Zhen Chao

Zhen Chao, Guoliang Chen

Recently, Bai and Benzi proposed a class of regularized Hermitian and skew-Hermitian splitting methods (RHSS) iteration methods for solving the nonsingular saddle point problem. In this paper, we apply this method to solve the singular saddle point problem from the Stokes equations. In the process of the semi-convergence analysis, we get that the RHSS method and the HSS method are unconditionally semi-convergent, which weaken the previous results. Then some spectral properties of the corresponding preconditioned matrix and a class of improved preconditioned matrix are analyzed. Finally, some numerical experiments on linear systems arising from the discretization of the Stokes equations are presented to illustrate the feasibility and effectiveness of this method and preconditioners.

Hwi Lee, Zhen Chao, Harris Cobb et al.

In this paper, a deep learning method for solving an improved one-dimensional Poisson-Nernst-Planck ion channel (PNPic) model, called the PNPic deep learning solver, is presented. In particular, it combines a novel local neural network scheme with an effective PNPic finite element solver. Since the input data of the neural network scheme only involves a small local patch of coarse grid solutions, which the finite element solver can quickly produce, the PNPic deep learning solver can be trained much faster than any corresponding conventional global neural network solvers. After properly trained, it can output a predicted PNPic solution in a much higher degree of accuracy than the low cost coarse grid solutions and can reflect different perturbation cases on the parameters, ion channel subregions, and interface and boundary values, etc. Consequently, the PNPic deep learning solver can generate a numerical solution with high accuracy for a family of PNPic models. As an initial study, two types of numerical tests were done by perturbing one and two parameters of the PNPic model, respectively, as well as the tests done by using a few perturbed interface positions of the model as training samples. These tests demonstrate that the PNPic deep learning solver can generate highly accurate PNPic numerical solutions.