Non-uniqueness result for a hybrid inverse problem

Hybrid inverse problems aim to combine two imaging modalities, one that displays large contrast and one that gives high resolution. Mathematically, quantitative reconstructions in such hybrid problems involve reconstructing coefficients in a partial differential equation (PDE) from point-wise functionals of the coefficients and the PDE solution. There are many settings in which such inverse problems are shown to be well posed in the sense that the reconstruction of the coefficients is unique and stable for an appropriate functional setting. In this paper, we obtain an example where uniqueness fails to hold. Such a problem appears as a simplified model in acousto-optics, a hybrid medical imaging modality, and is related to the inverse medium problem where uniqueness results were obtained. Here, we show that two different solutions satisfying the same measurements can be reconstructed. The result is similar in spirit to the Ambrosetti-Prodi non-uniqueness result in the analysis of semi-linear equations. Numerical simulations confirm the theoretical predictions.