  This paper addresses the numerical solution of nonlinear eigenvector problems
such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational
physics and chemistry. These problems characterize critical points of energy
minimization problems on the infinite-dimensional Stiefel manifold. To
efficiently compute minimizers, we propose a novel Riemannian gradient descent
method induced by an energy-adaptive metric. Quantified convergence of the
methods is established under suitable assumptions on the underlying problem. A
non-monotone line search and the inexact evaluation of Riemannian gradients
substantially improve the overall efficiency of the method. Numerical
experiments illustrate the performance of the method and demonstrates its
competitiveness with well-established schemes.

Numerical Analysis;Numerical Analysis