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PhD (Cotutelle between Toulouse University and Polytechnique Montréal): Randomized Algorithms for Preconditioning of Nonlinear Inverse Problems

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Required Education : Master 2 in applied mathematics or scientific computing

PhD (Cotutelle between Toulouse University and Polytechnique Montréal): Randomized Algorithms for Preconditioning of Nonlinear Inverse Problems

 

Context
Inverse problems arising from science and engineering applications often deal with large-scale nonlinear problems involving computationally expensive numerical simulations and massive data sets. Advances in data collection and numerical simulation have changed the dynamics of scientific research and motivated the need for randomized algorithms to surmount challenges of complexity, robustness and scalability. Randomized algorithms can significantly accelerate computations and are well suited for exascale platforms. Recently, thanks to the advantages brought by high performance computing, randomized algorithms are being explored to accelerate the convergence of iterative methods for solving large-scale nonlinear inverse problems, which are typically solved by considering a sequence of quadratic subproblems. In this PhD we will focus on using randomized algebra to further accelerate the solution of quadratic subproblems.

 

Goals
The history of the iterative solvers used to solve the quadratic problems can be exploited to build an effective preconditioner. For instance, in data assimilation problems, this technique is used to update a preconditioner in a deterministic setting. Recently, the potential of stochastic preconditioners based on randomized algorithms has been investigated for weak-constraint 4D-Var. Although there are some preliminary numerical results in this area, there is limited theoretical analysis on randomized preconditioners for linear systems in sequence. In this project, we will focus a class of preconditioners based on randomized algorithms with application to data assimilation.

We will first focus on improving the accuracy of the preconditioner by exploring the strategies to choose the best possible data set to construct the preconditioner for a fixed given computational budget. As a second part, we will focus on how to reduce the computational cost of constructing the preconditioner. As the last and complementary direction, we will analyze the convergence behavior of the iterative methods that use the novel preconditioners. In this project we will first focus on symmetric and positive definite systems as encountered in data assimilation (DA) problems. We will develop our methods within the Object-Oriented Prediction System (OOPS) developed by ECMWF and Météo-France. We will first focus on the use of randomized preconditioners for a single DA problem. Then we will consider constructing preconditioners for ensembles of variational DAs which is an important subject for Météo-France and ECMWF.

 

Profile
Education: Master 2 in applied mathematics or scientific computing

Skills: linear algebra, optimization, interest for coding (C++, Python)

 

Practicalities

The successful PhD candidate will be enrolled in a co-tutelle PhD program between Toulouse University and Polytechnique Montréal. He/She is expected to spend the first half of the PhD at CERFACS in Toulouse and the second half of the PhD at Polytechnique Montréal.

 

Academic collaboration

The academic partner of the PhD is the Polytechnique Montreal located in Montreal, Canada.

 

Contact
Selime Gürol – selime.gurol(at)cerfacs.fr

Youssef Diouane – youssef.diouane(at)polymtl.ca