| Abstract: |
| Typical PDE based nonlinear inverse problems involve a known forcing term and a feature (such as coefficient or inner geometry feature) to be recovered using overdetermined boundary data. However, this does not apply to passive inverse problems. In these problems, the forcing term in the PDE is unknown while the geometry of a feature has to be recovered. There are thus both linear and nonlinear unknowns. We focus on recovery of cracks in unbounded domains with propagating waves. We present recent results regarding the stability of the Hausdorff distance between cracks, (Triki and Volkov, 2025). Next, we examine the case where cracks are defined through a vector parameter $m$ while the forcing term for the PDE is still in an infinite dimensional space (Ganesh, Hawkins, and Volkov, 2026). We proved Lipschitz continuity of a related inverse operator if the forward operator is restricted to $m$ -dependent finite dimensional spaces. These finite dimensional spaces are spanned by $m$ -dependent singular functions which can be computed. This led us to build neural networks that can solve the crack inverse problem. The solution is computed in an efficient non-iterative way and is robust to noise. |
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