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Domain reconstruction using photothermal techniques. (English) Zbl 1147.65072

Summary: 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.

MSC:
65M32Inverse problems (IVP of PDE, numerical methods)
35K05Heat equation
35R30Inverse problems for PDE
44A10Laplace transform
65K10Optimization techniques (numerical methods)
49Q10Optimization of shapes other than minimal surfaces
35A22Transform methods (PDE)
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