Piecewise constant level set algorithm for an inverse elliptic problem   in nonlinear electromagnetism
  An inverse problem of identifying inhomogeneity or crack in the workpiece made of nonlinear magnetic material is investigated. To recover the shape from the local measurements, a piecewise constant level set algorithm is proposed. By means of the Lagrangian multiplier method, we derive the first variation w.r.t the level set function and obtain the descent direction by the adjoint variable method. Numerical results show the robustness and effectiveness of our algorithm applied to reconstruct some complex shapes. 
Xiangyin Kong, Zhengfang Zhang and Zhengda Huang
math-ph math.MP