2013年全国计算数学及其应用研讨会日程安排及报告摘要

会议报告摘要

（按报告出场顺序）

Some Aspects of Finite Element Approximation for Reissener-Mindlin Plates

石钟慈

中国科学院

The Reissner-Mindlin plate model is one of the most commonly used models of a moderatelythick to thin linearly elastic plate. However, a direct and seemingly reasonable finite element discretization usually yields very poor results, which is referred to LOCKING phenomenon.

In the past two decades, many efforts have been devoted to the design of locking free finite elements to resolve this model, most of these work focus on triangular or rectangular elements,  the latter may be extended to parallelograms, but very few on quadrilaterals.

In this talk we will give an overview of the recent development of low order quadrilateral elements and present our new results.

The Mathematical Problems of Liquid Crystals

张平文

北京大学

    Liquid crystals represent a vast and diverse class of anisotropic soft matter materials which are intermediate between isotropic liquids and crystalline solids. The mathematicians work on the theory of liquid crystals by its intriguing mathematical structure and connections to different branches of mathematics.

    A wide spectrum of mathematical problems of Liquid Crystals will be considered, related to the modeling of liquid crystals with varying detail and at different length scales. One set of problems to be considered is the relationship between these different levels of modeling; for example how one can make a rigorous passage from molecular/statistical descriptions to continuum theories. On the continuum level, we will address the dynamics of various liquid crystal phases and the systematic derivation of various equations of motion, and will aim to clarify the accuracy of different descriptions and their applicability to different liquid crystal systems. Special consideration will be given to the existence, uniqueness and regularity of the solutions of these equations, as well as the bifurcation and properties of equilibrium and periodic solutions and dynamic switching, and to how well the equations describe phase transitions.

    An important problem is to find effective approximations which can be used to overcome the challenges posed by the complexity of liquid crystal phases with biaxial and more complicated order, which are becoming important in emerging applications.

Polynomial Preserving Recovery for Gradient and Hessian Including Boundary 

张智民

北京计算科学研究中心

As an alternative to Superconvergent patch recovery (SPR, due to Zienkiewicz Zhu), Polynomial preserving recovery (PPR, due to Zhang-Naga) has been widely accepted by the scientific community, which is evidenced by its implementation into COMSOL Multiphysics, a fast developing commercial software.

In this talk, some recent development of PPR will be discussed with particular interests on the boundary strategy and Hessian recovery.

大规模计算中的若干困难和对策

陈传淼

湖南师范大学

大规模科学与工程计算中遇到若干困难：高精度，后误差估计及大规模计算。由于各种多重网格法是当前大规模计算的最佳算法，其产生的网格具有分块几乎均匀的特征，因此可提出一套完整的对策：用超收敛和外推提高精度，用外推法估计真实的误差，用新外推多网格法求解大规模方程组。我们在PC机上都可以求解上千万阶问题，具有快速和高精度的特点。

物体三维绕流的边界层方程和维数分裂方法

李开泰

西安交通大学

建立以物体表面为基础的半测地坐标系下，给出物体表面压力和流动速度梯度所满足的一组非线性偏微分方程（边界层方程），在次基础上，将外部用物体表面测地平行一系列曲面，将外部分割成m+1层，在每层交界面上，建立一组 2d-3C Navier-stokes 方程，在最外面是无限区域，用Oseen 方程来逼近，利用Oseen方程基本解，建立了在第m+1面上关于法向面应力满足的一组 2D-3C偏微分方程, 一共有m+1 组 2d-3C Navier-stokes 方程 ，她们的右边包含未知量。m+1 组的解，可构造原问题的逼近解。可构造一个二度平行算法来求解这组方程组。

A Knetic Scheme for the Baer-Nunziato Model of Compressible 

Multi-phase Flows

江  松

北京应用物理与计算数学研究所

In this talk we introduce a second-order kinetic scheme, based on the BGK-model, for the numerical solution of the Baer-Nunziato model of compressible multi-phase flows which consists of 7 equations and forms a non-conservative hyperbolic system. Also, a moving mesh strategy is used to achieve high accuracy. A number of  numerical experiments have been carried out to demonstrate accuracy and robustness of the scheme.

A Dimension Splitting Method for the 3D-PDEs

黄艾香

西安交通大学

As well known that there exist a lot of difficulties in numerical computation for the 3D PDEs, in particular for the 3D Navier-Stokes equations, such as nonlinearity, incompressible constraint condition, complex boundary geometry, and boundary layer.In order to overcome the last two difficulties, we proposed a “Dimension Splitting Method” , which is to split the three dimensional complex flow problem into a series of two dimensional subproblems, then obtain a nonlinear system with N 2D subproblem to approximate the original 3D problem.Our method is different from the classical domain decomposition method, we only solve a 2D sub-problem in each sub-domain without solving 3D sub-problem.

光学计算机在高性能计算中的应用潜力

张  武

上海大学

与“计算”密切相关的是“运算速度”和“数据位数”，增加数据位数一直是计算机专家和科学计算领域专家关注的重要问题之一。 光学计算设备具有数据宽度大这一优势，其在“计算”中的应用潜力已引起人们的关注。电子计算机数值运算器的位数逐步扩大的到目的64位；并设立了矢量运算指令和矢量寄存器，以应对大数据量计算。之所以不断增多数据位数，是因为已有计算机的强大能力使得我们意欲处理大规模计算问题，而新的更具挑战性问题的一个显著特点就是数据量非常之大，以至要用数万、数十万个电子计算核运行同样的程序来分组计算这巨大数量的数据。

上海大学三值光学计算机实验系统证明它有数千位的数据宽度，这意味着用这种计算机处理同样的数据量，程序可以少运行很多次，其效率必将明显提高。  因此研究众多数据位数带给高性能计算的新思想、新方法、新算法和新程序具有较高的理论价值和广泛的应用前景。

      本报告介绍光计算机研究的基本特点和主要优势，结合我们正在开展的高性能计算应用研究，分析光计算机作为一种可能高性能计算加速器，将在高性能计算中的应用潜力。

An Augmented Partial Newton Method for Finding Multiple Solutions

谢资清

湖南师范大学

Motivated by the idea to use a support spanned by previously found solutions in solving semilinear PDEs for multiple solutions, a new augmented singular transformation is developed to change the barrier (singularity) structure in the original problem setting for finding multiple solutions. Then an augmented partial Newton method is designed to solve the augmented problem on the solution set. Mathematical justification of the new formulation and method is established. Numerical results are presented with their profile and contour plots to illustrate the method.

Modulus-based Matrix Splitting Iteration Methods for Linear 

Complementarity Problems

白中治

中国科学院数学与系统科学研究院

In order to solve large sparse linear complementarity problems on parallel multiprocessorsystems, by making use of the modulus reformulation of the target problems and the multiple splittings of the system matrices we design the sequential  and the parallel modulus-based matrix splitting iteration methods, the modulus-based matrix splitting two-stage iteration methods and their relaxed variants. We prove the  asymptotic convergence of these matrix splitting iteration methods for the H-matrices of positive diagonal entries, and give numerical results to show the feasibility and effectiveness of the modulus-based matrix multisplitting iteration methods  when they are implemented in the parallel computational environments.

张量特征值问题的一些新进展

杨庆之

南开大学数学学院

张量特征值是矩阵特征值的推广，有几种不同的定义。在这个报告中，我将介绍非负张量H-特征值理论和算法方面的一些结果。特别是关于不可约非负张量谱半径的一些性质，求解一般非负张量谱半径的算法及收敛性结果。

Preconditioners for the Optimization Problems with Convection-Diffusion Equation Constraints

潘建瑜

华东师范大学

    Optimization problems with PDE constraints arise widely in many areas of the sciences and engineering. There are two common approaches to deal with this kind of problems: discretize-then-optimize and optimize-then-discretize. In both approaches, we need to solve a large scale system of linear equations. In this talk, we will discuss the preconditioned Krylov subspace methods for solving the linear system arising from the discretization of the optimization problems with convection-diffusion equation constraints. Several preconditioners are proposed and the numerical tests are carried out to show the performance of the preconditioners.

特殊矩阵的Hadamard积和Fan积的研究

陈果良

华东师范大学

In this  report, firstly, we established some new bounds for the spectral radius of the Hadamard product of two nonneagtive matrices. Secondly, the new lower bounds for the minimum eigenvalues of the Fan product of two M-matrices are given. Finally, we give some inequalities for the Hadamard product of an M-matrix and an inverse M-matrix.

基于压缩感知的信号恢复问题

廖安平

湖南大学

 信号的获取和处理是当今世界人们面临的一个重要问题，它主要包括信号的采集和恢复等过程。传统的采样，根据奈奎斯特-香农（Nyquist-Shannon）采样定理，采样频率不得低于信号最高频率的两倍，这就造成了信号采样、传输和存储的巨大压力。近年来，由Candes、Romberg、Tao和Donoho等人在2004年提出，2006年发表的压缩感知理论（Compressed Sensing, CS）为克服该困难提供了解决方案。本文首先对压缩感知理论的数学模型，即min||x||0, s.t. Ax=b. 进行了深入的探讨，得到了问题（P0）解的个数的一个上界，具体来说就是对于给定的A∈Rm×n, B∈R(A), 证明了                                                                      其中S为问题（P0）的解集；其次提出了感知矩阵的构造依据和构造方法；最后我们还就目前人们提出的一些恢复算法的不足和缺陷进行了探讨和分析。

Some Predictor-corrector-type Iterative Schemes for Solving Nonsymmetric Algebraic Riccati Equations Arising in Transport Theory

黄  娜

福建师范大学

    It is as well known that nonsymmetric algebraic Riccati equation arising in transport theory has the form of

                             (1)

where A,B,C,D are given real matrices with special structure. And the Riccati equation（1）can be translated to vector equations:

                              (2)

where e=(1,1,...,1)^T and P, Q are positive matrices related to the coefficient matrices A,B,C,D.

Therefore, we can solve the equation (1) by solving the vector equations (2). In view of this idea, we propose six predictor-corrector type iterative schemes to solve the vector equations (2), such as

We give the convergence of these schemes. Unlike the previous work, we prove that the proposed schemes converge to the minimal positive solution of the vector equations (2) by the initial vector (u0,v0)=(e,e). Moreover, we prove that all the sequence generated by the proposed iterative schemes are strictly and monotonically increasing, and bounded above. In addition, some numerical results are also presented in this report, which confirm the good theoretical properties of our approach.

A Generalized Preconditioned HSS Method for Singular Saddle Point Problems

晁  震

华东师范大学

 In this report, we generalize the GHSS method and present a generalized preconditioned Hermitian and skew-Hermitian splitting method (GPHSS) to solve singular saddle point problems. We prove the semi-convergence of GPHSS under some conditions, and weaken some semi-convergent conditions of GHSS, moreover, we analyze the spectral properties of the corresponding preconditioned matrix.

Nonconforming Mixed Finite Element Methods for Stationary 

Incompressible Magnetohydrodynamics

石东洋

郑州大学

In this report, we will study the approximation of nonconforming mixed finite element methods for stationary, incompressible magnetohydrodynamics (MHD) equations in 3D. A family of nonconforming finite elements are taken as the approximate spaces for the velocity field, the piecewise constant element for the pressure and the Nedelec's element with the lowest order for the magnetic field on hexahedra or tetrahedra. A new simple method is adopted to prove the discrete Poincare-Friedrichs inequality instead of the discrete Helmholtz decomposition method. The existence and uniqueness of the approximate solutions are shown. The convergence analysis is presented and the optimal order error estimates for the pressure in L2-norm, as well as those in a broken H1-norm for the velocity field and H(curl)-norm for the magnetic field are derived.

Numerical Methods for Real Options in Climate Risk Management

张书华

天津财经大学

A large numbers of industries will experience climate change related damages with the climate change processes over the coming years. Therefore, more and more people attribute the risk of extreme weather-related events to anthropogenic climate change. Making decisions when to invest in the long term flood risk related projection is complex. The complexity of the decisions mainly lies in the evolving nature of flood risk, particularly regard to the global climate change but also the future socio economic development scenarios. In this paper, we first regard the sea level and the temperature as the underlying asset and then develop a real option model to evaluate potential flood risk management opportunities. In the case of American option, we reformulate the problem to a linear parabolic variational inequality (VI) in two spatial dimensions and develop a power penalty method to solve it. We show that the nonlinear PDE is uniquely solvable and the solution of the PDE converges to that of the VI at the rate of order . A fitted finite volume method is designed to solve the nonlinear PDE in both case of European and American option, and some numerical experiments are performed to illustrate the usefulness of this method.

Fully Discrete A-  Finite Element Method for Maxwell's Equations with Nonlinear Conductivity

康 彤

中国传媒大学

This paper is devoted to the study of a fully discrete A-phi finite element scheme to solve nonlinear Maxwell's equations based on backward Euler discretization in time and nodal finite elements in space. The nonlinearity is due to a field-dependent conductivity with the power-law form , . The system under study is hyperbolic and due to the nonlinear conductivity it lacks strong estimates of the second time derivative. We design a nonlinear time-discrete scheme for approximation in suitable function spaces. We show the well-posedness of the problem, prove convergence of our semidiscrete scheme based on boundedness of the second derivative in the dual space and derive the error estimate. Convergence of the nonlinear term is based on the Minty-Browder technique. We also discuss the error estimate for the fully discretized problem and support the theoretical result by some numerical experiments.

相容与不相容矩阵不等式AXB≥C迭代解法

彭振赟

桂林电子科技大学

 给出了矩阵不等式有最小非负偏差对称解、最小非负剩余对称解和最小Frobenius范数解的充分必要条件；给出了计算最小非负偏差对称解、最小非负剩余对称解和最小Frobenius范数解的迭代方法；证明了迭代方法的收敛性。

多尺度方法及其理论的若干结果

何文明

温州大学

针对具有小周期结构的二阶椭圆问题，我们在本报告中给出了多尺度均匀化方法关于位移与应变的若干局部与全局误差估计；我们运用的一个基本工具是我们自己提出的格林函数以及导数格林函数的均匀化方法的若干误差估计。以上结果被用来给出多尺度有限元方法的理论分析。

A Coupled Parallel Solver for Three Dimensional Full-Stokes Ice Sheet Modeling

张  怀

中国科学院大学

 A three-dimensional full-Stokes computational model is considered for determining the dynamics, temperature, and thickness of ice sheets. The governing thermo-mechanical equations consist of the three-dimensional full-Stokes system with nonlinear rheology for the momentum, an advective-diffusion energy equation for temperature evolution, and a mass conservation equation for ice-thickness changes. Here, we discuss the variable resolution meshes, the finite element discretizations, and the parallel algorithms employed by the model components. The solvers are integrated through a well-designed coupler for the exchange of parametric data between components. The discretization utilizes high-quality, variable-resolution centroidal Voronoi Delaunay triangulation meshing and existing parallel solvers. We demonstrate the gridding technology, discretization schemes, and the efficiency and scalability of the parallel solvers through computational experiments using both simplified geometries arising from benchmark test problems and a realistic Greenland ice sheet geometry.