arXiv subjects
Publications and source records attributed to E. Weinan.
This article analyzes well-definedness and regularity of renormalized powers of Ornstein-Uhlenbeck processes and uses this analysis to establish local existence, uniqueness and regularity of strong solutions of stochastic Ginzburg-Landau equations with polynomial nonlinearities in two space dimensions and with quadratic nonlinearities in three space dimensions.
Topological landscape is introduced for networks with functions defined on the nodes. By extending the notion of gradient flows to the network setting, critical nodes of different indices are defined. This leads to a concise and hierarchical representation of the network. Persistent homology from computational topology is used to design efficient algorithms for performing such analysis. Applications to some examples in social and biological networks are demonstrated, which show that critical nodes carry important information about structures and dynamics of such networks.
We develop a technique for generating a set of optimized local basis functions to solve models in the Kohn-Sham density functional theory for both insulating and metallic systems. The optimized local basis functions are obtained by solving a minimization problem in an admissible set determined by a large number of primitive basis functions. Using the optimized local basis set, the electron energy and the atomic force can be calculated accurately with a small number of basis functions. The Pulay force is systematically controlled and is not required to be calculated, which makes the optimized local basis set an ideal tool for ab initio molecular dynamics and structure optimization. We also propose a preconditioned Newton-GMRES method to obtain the optimized local basis functions in practice. The optimized local basis set is able to achieve high accuracy with a small number of basis functions per atom when applied to a one dimensional model problem.
Kohn-Sham density functional theory is one of the most widely used electronic structure theories. In the pseudopotential framework, uniform discretization of the Kohn-Sham Hamiltonian generally results in a large number of basis functions per atom in order to resolve the rapid oscillations of the Kohn-Sham orbitals around the nuclei. Previous attempts to reduce the number of basis functions per atom include the usage of atomic orbitals and similar objects, but the atomic orbitals generally require fine tuning in order to reach high accuracy. We present a novel discretization scheme that adaptively and systematically builds the rapid oscillations of the Kohn-Sham orbitals around the nuclei as well as environmental effects into the basis functions. The resulting basis functions are localized in the real space, and are discontinuous in the global domain. The continuous Kohn-Sham orbitals and the electron density are evaluated from the discontinuous basis functions using the discontinuous Galerkin (DG) framework. Our method is implemented in parallel and the current implementation is able to handle systems with at least thousands of atoms. Numerical examples indicate that our method can reach very high accuracy (less than 1meV) with a very small number ($4\sim 40$) of basis functions per atom.
The least action principle is exploited as a simulation tool to find the optimal dynamic path for spatially extended systems driven by a small noise. Applications are presented for thermally activated switching of a spatially-extended bistable system as well as the switching dynamics of magnetic thin films. The issue of nucleation versus propagation is discussed and the scaling for the number of nucleation events as a function of the terminal time and other material parameters is computed.