Dimensional Reductions for the Computation of Time-Dependent Quantum Expectations
SIAM Journal on Scientific Computing
Randomized Algorithms for Matrices and Data
Foundations and Trends® in Machine Learning
Quantum algorithms for predicting the properties of complex materials
Proceedings of the 1st Conference of the Extreme Science and Engineering Discovery Environment: Bridging from the eXtreme to the campus and beyond
Divide and Conquer on Hybrid GPU-Accelerated Multicore Systems
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Numerical analysis of finite dimensional approximations of Kohn---Sham models
Advances in Computational Mathematics
Convergence analysis of Lanczos-type methods for the linear response eigenvalue problem
Journal of Computational and Applied Mathematics
Journal of Computational Physics
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The goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems. The paper is intended for a diverse scientific computing audience. For this reason, we assume the reader does not have an extensive background in the related physics. Our overview focuses on the nature of the numerical problems to be solved, their origin, and the methods used to solve the resulting linear algebra or nonlinear optimization problems. It is common knowledge that the behavior of matter at the nanoscale is, in principle, entirely determined by the Schrödinger equation. In practice, this equation in its original form is not tractable. Successful but approximate versions of this equation, which allow one to study nontrivial systems, took about five or six decades to develop. In particular, the last two decades saw a flurry of activity in developing effective software. One of the main practical variants of the Schrödinger equation is based on what is referred to as density functional theory (DFT). The combination of DFT with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials. This article will emphasize pseudopotential-density functional theory, but other techniques will be discussed as well.