{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T22:26:35Z","timestamp":1787783195385,"version":"build-2784847793"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>This paper considers the problem of reconstructing a piecewise smooth model function from given, measured data. The data are compared to a field which is given as a possibly nonlinear function of the model. A regularization functional is added which incorporates the a priori knowledge that the model function is piecewise smooth and may contain jump discontinuities. Regularization operators related to total variation (TV) are therefore employed.<\/jats:p>\n                  <jats:p>Two popular methods are modified TV and Huber's function. Both contain a parameter which must be selected. The Huber variant provides a more natural approach for selecting its parameter, and we use this to propose a scheme for both methods. Our selected parameter depends both on the resolution and on the model average roughness; thus, it is determined adaptively. Its variation from one iteration to the next yields additional information about the progress of the regularization process.<\/jats:p>\n                  <jats:p>The modified TV operator has a smoother generating function; nonetheless we obtain a Huber variant with comparable, and occasionally better, performance.<\/jats:p>\n                  <jats:p>For large problems (e.g., high resolution) the resulting reconstruction algorithms can be tediously slow. We propose two mechanisms to improve efficiency. The first is a multilevel continuation approach aimed mainly at obtaining a cheap yet good estimate for the regularization parameter and the solution. The second is a special multigrid preconditioner for the conjugate gradient algorithm used to solve the linearized systems encountered in the procedures for recovering the model function.<\/jats:p>","DOI":"10.1137\/040617261","type":"journal-article","created":{"date-parts":[[2006,3,24]],"date-time":"2006-03-24T21:00:17Z","timestamp":1143234017000},"page":"339-358","source":"Crossref","is-referenced-by-count":45,"title":["On Effective Methods for Implicit Piecewise Smooth Surface Recovery"],"prefix":"10.1137","volume":"28","author":[{"given":"U. M.","family":"Ascher","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"E.","family":"Haber","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H.","family":"Huang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/17\/3\/314"},{"key":"R2","unstructured":"U. Ascher and E. Haber,\n                      Computational methods for large distributed parameter estimation problems with possible discontinuities\n                      , in Proceedings of the Inverse Problems, Design and Optimization Symposium, M. Colaco, H. Orlande and G. 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