{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:33:50Z","timestamp":1787322830662,"version":"build-2736575974"},"reference-count":31,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2024,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We consider the task of estimating the unknown diffusion parameter in an elliptic PDE as a model problem to develop and test the effectiveness and robustness to noise of reconstruction schemes with sparsity regularization. To this end, the model problem is recast as a nonlinear infinite dimensional optimization problem, where the logarithm of the unknown diffusion parameter is modeled using a linear combination of the elements of a dictionary, i.e., a known bounded sequence of [Formula: see text] functions, with unknown coefficients that form a sequence in [Formula: see text]. We show that the regularization of this nonlinear optimization problem using a weighted [Formula: see text]-norm has minimizers that are finitely supported. We then propose modifications of well-known algorithms (ISTA and FISTA) to find a minimizer of this weighted [Formula: see text]-norm regularized nonlinear optimization problem that accounts for the fact that in general the smooth part of the functional being optimized is a functional only defined over [Formula: see text]. We also introduce semismooth methods (ASISTA and FASISTA) for finding a minimizer, which locally uses Gauss\u2013Newton type surrogate models that additionally are stabilized by means of a Levenberg\u2013Marquardt type approach. Our numerical examples show that the regularization with the weighted [Formula: see text]-norm indeed does make the estimation more robust with respect to noise. Moreover, the numerical examples also demonstrate that the ASISTA and FASISTA methods are quite efficient, outperforming both ISTA and FISTA.<\/jats:p>","DOI":"10.1137\/23m1565346","type":"journal-article","created":{"date-parts":[[2024,1,17]],"date-time":"2024-01-17T04:36:10Z","timestamp":1705466170000},"page":"61-90","source":"Crossref","is-referenced-by-count":0,"title":["Identification of Sparsely Representable Diffusion Parameters in Elliptic Problems"],"prefix":"10.1137","volume":"17","author":[{"given":"Luzia N.","family":"Felber","sequence":"first","affiliation":[{"name":"Departement Mathematik und Informatik, Universit\u00e4t Basel, Spiegelgasse 1, 4051 Basel, Switzerland."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Helmut","family":"Harbrecht","sequence":"additional","affiliation":[{"name":"Departement Mathematik und Informatik, Universit\u00e4t Basel, Spiegelgasse 1, 4051 Basel, Switzerland."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marc","family":"Schmidlin","sequence":"additional","affiliation":[{"name":"Departement Mathematik und Informatik, Universit\u00e4t Basel, Spiegelgasse 1, 4051 Basel, Switzerland."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,1,17]]},"reference":[{"key":"ref1","first-page":"163","volume":"32","author":"Assmann U.","year":"2013","journal-title":"J. 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