### 2015年西安交通大学—厦门大学数学联合学术报告会

 时间：2015年4月24日（星期五）；地点：海韵园实验楼108报告厅 上        午 9:00-9:30 开幕式，合影 时  间 报告人 单  位 报告题目 主持人 9:30-10:00 侯延仁 西安交通大学 Convergence   Analysis of An Expandable Local and Parallel Two-Grid Finite Element Scheme 许传炬 10:00-10:30 陈黄鑫 厦门大学 Absolutely stable hp-HDG methods for   the indefinite Helmholtz and Maxwell equations with high wave number 10:30-11:00 晏文璟 西安交通大学 Shape design in   the incompressible viscous fluids 11:00-11:30 孙  剑 西安交通大学 基于视觉先验建模的图像处理算法研究 中午 12:00-13:30 午   餐 下     午 13:30-14:00 李  安 厦门大学 Necessary optimality conditions for   optimal control problems with a geometric constraint 陈志平 14:00-14:30 徐凤敏 西安交通大学 Efficient   Projected Gradient Methods for A Class of $L_0$ Constrained Optimization 14:30-15:00 陈竑焘 厦门大学 A Multigrid Method Based On   Shifted-Inverse Power Technique for Eigenvalue  Problems 18:30 晚   餐

Convergence Analysis of An Expandable Local and Parallel Two-Grid Finite Element Scheme

Yanren Hou（侯延仁）

Xi’an Jiaotong University

In this talk, we will do some convergence analysis of an expandable local and parallel two-grid finite element scheme for elliptic problems by taking example of Poisson equation. Compared with the usual local and parallel finite element schemes, the scheme discussed here can be easily implemented in a large parallel computer system with vast of CPUs. Convergence results base on $H^1$ and $L^2$ a priori error estimation of the scheme are obtained, which show that the scheme can reach the optimal convergence orders within $|\ln H|^2$ or $|\ln H|$ two-grid iterations if the coarse mesh size $H$ and the fine mesh size $h$ are properly configured in 2-D or 3-D case, respectively. Some numerical results are provided to support the analysis results.

Absolutely stable hp-HDG methods for the indefinite Helmholtz and Maxwell equations with high wave number

Huangxin Chen（陈黄鑫）

Xiamen University

In this talk we will introduce hybridizable discontinuous Galerkin (HDG) methods for the indefinite Helmholtz and time-harmonic Maxwell equations with high wave number. For the Helmholtz equation, the HDG method is directly applied to the first-order system of the equation. For the time-harmonic Maxwell equation, the divergence-free condition is enforced on the electric field, then a Lagrange multiplier is introduced, and the problem becomes in solving a mixed curl-curl formulation of the Maxwell’s problem. The methods are shown to be absolutely stable HDG methods for the indefinite Helmholtz and Maxwell equations with high wave number. By exploiting the duality argument, the dependences of convergence of the HDG methods on wave number κ, mesh size h and polynomial order p are obtained.

Wenjing Yan（晏文璟）

Xi’an Jiaotong University

This talk is concerned with the shape design in the incompressible viscous fluids. Generally, there are two kinds of shape design in fluids. In the first part, the shape reconstruction of a boundedand smooth domain from observed information in the Navier-Stokesflow is considered. The continuous dependence of the solution with respect to theboundary is established, and the domain derivative of corresponding equations is derived. By the boundary parametrization method, a regularized Gauss-Newton scheme is employed to the shape inverse problem.In the second part, the shape optimization problem of a body immersedin the incompressible fluid governed by Navier-Stokes equations is addressed. Then the structure of shape gradient for the cost functionalis derived by the differentiability of a saddle point formulation involving a Lagrangian functional with the function space parametrizationtechnique. Moreover, a gradient-type algorithm is applied to solve theshape optimization problem. The numerical examples indicate that theproposed algorithm is feasible and effective.

Jian Sun（孙剑）

Xi’an Jiaotong University

A Multigrid Method Based On Shifted-Inverse Power Technique for Eigenvalue  Problems

Hongtao Chen（陈竑焘）

Xiamen University

A multigrid method is proposed in this talk to solve eigenvalue problems by the finite element method based on the shifted-inverse power iteration technique. With this scheme, solving eigenvalue problem is transformed to a series of nonsingular solutions of boundary value problems on multilevel meshes. Since replacing the difficult eigenvalue solving by the easier solution of boundary value problems,the multigrid way can improve the overall efficiency of the eigenvalue problem solving.Some numerical experiments are presented to validate the efficiency of this new method.

Efficient Projected Gradient Methods for A Class of $L_0$ Constrained Optimization

Fengmin Xu（徐凤敏）

Xi’an Jiaotong University

Sparse optimization has attracted increasing attentions in numerous areas such as compressed sensing, finance optimization and image processing. In this paper, we consider a special class of $L_0$ constrained optimization, which involves box constraints and a singly linear constraint. An efficient approach is provided for calculating the projection over the feasibility set after a careful analysis on the projection subproblem. Then we present several types of projected gradient methods for the special class of $L_0$ constrained optimization. Global convergence of the methods are established under suitable assumptions. The computational results on signal recovery and enhanced indexation demonstrate that the proposed projected gradient methods are efficient in terms of both solution quality and speed.This is a joint work with Yu-Hong Dai, Zhihua Zhao and ZongbenXu.

Necessary optimality conditions for optimal control problems with a geometric constraint

An Li（李安）

Xiamen University

We study an optimal control problem in which the state and control are subject to a geometric constraint. The geometric constraint is very general and in particular it subsumes the  mixed equality and inequality constraints. In most of the existing results, the necessary optimality condition for optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz Condition (MFC). In this paper we derive necessary optimality conditions under weaker constraint qualifications based on pseudo-Lipschitz continuity and calmness of certain set-valued maps. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold.