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An inverse obstacle problem for the wave equation in a finite time domain

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  • We consider an inverse obstacle problem for the acoustic transient wave equation. More precisely, we wish to reconstruct an obstacle characterized by a Dirichlet boundary condition from lateral Cauchy data given on a subpart of the boundary of the domain and over a finite interval of time. We first give a proof of uniqueness for that problem and then propose an "exterior approach" based on a mixed formulation of quasi-reversibility and a level set method in order to actually solve the problem. Some 2D numerical experiments are provided to show that our approach is effective.

    Mathematics Subject Classification: Primary: 35R25, 35R30, 35R35; Secondary: 65M60.

    Citation:

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  • Figure 1.  Notations

    Figure 2.  Illustration of the non-monotonicity of the mapping $ O \mapsto D(\Omega,\Gamma) $

    Figure 3.  Radial case. Discrepancy $ |u_ \varepsilon -u| $ as a function of $ |x| $, for $ t = 2.5 $, $ t = 3 $, $ t = 3.5 $, $ t = 4 $ and $ t = 4.5 $

    Figure 4.  Two discs. Left: function $ u_ \varepsilon $. Right: function $ |u_ \varepsilon -u| $

    Figure 5.  Validation of the level set method ($ T = 25 $)

    Figure 6.  Two discs and exact data. Top left: $ T = 10 $. Top right: $ T = 15 $. Bottom: $ T = 25 $

    Figure 7.  Two discs and noisy data. Top left: $ \delta = 0 $ (exact data). Top right: $ \delta = 0.02 $. Bottom: $ \delta = 0.05 $

    Figure 8.  Partial (exact) data and one disc. Left: obstacle located far away from $ \partial G \setminus \overline{\Gamma} $. Right: obstacle located close to $ \partial G \setminus \overline{\Gamma} $

    Figure 9.  Boomerang obstacle. Left: $ \delta = 0 $ (exact data). Right: $ \delta = 0.02 $

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