Qiya Hu

Qiya Hu, Rongrong Song

In this paper we are concerned with the plane wave method for the discretization of time-harmonic Maxwell's equations in three dimensions. As pointed out in [6], it is difficult to derive a satisfactory L2 error estimate of the standard plane wave approximation of the time-harmonic Maxwell's equations. We propose a variant of the plane wave least squares (PWLS) method and show that the new plane wave approximations possess the desired L2 error estimate. Moreover, the numerical results indicate that the new approximations have sightly smaller L2 errors than the standard plane wave approximations. More importantly, the results are derived for more general models in layered media.

Qiya Hu, Shaoliang Hu

In this paper we propose a variant of the substructuring preconditioner for solving three-dimensional elliptic-type equations with strongly discontinuous coefficients. In the proposed preconditioner, we use the simplest coarse solver associated with the finite element space induced by the coarse partition, and construct vertex-related inexact interface solvers based on overlapping domain decomposition with small overlaps. This new preconditioner has an important merit: its construction and efficiency do not depend on the concrete form of the considered elliptic-type equations. % in the sense that they not only are cheap but also are easy to implement. We apply the proposed preconditioner to solve the linear elasticity problems and Maxwell's equations in three dimensions. Numerical results show that the convergence rate of PCG method with the preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficients in the considered equations.

Qiya Hu, Xuan Li

In this paper we are concerned with fast algorithms for the systems arising from the plane wave discretizations for two-dimensional Helmholtz equations with large wave numbers. We consider the plane wave weighted least squares (PWLS) method and the plane wave discontinuous Galerkin (PWDG) method. The main goal of this paper is to construct multilevel parallel preconditioners for solving the resulting Helmholtz systems. To this end, we first build a multilevel overlapping space decomposition for the plane wave discretization space based on a multilevel overlapping domain decomposition method. Then, corresponding to the space decomposition, we construct an additive multilevel preconditioner for the underlying Helmholtz systems. Further, we design both additive and multiplicative multilevel preconditioners with smoothers, which are different from the standard multigrid preconditioners. We apply the proposed multilevel preconditioners with a {\it constant} coarsest mesh size to solve two dimensional Helmholtz systems generated by PWLS method or PWDG method, and we find that the new preconditioners possess nearly stable convergence, i.e., the iteration counts of the preconditioned iterative methods (PCG or PGMRES) with the preconditioners increase very slowly when the wave number increases (and the fine mesh size decreases).

Qiya Hu, Shaoliang Hu

In this paper we propose two variants of the substructuring preconditioner for solving three-dimensional elliptic-type equations with strongly discontinuous coefficients. In the new preconditioners, we use the simplest coarse solver associated with the finite element space induced by the coarse partition, and construct novel interface solvers based on some new observations. The resulting preconditioners share the merits of the non-overlapping domain decomposition method (DDM) and the overlapping DDM in the sense that they not only are cheap but also are easy to implement. We apply the proposed preconditioners to solve the linear elasticity problems and Maxwell's equations in three dimensions. Numerical results show that the convergence rate of PCG method with the preconditioners are nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficients in the considered equations.

Qiya Hu, Yuhan Luo

It is known that the weighted $L^2$ projection operator exhibits approximation properties different from those of the classical $L^2$ projection, in the sense that the $L^2$ error of the weighted $L^2$ projection of an $H^1$ function generally cannot be bounded by the $H^1$ semi-norm of the function. In this paper, we establish sharper $L^2$ error estimates for the weighted $L^2$ projection of an $H^1$ function under general weight distributions. These new estimates show that the $L^2$ errors of the weighted $L^2$ projection can be controlled by the $H^1$ semi-norm of the function, except when the weight distribution is highly irregular, such as those resembling a ``checkerboard" pattern. These results can be applied to more refined analyses of domain decomposition methods and multigrid methods for certain partial differential equations with large jump coefficients.

Liqi Wang, Qiya Hu

Dropout as a regularization technique is widely used in fully connected layers while is less effective in convolutional layers. Therefore more structured forms of dropout have been proposed to regularize convolutional networks. The disadvantage of these methods is that the randomness introduced causes inconsistency between training and inference. In this paper, we apply a mutual learning training strategy for convolutional layer regularization, namely R-Block, which forces two outputs of the generated difference maximizing sub models to be consistent with each other. Concretely, R-Block minimizes the losses between the output distributions of two sub models with different drop regions for each sample in the training dataset. We design two approaches to construct such sub models. Our experiments demonstrate that R-Block achieves better performance than other existing structured dropout variants. We also demonstrate that our approaches to construct sub models outperforms others.

Bin Zheng, Qiya Hu, Jinchao Xu

In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem involving both curl^2 and curl^4 operators. The element has a very small number of degrees of freedom and it imposes the inter-element continuity along the tangential direction which is appropriate for the approximation of magnetic fields. We also provide explicit formulae of basis functions for this element.