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

Preview Full PDF 231 1054
Export citation
  • 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

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@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} }
TY - JOUR T1 - Efficient Reconstruction Methods for Nonlinear Elliptic Cauchy Problems with Piecewise Constant Solutions AU - Herbert Egger & Antonio Leitao JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 729 EP - 749 PY - 2009 DA - 2009/01 SN - 1 DO - http://dor.org/10.4208/aamm.09-m09S03 UR - https://global-sci.org/intro/aamm/8394.html KW - Nonlinear Cauchy problems KW - Elliptic operators KW - Level-set methods AB -

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.

Herbert Egger & Antonio Leitao. (1970). Efficient Reconstruction Methods for Nonlinear Elliptic Cauchy Problems with Piecewise Constant Solutions. Advances in Applied Mathematics and Mechanics. 1 (6). 729-749. doi:10.4208/aamm.09-m09S03
Copy to clipboard
The citation has been copied to your clipboard