Colloque - Benjamin Stamm : The Reduced Basis Method for Ground-State Eigenvalue Computations

Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Benjamin Stamm : The Reduced Basis Method for Ground-State Eigenvalue ComputationsBenjamin StammProfessor, Head of Group Institute of Applied Analysis and Numerical Simulation, Chair of Numerical Mathematics for High Performance Computing, Faculty of Mathematics, University of Stuttgart, GermanyRésuméIn this talk, we present a reduced basis method for the computation of the lowest eigenvalue(s) of a parametrized family of eigenvalue problems motivated by, but not restricted to, many-body quantum problems. Within the reduced basis methods approach, an effective low-dimensional subspace of the high-dimensional Hilbert space is constructed in order to investigate, e.g., the ground-state phase diagram. The basis of this subspace is built from solutions of snapshots, i.e., ground states corresponding to particular and well-chosen parameter values. We highlight the difficulties due to the nature of eigenvalue problems such as degeneracy of eigenstates and vanishing gaps in error certification, and provide some computational strategies with guarantees as potential remedies. We will discuss also recent progress in local error estimation of the Taylor Reduced Basis Method for eigenvalue problems, i.e. if perturbative modes with respect to the parameter are included in the basis as well. Numerical experiments will accompany in both cases the theoretical results.