Article Contents
Article Contents

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

• 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:

• 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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