<p>In the context of a collaboration between OPIS and IDEFIX Inria teams, the aim of this internship is to investigate&nbsp;the design and convergence analysis of fixed-point algorithms for the resolution of inverse problems involving PDEs.</p>
<p><strong>Subject:</strong>&nbsp; <br />Inverse problems are central to many applications like medical imaging, geophysical inspection, and non-destructive testing, where the goal is to identify some parameters (such&nbsp;as physical properties or defect geometries) from given measurements. Unlike forward&nbsp;problems, which involve solving linear PDEs with a unique solution, inverse problems are&nbsp;often nonlinear and ill-posed. Commonly used inversion algorithms in engineering often&nbsp;rely on the linearization of the inverse problem, leading for example to methods like migration or time reversal for inverse wave problems. While these methods are fast and do&nbsp;not require iterative procedures, they suffer from biases due to linearization and can fail&nbsp;in complex environments. A more conventional/traditional approach involves formulating&nbsp;the inverse problem as a minimization of a cost functional that measures data fidelity by&nbsp;comparing it with the PDE solution for a given parameter. However, solving this is often&nbsp;more computationally expensive than solving the PDE itself, making it impractical for&nbsp;real-time applications or large-scale problems. While efforts have been made to enhance&nbsp;the performance of these methods, they often treat forward iterative<br />solvers independently of inversion algorithms, or incorporate the forward map into the cost&nbsp;functional leading to suboptimal solutions for the forward problem.</p>
<p>Preliminary results have been obtained, showing the suitability of proximal fixed point techniques to address the problem. The goal of the internship is to explore consensus-based formulation and implementation of the methods.&nbsp;</p>
<p>&nbsp;</p>