Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Abstract

This article is concerned with the numerical solution of subspace optimization problems, consisting of minimizing a smooth functional over the set of orthogonal projectors of fixed rank. Such problems are encountered in particular in electronic structure calculation (Hartree-Fock and Kohn-Sham Density Functional Theory-DFT-models). We compare from a numerical analysis perspective two simple representatives, the damped self-consistent field (SCF) iterations and the gradient descent algorithm, of the two classes of methods competing in the field: SCF and direct minimization methods. We derive asymptotic rates of convergence for these algorithms and analyze their dependence on the spectral gap and other properties of the problem. Our theoretical results are complemented by numerical simulations on a variety of examples, from toy models with tunable parameters to realistic Kohn-Sham computations. We also provide an example of chaotic behavior of the simple SCF iterations for a nonquadratic functional.

A robust and efficient line search for self-consistent field iterations

“…By analogy with Hamiltonian mixing, Theorem 1 guarantees that global convergence can always be ensured by selecting α small enough. In this respect our results from Section II C strengthen a number of previous results 15,18,22 , which established local convergence for sufficiently small α.…”

“…We use similar notation to those in Cancès et al 19 , extend the analysis in that paper to the finite-temperature case, and introduce the potential mixing algorithm. We work in the grand-canonical ensemble: we fix a chemical potential (or Fermi level) µ and an inverse temperature β.…”

Implicit algorithms for eigenvector nonlinearities

“…The convergence of SCF and its variants has been studied in a number of works which can be classified into two broad categories: the optimization-based approach of looking at (1) as the optimality conditions of a minimization problem [8,[18][19][20] or different matrix analysis-based approaches [34,35]. For a discussion of similarities and differences among the two approaches, see [9]. Strategies for accelerating the convergence of SCF have also been studied well, e.g., [24,25].…”

Practical error bounds for properties in plane-wave electronic structure calculations

“…The unknown is a subspace of dimension N el , the number of electrons in the system; this subspace can be conveniently described using either the orthogonal projector on it (density matrix formalism) or an orthonormal basis of it (orbital formalism). This problem is well-known in the literature and the interested reader is referred to [7] and references therein for more information on how it is solved in practice.…”