The Reduced Basis Method for Ground-State Eigenvalue Computations

Abstract

In this talk, we present a reduced basis method for computing 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, for example, the ground-state phase diagram. The basis of this subspace is built from solutions of snapshots, i.e., ground states corresponding to specific and well-chosen parameter values. We highlight the difficulties arising from 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 also discuss recent progress in local error estimation of the Taylor Reduced Basis Method for eigenvalue problems, i.e., when perturbative modes with respect to the parameter are included in the basis as well. Numerical experiments will accompany the theoretical results in both cases.

Benjamin Stamm

Prof. Dr. Benjamin Stamm is a professor in the Department of Mathematics at the University of Stuttgart, where he joined in August 2022. He is the Chair of Numerical Mathematics for High Performance Computing. Before joining the University of Stuttgart, he held academic positions at RWTH Aachen University and Sorbonne Université/UPMC Paris 6, following research stays at the University of California, Berkeley, and Brown University.

He earned both his Ph.D. and Master’s degree in Mathematics from the École Polytechnique Fédérale de Lausanne (EPFL).

His research lies at the interface of numerical analysis, scientific computing, and simulation, with a particular focus on efficient discretizations and solvers for partial differential equations, eigenvalue problems, reduced-basis methods, a posteriori error estimation, error certification, and high-performance computing. His work develops scalable numerical methods for complex problems arising in the natural sciences, especially in computational chemistry, physics, and materials science in an interdisciplinary context.

Recent contributions include work on reduced-basis methods for eigenvalue problems, certified model order reduction, Kohn–Sham equations, Schrödinger eigenstates, and molecular simulation.