Sparse Representation of Wannier Functions from $L_1$ Regularized Optimization

Wannier functions are real-space localized wavefunctions for electronic systems. The construction of Wannier functions from Schr\"{o}dinger equation has been extensively studied in the community of condensed matter physics. The most widely used scheme is to minimize spread functional, i.e., second moment, with respect to the gauge matrices, resulting in the maximally localized Wannier functions (MLWFs) \cite{PhysRevB.56.12847,PhysRevB.65.035109,RevModPhys.84.1419}.The recent advancement in machine learning, image processing and compressive sensing has brought tremendous computational efficiency to study complex many-particle systems, in terms of solving non-convex and non-differentiable optimization problems. Inspired by the variable selection property of Least Absolute Shrinkage and Selection Operator (LASSO) \cite{10.2307/2346178,hastie01statisticallearning} in machine learning, we calculate Wannier functions by minimizing total energy plus an $L_1$ regularization term. The resulting "compressed" Wannier functions have compact support, which means they are localized only within a finite region and are strictly zero outside.Another question considered in this dissertation is how to calculate symmetry-adapted Wannier functions. We show that induced group representation theory fits nicely into our variational formula. Same algorithms work equally well for both metals and insulators, in a sense that no prior disentanglement procedure is needed for our calculations. When symmetry constraints are incorporated, it suffices to use Gaussians at Wannier centers as initial trial functions for all the orbitals. Such algorithms have been implemented in planewaves-pseudopotential codes for real materials simulations