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Accueil du site > English > Research Topics > Topic 5 - Modelization, Algorithms and High Performance Computing > APO team

APO team

Parallel Algorithms and Optimization
Head : Patrick Amestoy

 

The two main focuses of our research activity are numerical optimization and sparse linear algebra.

 

Numerical optimization

Our first activity in optimization is the analysis and control of dynamical systems, that is of systems governed by ODE and equipped with a cost functional. We are interested in the numerical solution of optimal control problems, mostly via indirect methods.

In the framework of grants with the French Space Agency (CNES), low thrust orbit transfer is a privileged application of the team, with an emphasis on time or consumption minimization. The advantage of electro-ionic propulsion over the traditional chemical one is an increased efficiency. The counterpart is a very low thrust that demands to solve optimal control problems with very long transfer times. In the case of the minimization of consumption, the control is discontinuous and it is crucial to deal properly with the numerous switchings occuring. Besides, technological constraints on the engines result in logical constraints on the system. These have first been studied on the linear quadratic regulator problem by means of a direct method, leading to a large scale mixed-integer programming problem. Some techniques used in this context such as branch-and-bound or interval analysis, also pertain to our second activity in the field which is global optimization. The motivation is to design efficient algorithms yielding global minimizers, and a key step in this direction is a refined computation of bounds on the criteria. Eventually, we are also beginning to consider applications of optimization in computational biology.

 

Sparse linear algebra

In sparse linear algebra, we consider both direct methods (LU, LDLT and QR factorizations) of the original system of equations and iterative methods. In this context, it is often critical to preprocess (reordering, numerical scaling) and/or to precondition the original matrix to reduce time and memory requirements. Furthermore, to solve more challenging large real problems with hundreds of millions of unknowns, our algorithms must also exploit efficiently large scale parallel distributed memory computers while controlling the accuracy of the solution. In that respect, not only we aim at continuing to improve our direct and preconditioned iterative schemes but also to mix the two approaches in order to study the relevance of parallel hybrid solvers. Finally in the context of the GRID-TLSE project (started in January 2003 within the framework of the french inititative ACI-Grid), our group has been involved in the design and the development of a Web site dedicated to sparse linear algebra. Given a set of criteria choosing the best sparse solver to solve a problem is often a complex and time consuming task. In this context, the computational Grid is used through our Web site to help the user find the best solution to his problem.

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