Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients

Tony Chan, Xue Cheng Tai*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

154 Scopus citations

Abstract

We propose a level set approach for elliptic inverse problems with piecewise constant coefficients. The geometry of the discontinuity of the coefficient is represented implicitly by level set functions. The inverse problem is solved using a variational augmented Lagrangian formulation with total variation regularization of the coefficient. The corresponding Euler-Lagrange equation gives the evolution equation for the level set functions and the constant values of the coefficients. We use a multiple level set representation which allows the coefficient to have multiple constant regions. Knowledge of the exact number of regionsis not required, only an upper bound is needed. Numerical experiments show that the method can recover coefficients with rather complicated geometries of discontinuities under moderate amount of noise in the observation data. The method is also robust with respect to the initial guess for the geometry of the coefficient discontinuities.

Original languageEnglish (US)
Pages (from-to)40-66
Number of pages27
JournalJournal of Computational Physics
Volume193
Issue number1
DOIs
StatePublished - Jan 1 2004

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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