Volume 1, Issue 6
Efficient Reconstruction Methods for Nonlinear Elliptic Cauchy Problems with Piecewise Constant Solutions

Herbert Egger & Antonio Leitao

Adv. Appl. Math. Mech., 1 (2009), pp. 729-749.

Published online: 2009-01

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  • Abstract

In this article, a level-set approach for solving nonlinear elliptic Cauchy problems with piecewise constant solutions is proposed, which allows the definition of a Tikhonov functional on a space of level-set functions. We provide convergence analysis for the Tikhonov approach, including stability and convergence results. Moreover, a numerical investigation of the proposed Tikhonov regularization method is presented. Newton-type methods are used for the solution of the optimality systems, which can be interpreted as stabilized versions of algorithms in a previous work and yield a substantial improvement in performance. The whole approach is focused on three dimensional models, better suited for real life applications.

  • Keywords

Nonlinear Cauchy problems Elliptic operators Level-set methods

  • AMS Subject Headings

65J20 35J60

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COPYRIGHT: © Global Science Press

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@Article{AAMM-1-729, author = {Herbert Egger and Antonio Leitao}, title = {Efficient Reconstruction Methods for Nonlinear Elliptic Cauchy Problems with Piecewise Constant Solutions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2009}, volume = {1}, number = {6}, pages = {729--749}, abstract = {

In this article, a level-set approach for solving nonlinear elliptic Cauchy problems with piecewise constant solutions is proposed, which allows the definition of a Tikhonov functional on a space of level-set functions. We provide convergence analysis for the Tikhonov approach, including stability and convergence results. Moreover, a numerical investigation of the proposed Tikhonov regularization method is presented. Newton-type methods are used for the solution of the optimality systems, which can be interpreted as stabilized versions of algorithms in a previous work and yield a substantial improvement in performance. The whole approach is focused on three dimensional models, better suited for real life applications.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.09-m09S03}, url = {http://global-sci.org/intro/article_detail/aamm/8394.html} }
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