Direct Minimization for Calculating Invariant Subspaces in Density Functional Computations of the Electronic Structure (Journal of Computational Mathematics, 2018)
In this article, we analyse three related preconditioned steepest descent algorithms,which are partially popular in Hartree-Fock and Kohn-Sham theory as well as invariantsubspace computations, from the viewpoint of minimization of the corresponding functionals, constrained by orthogonality conditions. We exploit the geometry of the admissiblemanifold, i.e., the invariance with respect to unitary transformations, to reformulate theproblem on the Grassmann manifold as the admissible set. We then prove asymptoticallinear convergence of the algorithms under the condition that the Hessian of the corresponding Lagrangian is elliptic on the tangent space of the Grassmann manifold at theminimizer.