In many applications, once a system is reasonably modeled, arises the question of optimizing some function of the state of the system. Therefore optimization is a central algorithmic area in numerical simulation.
This project aims at developing efficient methods and codes for possibly large-scale linear and nonlinear optimization problems. For such demanding problems, derivative based techniques will be considered, but not only. The most advanced algorithms we will consider in this area are parallel optimization algorithms either based on multigrid or domain decomposition approaches. However, we consider that, provided a suitable model reduction approach can be proposed for a given application, derivative free techniques can also be employed, which is interesting in situations where the gradient of the function to minimize is either unavailable or computationally expensive.
Particular attention will be paid to linear and nonlinear parameter estimation and more specifically to data assimilation problems. These problems arise in various fields such as signal processing, geophysics, space dynamics, and meteorology. The algorithms used are based on very general results of probability therory and physics. However, what distinguishes these applications is that they usually exhibit a strong structure that has to be exploited to obtain efficient algorithms. Our treatment will use recent advances made in stochastic filtering to design fast and robust algorithms that might include : fast and robust parameter estimation, observability and sensitivity analysis, covariance estimation, regularization techniques for underdetermined problems, design of efficient preconditioners, and study of stopping criteria for iterative solvers.
Properties of all these algorithms will be analyzed both theoretically and experimentally on practical problems.
This activity has developed with a strong partnership with CERFACS, Toulouse. In particular several Thesis and Postdoc grants have been jointly supervised by reseracher from IRIT and from CERFACS. A natural evolution of this activity will give rise to a CERFACS-IRIT joint laboratory in 2013, which will be a very interesting place for academic and industrial research and will enable the development of fruitful collaborative research projects between CERFACS and IRIT. The activity is also strongly supported by the RTRA-STAE foundation, in the framework of the ADTAO and FILAOS projects. These two projects are focus on the design of parallel algorithms for solving inverse problems that present some difficulties because they are non-Gaussian, they are strongly nonlinear or involve a large number of degrees of freedom. The APO team is also in charge of the supervision of the MoMa activity group on Data Assimilation for the RTRA-STAE foundation. This project is also one of the activities of the CIMI Labex that is an innovative center at the interface if IRIT and of the Mathematical institute of Toulouse.