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Volume 4, Issue 1
A Binary Level Set Model for Elliptic Inverse Problems with Discontinuous Coefficients

L. K. Nielsen, X.-C. Tai, S. I. Aanonsen & M. Espedal

Int. J. Numer. Anal. Mod., 4 (2007), pp. 74-99.

Published online: 2007-04

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  • Abstract

In this paper we propose a variant of a binary level set approach for solving elliptic problems with piecewise constant coefficients. The inverse problem is solved by a variational augmented Lagrangian approach with a total variation regularisation. In the binary formulation, the sought interfaces between the domains with different values of the coefficient are represented by discontinuities of the level set functions. The level set functions shall only take two discrete values, i.e. 1 and -1, but the minimisation functional is smooth. Our formulation can, under moderate amount of noise in the observations, recover rather complicated geometries without requiring any initial curves of the geometries, only a reasonable guess of the constant levels is needed. Numerical results show that our implementation of this formulation has a faster convergence than the traditional level set formulation used on the same problems.

  • Keywords

inverse problems, parameter identification, elliptic equation, augmented Lagrangian optimisation, level set methods, total variation regularisation.

  • AMS Subject Headings

49Q10, 35R30, 65J20, 74G75

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-4-74, author = {L. K. Nielsen, X.-C. Tai, S. I. Aanonsen and M. Espedal}, title = {A Binary Level Set Model for Elliptic Inverse Problems with Discontinuous Coefficients}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2007}, volume = {4}, number = {1}, pages = {74--99}, abstract = {

In this paper we propose a variant of a binary level set approach for solving elliptic problems with piecewise constant coefficients. The inverse problem is solved by a variational augmented Lagrangian approach with a total variation regularisation. In the binary formulation, the sought interfaces between the domains with different values of the coefficient are represented by discontinuities of the level set functions. The level set functions shall only take two discrete values, i.e. 1 and -1, but the minimisation functional is smooth. Our formulation can, under moderate amount of noise in the observations, recover rather complicated geometries without requiring any initial curves of the geometries, only a reasonable guess of the constant levels is needed. Numerical results show that our implementation of this formulation has a faster convergence than the traditional level set formulation used on the same problems.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/852.html} }
TY - JOUR T1 - A Binary Level Set Model for Elliptic Inverse Problems with Discontinuous Coefficients AU - L. K. Nielsen, X.-C. Tai, S. I. Aanonsen & M. Espedal JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 74 EP - 99 PY - 2007 DA - 2007/04 SN - 4 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/852.html KW - inverse problems, parameter identification, elliptic equation, augmented Lagrangian optimisation, level set methods, total variation regularisation. AB -

In this paper we propose a variant of a binary level set approach for solving elliptic problems with piecewise constant coefficients. The inverse problem is solved by a variational augmented Lagrangian approach with a total variation regularisation. In the binary formulation, the sought interfaces between the domains with different values of the coefficient are represented by discontinuities of the level set functions. The level set functions shall only take two discrete values, i.e. 1 and -1, but the minimisation functional is smooth. Our formulation can, under moderate amount of noise in the observations, recover rather complicated geometries without requiring any initial curves of the geometries, only a reasonable guess of the constant levels is needed. Numerical results show that our implementation of this formulation has a faster convergence than the traditional level set formulation used on the same problems.

L. K. Nielsen, X.-C. Tai, S. I. Aanonsen and M. Espedal. (2007). A Binary Level Set Model for Elliptic Inverse Problems with Discontinuous Coefficients. International Journal of Numerical Analysis and Modeling. 4 (1). 74-99. doi:
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