Xiaoying Dai

Xiaoying Dai, Yvon Maday

The parareal in time algorithm allows to perform parallel simulations of time dependent problems. This algorithm has been implemented on many types of time dependent problems with some success. Recent contributions have allowed to extend the domain of application of the parareal in time algorithm so as to handle long time simulations of Hamiltonian systems. This improvement has managed to avoid the fatal large lack of accuracy of the plain parareal in time algorithm consequence of the fact that the plain parareal in time algorithm does not conserve invariants. A somehow similar difficulty occurs for problems where the solution lacks regularity, either initially or in the evolution, like for the solution to hyperbolic system of conservation laws. In this paper we identify the problem of lack of stability of the parareal in time algorithm and propose a simple way to cure it. The new method is used to solve a linear wave equation and a non linear Burger's equation, the results illustrate the stability of this variant of the parareal in time algorithm.

Huajie Chen, Xiaoying Dai, Xingao Gong et al.

The Kohn-Sham equation is a powerful, widely used approach for computation of ground state electronic energies and densities in chemistry, materials science, biology, and nanosciences. In this paper, we study the adaptive finite element approximations for the Kohn-Sham model. Based on the residual type a posteriori error estimators proposed in this paper, we introduce an adaptive finite element algorithm with a quite general marking strategy and prove the convergence of the adaptive finite element approximations. Using D{\" o}rfler's marking strategy, we then get the convergence rate and quasi-optimal complexity. We also carry out several typical numerical experiments that not only support our theory,but also show the robustness and efficiency of the adaptive finite element computations in electronic structure calculations.

Xiaoying Dai, Lianhua He, Aihui Zhou

In this paper, we study an adaptive finite element method for multiple eigenvalue problems of a class of second order elliptic equations. By using some eigenspace approximation technology and its crucial property which is also presented in this paper, we extend the results in \cite{dai-xu-zhou08} to multiple eigenvalue problems, we obtain both convergence rate and quasi-optimal complexity of the adaptive finite element eigenvalue approximation.

Yan Pan, Xiaoying Dai, Stefano de Gironcoli et al.

Motivated by the recently proposed parallel orbital-updating approach in real space method, we propose a parallel orbital-updating based plane-wave basis method for electronic structure calculations, for solving the corresponding eigenvalue problems. In addition, we propose two new modified parallel orbital-updating methods. Compared to the traditional plane-wave methods, our methods allow for two-level parallelization, which is particularly interesting for large scale parallelization. Numerical experiments show that these new methods are more reliable and efficient for large scale calculations on modern supercomputers

Xiaoying Dai, Zhuang Liu, Xin Zhang et al.

In this paper, we propose a parallel optimization method for electronic structure calculations based on a single orbital-updating approximation. It is shown by our numerical experiments that the method is efficient and reliable for atomic and molecular systems of large scale over supercomputers.

Xiaoying Dai, Zhuang Liu, Liwei Zhang et al.

In this paper, we study a conjugate gradient method for electronic structure calculations. We propose a Hessian based step size strategy, which together with three orthogonality approaches yields three algorithms for computing the ground state energy of atomic and molecular systems. Under some mild assumptions, we prove that our algorithms converge locally. It is shown by our numerical experiments that the conjugate gradient method is efficient.

Xiaoying Dai, Xingao Gong, Aihui Zhou et al.

In this paper, we propose an orbital iteration based parallel approach for electronic structure calculations. This approach is based on our understanding of the single-particle equations of independent particles that move in an effective potential. With this new approach, the solution of the single-particle equation is reduced to some solutions of independent linear algebraic systems and a small scale algebraic problem. It is demonstrated by our numerical experiments that this new approach is quite efficient for full-potential calculations for a class of molecular systems.