Reduced basis method for quantum models of crystalline solids

Abstract

Electronic structure problems in solids usually involve repetitive determination of quantities of interest, evaluation of which requires the solution of an underlying partial differential equation. We present in this thesis the application of the reduced basis method in accurate and rapid evaluations of outputs associated with some nonlinear eigenvalue problems related to electronic structure calculations. The reduced basis method provides a systematic procedure by which efficient basis sets and computational strategies can be constructed. The essential ingredients are (i) rapidly convergent global reduced basis approximation spaces; (ii) an offline-online computational procedure to decouple the generation and projection stages of the approximation process; and (iii) inexpensive a posteriori error estimation procedure for outputs of interest. We first propose two strategies by which we can construct efficient reduced basis approximations for vectorial eigensolutions - solutions consisting of several eigenvectors. The first strategy exploits the optimality of the Galerkin procedure to find a solution in the span of all eigenvectors at N judiciously chosen samples in the parameter space.

(cont.) The second strategy determines a solution in the span of N vectorial basis functions that are pre-processed to better represent the smoothness of the solution manifold induced by the parametric dependence of the solutions. We deduce from numerical results conditions in which these approximations are rapidly convergent. For linear eigenvalue problems, we construct a posteriori asymptotic error estimators for our reduced basis approximations - extensions on existing work in algebraic eigenvalue problems. We further construct efficient error estimation procedures that allow efficient construction of reduced basis spaces based on the "greedy" sampling procedure. We extend our methods to nonlinear eigenvalue problems, utilizing the empirical interpolation method. We also provide a more efficient construction procedure for the empirical interpolation method. Finally, we apply our methods to two problems in electronic structure calculations - band structure calculations and electronic ground state calculations. Band structure calculations involve approximations of linear eigenvalue problems; we demonstrate the applicability of our methods in the many query limit with several examples related to determination of spectral properties of crystalline solids.

(cont.) Electronic ground state energy calculations based on Density Functional Theory involve approximations of nonlinear eigenvalue problems; we demonstrate the potential of our methods within the context of geometry optimization.