Speaker
Description
We deal with efficient and certified numerical approximation of parametric Hermitian eigenproblems. For this aim, we rely on projection-based model order reduction, i.e. we approximate a possibly large-scale problem with one of a much smaller dimension by projecting it on a suitable subspace. Such a space is constructed by means of weak-greedy type strategies applied either on a continuum or discrete parameter domain. After discussing the connections with the reduced basis method for source problems, we introduce a novel posteriori error estimate for the eigenspace associated with the smallest eigenvalue. It turns out that the approximation of the difference between the second smallest and the smallest eigenvalues, the so-called spectral gap, is crucial for the reliability of the error estimate. Therefore, we propose new efficiently computable upper and lower bounds for the spectral gap, which allow for an approximation through a greedy procedure. Our framework is well-suited to tackle the cases where the smallest eigenvalue is not simple. Our work is motivated by a particular application, which is the repeated evaluation of the ground state of parametric quantum spin system models. In this framework, the ground state corresponds to the eigenspace associated with the smallest eigenvalues of the QSS Hamiltonian. Besides that, finding the ground state of a parametric Hermitian eigenproblem is omnipresent in physics, chemistry, and engineering.
