{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:27:01Z","timestamp":1787329621815,"version":"3.56.0"},"reference-count":16,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>Regularization is used in order to obtain a reasonable estimate of the solution to an ill-posed inverse problem. One common form of regularization is to use a filter to reduce the influence of components corresponding to small singular values, perhaps using a Tikhonov least squares formulation. In this work, we break the problem into subproblems with narrower bands of singular values using spectrally defined windows, and we regularize each subproblem individually. We show how to use standard parameter-choice methods, such as the discrepancy principle and generalized cross-validation, in a windowed regularization framework. A perturbation analysis gives sensitivity estimates. We demonstrate the effectiveness of our algorithms on deblurring images and on the backward heat equation.<\/jats:p>","DOI":"10.1137\/100809787","type":"journal-article","created":{"date-parts":[[2011,11,2]],"date-time":"2011-11-02T18:13:12Z","timestamp":1320257592000},"page":"3175-3200","source":"Crossref","is-referenced-by-count":9,"title":["Windowed Spectral Regularization of Inverse Problems"],"prefix":"10.1137","volume":"33","author":[{"given":"Julianne","family":"Chung","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Glenn","family":"Easley","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dianne P.","family":"O'Leary","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,11,1]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/18\/4\/314"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1109\/18.382009"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1080\/00401706.1979.10489751"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/BF02149761"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"P. C. Hansen,\n                      Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion\n                      , SIAM, Philadelphia, 1998.","DOI":"10.1137\/1.9780898719697"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"C. Leung and W. Lu,\n                      A multiple-parameter generalization of the Tikhonov-Miller regularization method for image restoration\n                      , in Signals, Systems and Computers, IEEE Conference Record of the 27th Asilomar Conference, 1993, pp. 856\u2013860.","DOI":"10.1109\/ACSSC.1993.342448"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"S. Mallat,\n                      A Wavelet Tour of Signal Processing\n                      , Academic Press, San Diego, CA, 1998.","DOI":"10.1016\/B978-012466606-1\/50008-8"},{"key":"R8","unstructured":"K. Modarresi and G. Golub,\n                      Multi-level approach to numerical solution of inverse problems\n                      , in CSC 2007: SIAM Workshop on Combinatorial Scientific Computing, Costa Mesa, CA, 2007; also available online from http:\/\/www.tau.ac.il\/~stoledo\/CSC07\/Kourosh1.pdf."},{"key":"R9","unstructured":"K. Modarresi and G. Golub,\n                      Using multiple generalized cross-validation as a method for varying smoothing effects\n                      , in CSC 2007: SIAM Workshop on Combinatorial Scientific Computing, Costa Mesa, CA, 2007; also available online from http:\/\/www.tau.ac.il\/~stoledo\/CSC07\/Kourosh2.pdf."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2003.821103"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827598341384"},{"key":"R12","unstructured":"A. Papoulis and S. U. Pillai,\n                      Probability, Random Variables and Stochastic Processes\n                      , McGraw\u2013Hill, New York, 2001."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/24\/3\/034005"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"I. Stephanakis and S. Kollias,\n                      Generalized-cross-validation estimation of the regularization parameters of the subbands in wavelet domain regularized image restoration\n                      , in Signals, Systems and Computers, IEEE Conference Record of the 32nd Asilomar Conference, Vol. 2, 1998, pp. 938\u2013940.","DOI":"10.1109\/ACSSC.1998.751400"},{"key":"R15","unstructured":"M. Vetterli and J. Kova\u010devi\u010d,\n                      Wavelets and Subband Coding\n                      , Prentice\u2013Hall, Englewood Cliffs, NJ, 1995."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/s00190-006-0025-0"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/100809787","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:23:34Z","timestamp":1787325814000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/100809787"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":16,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1137\/100809787"],"URL":"https:\/\/doi.org\/10.1137\/100809787","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1]]}}}