{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:19:31Z","timestamp":1787329171258,"version":"build-2736575974"},"reference-count":33,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We develop a local regularization theory for the nonlinear inverse autoconvolution problem. Unlike classical regularization techniques such as Tikhonov regularization, this theory provides regularization methods that preserve the causal nature of the autoconvolution problem, allowing for fast sequential numerical solution (${\\cal O}(rN^2 - r^2N)$ flops, where $r\\ll N$ for the method discussed in this paper as applied to the nonlinear problem; in comparison, the cost for Tikhonov regularization applied to a general linear problem is ${\\cal O}(N^3)$ flops). We prove the convergence of the regularized solutions to the true solution as the noise level in the data shrinks to zero and supply convergence rates for the cases of both $L_2$ and continuous data. We propose several regularization methods and provide a theoretical basis for their convergence; of note is that this class of methods does not require an initial guess of the unknown solution. Our numerical results confirm the effectiveness of the methods, with results comparing favorably to numerical examples found in the literature for the autoconvolution problem (e.g., [G. Fleischer, R. Gorenflo, and B. Hofmann, ZAMM Z. Angew. Math. Mech., 79 (1999), pp. 149\u2013159] for examples using Tikhonov regularization with total variation constraints and [J. Janno, Inverse Problems, 16 (2000), pp. 333\u2013348] for examples using the method of Lavrent'ev); this especially seems to be true when it comes to the recovery of sharp features in the unknown solution. We also show the effectiveness of our method in cases not covered by the theory.<\/jats:p>","DOI":"10.1137\/070679247","type":"journal-article","created":{"date-parts":[[2008,2,21]],"date-time":"2008-02-21T18:06:05Z","timestamp":1203617165000},"page":"832-868","source":"Crossref","is-referenced-by-count":22,"title":["Local Regularization for the Nonlinear Inverse Autoconvolution Problem"],"prefix":"10.1137","volume":"46","author":[{"given":"Zhewei","family":"Dai","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Patricia K.","family":"Lamm","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,2,20]]},"reference":[{"key":"R1","unstructured":"J. Baumeister,\n                      Deconvolution of appearance potential spectra\n                      , in Direct and Inverse Boundary Value Problems, R. Kleinman, R. Kress, adn E. 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Cui,\n                      Local Regularization Methods for n-Dimensional First-Kind Integral Equations\n                      , Ph.D. thesis, Michigan State University, East Lansing, MI, 2005."},{"key":"R8","unstructured":"Z. Dai,\n                      Local Regularization for the Autoconvolution Problem\n                      , Ph.D. thesis, Department of Mathematics, Michigan State University, East Lansing, MI, 2005."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/5\/4\/007"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"H. W. Engl, M. Hanke, and A. Neubauer,\n                      Regularization of Inverse Problems\n                      , Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.","DOI":"10.1007\/978-94-009-1740-8"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1521-4001(199903)79:3<149::AID-ZAMM149>3.0.CO;2-N"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/12\/4\/006"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/S0362-546X(96)00334-3"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/10\/2\/011"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.19970770521"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/16\/2\/305"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139994274496"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1995.1368"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/13\/2\/012"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"P. K. Lamm,\n                      Variable-smoothing regularization methods for inverse problems\n                      , in Theory and Practice of Control and Systems, A. Conte and A. M. Perdon, eds., World Scientific, Singapore, 1999.","DOI":"10.1142\/9789814447317_0055"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"P. K. Lamm,\n                      A survey of regularization methods for first-kind Volterra equations\n                      , in Surveys on Solution Methods for Inverse Problems, D. Colton, H. W. Engl, A. Louis, J. R. McLaughlin, and W. Rundell, eds., Springer, Vienna, New York, 2000, pp. 53\u201382.","DOI":"10.1007\/978-3-7091-6296-5_4"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/19\/1\/311"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/21\/3\/001"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/21\/5\/016"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/S003614299528081X"},{"key":"R26","unstructured":"P. K. Lamm and X. Luo,\n                      Local Regularization Methods for Nonlinear Hammerstein Equations\n                      , preprint, 2006."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/16\/2\/308"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1081\/NFA-100108315"},{"key":"R29","unstructured":"R. Miller,\n                      Nonlinear Volterra Integral Equations\n                      , W. A. 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