Master Internship - Fixed Point Algorithms for Inverse Wave Problems

The Inria Saclay-Île-de-France Research Centre was established in 2008. It has developed as part of the Saclay site in partnership with Paris-Saclay University and with the Institut Polytechnique de Paris .

The centre has 40 project teams , 32 of which operate jointly with Paris-Saclay University and the Institut Polytechnique de Paris; Its activities occupy over 600 people, scientists and research and innovation support staff, including 44 different nationalities.

In the context of a collaboration between OPIS and IDEFIX Inria teams, the aim of this internship is to investigate the design and convergence analysis of fixed-point algorithms for the resolution of inverse problems involving PDEs.

Subject: 
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 as physical properties or defect geometries) from given measurements. Unlike forward problems, which involve solving linear PDEs with a unique solution, inverse problems are often nonlinear and ill-posed. Commonly used inversion algorithms in engineering often 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 not require iterative procedures, they suffer from biases due to linearization and can fail in complex environments. A more conventional/traditional approach involves formulating the inverse problem as a minimization of a cost functional that measures data fidelity by comparing it with the PDE solution for a given parameter. However, solving this is often more computationally expensive than solving the PDE itself, making it impractical for real-time applications or large-scale problems. While efforts have been made to enhance the performance of these methods, they often treat forward iterative
solvers independently of inversion algorithms, or incorporate the forward map into the cost functional leading to suboptimal solutions for the forward problem.

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. 

Missions: The recruited student will first perform a bibliographical study on the topic. Then, in a second step, the student will propose a mathematical formulation of the problem, and a suitable distributed fixed-point algorithm to solve it. Third, the convergence of the approach will be established.

Environment: The intern will be supervised by Emilie Chouzenoux (Head of OPIS team, Inria Saclay). The intern student will join the Inria Saclay team OPIS (https://opis-inria.eu/). He/she will be located in the Centre de la Vision Numérique, in CentraleSupélec campus, Saclay, France. He/she will enjoy an international and creative environment where research seminars and reading groups take place very often. Informatic material expenses will be covered within the limits of the scale in force. The student will participate to monthly collaborative meetings researchers from OPIS team (Emilie Chouzenoux, Jean-Christophe Pesquet), and IDEFIX team (Lorenzo Audibert, Houssem Haddar).

Organization: The proposed offer is dedicated to internship of Master 2 / Engineering students. The starting/end dates are flexible, with a minimum duration of 5 months.

Main activities :

Bibliographical study

Optimization problem formulation and resolution

Convergence Analysis

Scientific meetings

Writing of scientific reports

Languages : The candidate must be fluent in english and/or french languages.

• Subsidized meals
• Partial reimbursement of public transport costs
• Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
• Possibility of teleworking and flexible organization of working hours
• Professional equipment available (videoconferencing, loan of computer equipment, etc.)
• Social, cultural and sports events and activities
• Access to vocational training
• Social security coverage

Gratification