"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1124036","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:15:38Z","timestamp":1787336138000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1124036"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1124036"],"URL":"https:\/\/doi.org\/10.1137\/17m1124036","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}