Abstract : With the rapid development of the research in the electronic structure calculations, new mechanisms and phenomenon are constantly being discovered, which bring the challenging on mathematical modeling and related analysis. Meanwhile, the development of the computational hardware also provides chances to the new algorithm design and analysis. In this mini-symposium, experts from mathematics, physics, etc., would share their recent progress in the research of the electronic structure calculations, and will discuss together towards the potential applications in areas such as computational quantum chemistry, nano-optics.
[04686] Numerical method for the Elasticity Transmission Eigenvalues
Format : Talk at Waseda University
Author(s) :
xia ji (Beijing institute of technology )
Abstract : We develop a discontinuous Galerkin method computing a few smallest elasticity transmission eigenvalues, which are of practical importance in inverse scattering theory. For high order problems, DG methods are competitive since they use simple basis functions, the numerical implementation is much easier compared with classical conforming finite element methods. In this talk, we propose an interior penalty discontinuous Galerkin method using C0 Lagrange elements (C0IP) for the transmission eigenvalue problem for elastic waves and prove the optimal convergence. The method is applied to several examples and its effectiveness is validated.
[03287] A multi-mesh adaptive finite element method for Kohn--Sham equation
Format : Talk at Waseda University
Author(s) :
Yang Kuang (Guangdong University of Technology)
Abstract : We present a multi-mesh adaptive finite element method for solving the Kohn–Sham (KS) equation. Specifically, the KS equation and the Poisson equation corresponding to the Hartree potential is solved in two different adaptive meshes on the same computational domain. With the presented method, we are able to evaluate the Hartree potential and Hartree energy more accurately, so as to reach a better accuracy with less computational cost in the all-electron calculations.
[03432] Mathematical theory and numerical methods for Bose-Einstein condensation with higher order interactions
Format : Talk at Waseda University
Author(s) :
Xinran Ruan (Capital Normal University)
Weizhu Bao (National University of Singapore)
Yongyong Cai (Beijing Normal University)
Abstract : The binary interaction in Gross-Pitaevskii equation, which is a big success in describing Bose-Einstein condensate, is typically chosen as Fermi contact interaction. However, in certain cases, higher order interaction (HOI) needs to be included.
In the talk, I will show new phenomenon introduced by HOI, such as the non-Gaussian type approximations in dimension reduction problems，new types of Thomas-Fermi approximations. Besides, two algorithms for computing ground states, which overcome the stability issue, will be presented.
[03403] Solving Schrodinger equation using tensor neural network
Format : Talk at Waseda University
Author(s) :
Hehu Xie (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Abstract : In this talk, we introduce a prototype of machine learning method to solve high dimensional partial differential equations by the tensor neural network. Based on the tensor product structure, we can do the direct numerical integration by using fixed quadrature points for the functions constructed by the tensor neural network within tolerable computational complexity. The corresponding machine learning method is built for solving high dimensional Schrodinger equation with high accuracy. Some numerical examples are provided to validate the accuracy and efficiency of the proposed algorithms. This work is collaborated with Yifan Wang and Pengzhan Jin.
[04589] Computations of the ground states and collective excitations of Bose-Einstein condensates
Author(s) :
Manting Xie (Tianjin University)
Hehu Xie (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Yong Zhang (Tianjin University)
Abstract : In this talk, we'll show some efficient and robust numerical methods to study the ground states and collective excitations of Bose-Einstein condensates.
[03384] Multilevel correction adaptive finite element method for Kohn-Sham equation
Format : Talk at Waseda University
Author(s) :
Fei Xu (Beijing University of Technology)
Abstract : An efficient adaptive finite element method is proposed for solving the ground state solution of the Kohn-Sham equation which is based on the combination of the multilevel correction scheme and the adaptive refinement technique. The multilevel correction adaptive finite element method can transform the Kohn-Sham equation into a series of linear boundary value problems on the adaptive partitions and a series of Kohn-Sham equations on very low dimensional spaces. Further, the Hartree potential and the exchange-correlation potential are treated individually, and the algorithm can be performed in an eigenpairwise approach. Therefore, the presented adaptive method for Kohn-Sham equation can arrive at the similar efficiency as that of the adaptive finite element method for the associated linear boundary value problems.
[03227] The Wigner function of ground state and one-dimensional numerics
Format : Talk at Waseda University
Author(s) :
Hongfei Zhan (Peking University)
Abstract : In this talk, the ground state Wigner function of a many-body system is explored theoretically and numerically. An eigenvalue problem for Wigner function is derived based on the energy operator of the system. On the other hand, a numerical method is designed for solving proposed eigenvalue problem in one dimensional case. Results of several numerical experiments verify our method, where the potential application for large scale system is demonstrated by examples with density functional theory.
