A posteriori analysis for nonlinear eigenvalue problems, application toelectronic structure calculations

Abstract

Many mathematical models aiming at the determination of electronic structures give rise tononlinear eigenvalue problems whose resolutions require very large computational resources [1]. Thecomplexity of these computations reflects, among others, the chosen discretization and the chosen(possibly iterative) algorithm. The a posteriori analysis of such problems enables to reduce thecomputations involved to solve the problem by first giving a guaranteed upper bound on the totalerror and second by separating the error components stemming from the different sources andcontrolling each of them. This makes possible to iteratively fit these discretization parameters leadingto small error at low computational cost [3].We shall first present a full a posteriori analysis for a simple but representative one-dimensional Gross-Pitaevskii type equation, in a periodic setting with planewave (Fourier)approximation. The nonlinear discretized problem is solved with a Self-Consistent Field (SCF)algorithm, which consists in solving a linear eigenvalue problem at each step. To start with, weprovide a computable upper bound of the energy error. We then separate this bound into twocomponents, one of them depending mainly on the dimension of the discretized space, the other oneon the number of iterations done in the SCF algorithm. This enables to adaptatively choose betweenrefining the discretization and performing SCF iterations, as we try to balance the error components.We also illustrate numerically the coherent performances of this a posteriori analysis. We thenpostprocess the approximate solution (eigenvalue and eigenfunction) using a linear perturbationtheory based on residual computation [2]. This theoretically and numerically reduces the errorsignificantly both for the eigenvalue and the eigenfunction.This work is a first step towards an a posteriori analysis and postprocess of more complexelectronic structure models like Hartree-Fock or Kohn-Sham models. We shall present the firstresults for the extension of our results in this framework.