A hyperbolic seminar on dynamical low-rank approximation - Gianluca Ceruti (EPFL)

07.06.2022 14:00 – 15:00

In the present seminar, we begin with a recapitulation of the dynamical low-rank approximation for matrices of fixed rank-r together with the derivation of the so-called matrix projector-splitting integrator. We show that the matrix projector splitting integrator satisfies two remarkable properties: It reproduces rank-r time-dependent matrices exactly, and it is robust with respect to presence of small singular values in the approximation or the solution. Furthermore, two major built-in drawbacks of the matrix projector-splitting integrator are discussed: The integrator contains a backward substep, which is a major source of instability for dissipative problems, and it does not allow for an adaptive choice of the rank. Therefore, a novel unconventional integrator is introduced. The so-called unconventional integrator is shown to maintain the remarkable properties of the original matrix projector-splitting integrator meanwhile overcoming the aforementioned drawbacks. Finally, the extension to a continuous L2-setting of the dynamical low-rank approximation together with the matrix projector-splitting integrator for matrices is introduced. In the spirit of the Von Neumann stability analysis, a stability analysis of dynamical low-rank approaches for linear hyperbolic problems is discussed and the application to Burgers’ equation with 2 uncertainties is illustrated.

This seminar is based upon recent joint collaborations with Christian Lubich, Hanna Walach, Jonas Kusch, Lukas Einkemmer, and Martin Frank.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Séminaire d'analyse numérique

entrée libre