Electrical Impedance Tomography
SIAM Review
Wavelet and Fourier Methods for Solving the Sideways Heat Equation
SIAM Journal on Scientific Computing
Incorporating topological derivatives into level set methods
Journal of Computational Physics
On the numerical inversion of the Laplace transform of certain holomorphic mappings
Applied Numerical Mathematics
Determining Planar Multiple Sound-Soft Obstacles from Scattered Acoustic Fields
Journal of Mathematical Imaging and Vision
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A numerical method to detect objects buried in a medium by surface thermal measurements is presented. We propose a new approach combining the use of topological derivatives and Laplace transforms. The original optimization problem with time-dependent constraints is replaced by an equivalent problem with stationary constraints by means of Laplace transforms. The first step in the reconstruction scheme consists in discretizing the inversion formula to produce an approximate optimization problem with a finite set of constraints. Then, an explicit expression for the topological derivative of the approximate shape functional is given. This formula is evaluated at low cost using explicit expressions of the forward and adjoint fields involved. We apply this technique to a simple shape reconstruction problem set in a half space. Good approximations of the number, location and size of the obstacles are obtained. The description of their shapes can be improved by more expensive hybrid methods combining time averaging with topological derivative based iterative schemes.