Section: New Results

Application of greedy algorithms

Participants : Sébastien Boyaval, Eric Cancès, Virginie Ehrlacher, Tony Lelièvre.

Model reduction techniques are very important tools for applications. They consist in deriving from a high-dimensional problem, a low-dimensional model, which very quickly gives reliable results. In particular, the team is interested in two techniques: Proper Generalized Decomposition (greedy algorithms) and Reduced Basis techniques.

Eric Cancès, Virginie Ehrlacher and Tony Lelièvre have extended a greedy algorithm suggested for the resolution of high-dimensional eigenvalue problems in order to approximate the solution of the many-body Schrödinger electronic problem. The main technical difficulty in the extension of these algorithms lies in the antisymmetry of the wavefunction of the electrons. To deal with this difficulty, an approximation of the wavefunction is computed as a sum of Slater determinants, each Slater determinant function being computed in an iterative way.

Virginie Ehrlacher has obtained preliminary encouraging results on greedy algorithms for parametric eigenvalue problems. The method has been applied to the computation of the first buckling mode of a plate in the presence of a defect, the position of the defect playing the role of a parameter entering the eigenvalue problem defining the first buckling mode of the plate.

A new numerical method for the construction of an efficient reduced-order model for the solution of the Vlasov equation, arising in plasma physics or in the modeling of electron transport in semiconductors, has been tested by Damiano Lombardi (REO Inria team) and Virginie Ehrlacher. This method is based on the use of an analytic Lax Pair for the Vlasov equations and is inspired by previous works done on transport equations by Jean-Frederic Gerbeau, Damiano Lombardi and Elisa Schenone. Encouraging preliminary numerical results have been obtained.