国家天元数学西北中心“数值代数与科学计算国际会议”定于2019年6月8日至6月9日在西安召开。本次会议旨在为从事计算数学、数值代数及相关研究的专家学者搭建学科发展和合作的交流平台，借助学术报告等形式，探讨计算数学理论、方法及应用方面的发展、以及与其他工程和科学领域的交叉、融合与创新，推进数学学科发展建设。
学术委员会：
主席：江松（北京应用物理与计算数学研究所）
委员（按姓氏拼音顺序）：
白中治（中国科学院）
韩渭敏（爱荷华大学）
何晓明（密苏里科技大学）
何银年（西安交通大学）
侯延仁（西安交通大学）
黄艾香（西安交通大学）
李开泰（西安交通大学）
林 涛（弗吉尼亚理工大学）
梅立泉（西安交通大学）
聂玉峰（西北工业大学）
温瑞萍（太原师范学院）
伍渝江（兰州大学）
会议日程表
Session 1
地点：西北大学宾馆1100会议室
日期 |
主持人 |
时间 |
报告人 |
题目 |
6 月 8 日 |
李长宏 张 瑞 |
8:30-9:00 |
开幕式 |
韩渭敏
|
9:00-9:40 |
黄艾香 |
A Dimension Splitting Method for the 3D-PDEs |
9:40-10:20
|
李开泰 |
血管里的血液流动和血管壁的弹性壳体的耦合系统的球单元有限元逼近 |
|
10:20-10:40 |
茶 歇 |
聂玉峰 |
10:40-11:20 |
白中治
|
Banded M-splitting Iteration Methods for Spatial Fractional Diffusion Equations |
11:20-12:00 |
韩渭敏 |
Numerical Analysis of Hemivariational Inequalities Arising in Mechanics |
午 餐 |
王冬岭 |
14:30-15:10 |
林 涛
|
Immersed finite Element Methods: Development, Analysis, and Applications |
15:10-15:50
|
何银年 |
Decoupled Finite Element Methods for the 3D Primitive Equations of Ocean
|
|
15:50-16:10 |
茶 歇 |
郭英文 |
16:10-16:50
|
侯延仁 |
Time Filter for the Unsteady Coupled Stokes/Darcy Model |
16:50-17:30 |
梅立泉 |
等离子体物理中非线性色散方程的有限元数值解法
|
17:30-18:10
|
何晓明 |
A non-iterative multi-physics domain decomposition method for coupled free flow and porous media flow problem |
晚 餐
|
Session 2
地点：西安交通大学数学楼2-1会议室
日期 |
主持人 |
时间 |
报告人 |
题目 |
6 月 8 日 |
温瑞萍 |
15:00-15:40 |
伍渝江
|
Minimum residual HSS iteration method for non-Hermitian positive definite complex linear systems
|
15:40-16:20
|
黄玉梅 |
Weighted Nuclear Norm Minimization Based Regularization Method for Image Restoration
|
16:20-17:00 |
吴钢
|
Randomized GLRAM-type algorithms for high dimensionality reduction and image reconstruction
|
17:00-17:30 |
|
自由讨论
|
晚 餐 |
Session 3
地点：西北大学非线性科学研究中心学术报告厅
日期 |
主持人 |
时间 |
报告人 |
题目 |
6 月 9 日 |
刘海峰
|
8:30-9:10 |
温瑞萍
|
Toeplitz矩阵重建的几种算法
|
9:10-9:50
|
汪 祥
|
Numerical methods for solving linear complementarity problems
|
|
9:50-10:10 |
茶 歇 |
侯江勇
|
10:10-10:50 |
张国栋
|
Analysis of symmetric schemes and robust preconditioners for incompressible MHD system
|
10:50-11:30 |
王淑琴
|
Numerical analysis of the stabilized characteristic mixed IPDG method for the incompressible Navier-Stokes equations
|
11:30-12:10
|
李 瑞
|
自由流和裂缝多孔介质流区域耦合问题建模及其间断Galerkin有限元方法
|
午 餐 |
冉育红 |
14:30-17:30
|
自由讨论
|
晚 餐 |
报告摘要
Banded M-splitting Iteration Methods for Spatial Fractional Diffusion Equations
白中治
（中国科学院）
For solving time-dependent one-dimensional spatial-fractional diffusion equations of variable coefficients, we establish a banded M-splitting iteration method applicable to compute approximate solutions for the corresponding discrete linear systems resulting from certain finite difference schemes at every temporal level, and demonstrate its asymptotic convergence without imposing any extra condition. Also, we provide a multistep variant for the banded M-splitting iteration method, and prove that the computed solutions of the discrete linear systems by employing this iteration method converge to the exact solutions of the spatial fractional diffusion equations. Numerical experiments show the accuracy and efficiency of the multistep banded M-splitting iteration method.
Numerical Analysis of Hemivariational Inequalities Arising in Mechanics
韩渭敏
（爱荷华大学）
Inequality problems in mechanics can be divided into two main categories: that of variational inequalities which is concerned with nonsmooth and convex functionals (potentials), and that of hemivariational inequalities which is concerned with nonsmooth and nonconvex functionals (superpotentials). While variational inequalities have been studied extensively, the study of hemivariational inequalities is more recent. Through the formulation of hemivariational inequalities, problems involving nonmonotone, nonsmooth and multivalued constitutive laws, forces, and boundary conditions can be treated successfully. In the recent years, substantial progress has been made on numerical analysis of hemivariational inequalities. In this talk, a summarizing account will be given on recent and new results on the numerical solution of hemivariational inequalities with applications in contact mechanics.
A non-iterative multi-physics domain decomposition method for coupled free flow and porous media flow problem
何晓明
（密苏里科技大学）
The Stokes-Darcy and Navier-Stokes-Darcy model have attracted significant attention in the past ten years since they arise in many applications involving with coupled free flow and porous media flow such as surface water flows, groundwater flows in karst aquifers, petroleum extraction and industrial filtration. They have higher fidelity than either the Darcy or Navier-Stokes systems on their own, but coupling the two constituent models leads to a very complex system. This presentation discusses a series of works for the non-iterative multi-physics domain decomposition method to solve this type of problems, including both the algorithm development and analysis. The key ideas are to (1) decouple the free flow and porous media flow through Robin type boundary conditions which directly arise from a direct re-organization of the three interface conditions; (2) use the information from the previous time steps to directly predict the interface information for the current step without an iteration for domain decomposition. Related Ritz projections are analyzed. Optimal convergence is proved for the finite element solution with the k-step back backward differentiation scheme in temporal discretization (k less than or equal to 5). Numerical results are presented to illustrate the features of the proposed method.
Decoupled Finite Element Methods for the 3D Primitive Equations of Ocean
何银年
（西安交通大学）
In this paper, two decoupled finite element methods are proposed for solving the 3D primitive equations of ocean. Based on the finite element approximation, optimal error estimates are given under the convergence condition. And the detailed algorithms are given in the section of numerical tests. Further, numerical calculations are implemented to validate the theoretical analysis and more calculations are implemented for a more meaningful problem. For both theoretical and numerical points of view, the proposed decoupled finite element methods are the effective strategies to solve the 3D primitive equations of ocean.
Time Filter for the Unsteady Coupled Stokes/Darcy Model
侯延仁
（西安交通大学）
In this talk, we apply the time filter technique for increasing the accuracy of the fully discrete backward Euler scheme for the unsteady Stokes/Darcy model. Roughly speaking, by adding a single line of code to the backward Euler scheme, we can improve the first order scheme into a second order scheme. The stability and the error estimations for the proposed scheme are obtained. The analysis results show that the scheme is unconditional stable and second order accurate in time in L^2 norm case while it is still a first order scheme in H^1 norm sense. Finally, a simple numerical experiment is carried out to verify the analysis results.
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, such as nonlinearity; incompressible constraint condition; complex boundary geometry; 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 domian decompostion method, we only solve a 2D sub-problem in each sub-domian without solving 3D sub-problem.
Weighted Nuclear Norm Minimization Based Regularization Method for Image Restoration
黄玉梅
（兰州大学）
Regularization methods have been substantially applied in image restoration due to the ill-posedness of the image restoration problem. Different assumptions or priors on images are applied in the construction of image regularization methods. In recent years, low-rank matrix approximation has been successfully introduced in image denoising and significant denoising effects have been achieved. The computation of low-rank matrix minimization is a NP hard problem and it is often replaced with the matrix's weighted nuclear norm minimization. Nonlocal image denoising methods assume that an image contains an extensive amount of self-similarity. Based on such assumption, in this talk, we develop a model for image restoration by using weighted nuclear norm to be the regularization term. An alternating iterative algorithm is designed to solve the proposed model and we also present the convergence analyses of the algorithm. Numerical experiments show that the proposed method can recover the images much better than the existing regularization methods in terms of both recovered quantities and visual qualities.
血管里的血液流动和血管壁的弹性壳体的耦合系统的球单元有限元逼近
李开泰
（西安交通大学）
血管里的血液流动和血管壁的弹性壳体的耦合系统的球单元有限元逼近，血管壁视为三维弹性壳体，服从三维弹性方程，血液流动视为不可压缩的牛顿流体，服从Navier_stokes方程，连接边界服从速度和法向应力连续条件。应用有限元方法求解，有限元单元为三维球单元。血管壁作为弹性壳体，在厚度方向只用一个球单元，在血液流内，元素的几何都是三维球，但几何尺寸
是不均匀的。尚未进行数值分析。
自由流和裂缝多孔介质流区域耦合问题建模及其间断Galerkin有限元方法
李瑞
（陕西师范大学）
自由流和多孔介质流耦合问题在工程中有着重要的应用，例如地表水和地下水的交互、地下水在喀斯特岩溶含水层的流动、血液在血管和器官之间的流动、工业过滤、污染物的运移、石油开采、地热能的研究、二氧化碳的封存、海绵城市的建设等。而在实际多孔介质区域, 地质结构复杂多变, 基质中通常含有裂缝。裂缝的几何区域狭长, 分布具有随机性, 大小跨越多个数量级, 且填充严重, 既可作为高导流通道, 也可成为流动屏障. 自由流和裂缝多孔介质区域耦合问题流体流动通道尺度差异大，多种流动形式共存，不仅存在渗流，还存在自由流，以及三个区域之间的流体交换，同时，裂缝的存在加剧了孔隙介质的非均匀性，各向异性和不连续性，从而导致基质中流体的速度和压力不连续。本文针对不可压缩单相流体在自由流区域和裂缝多孔介质区域中的耦合流动问题，结合数学建模、理论分析和数值模拟展开研究。我们拟利用区域耦合的建模框架结合数学上的理论推导，建立描述自由流和裂缝多孔介质流区域耦合问题的可计算数学模型。为了准确模拟自由流和裂缝多孔介质流区域耦合问题的流体流动，一方面要求数值方法可以精准捕捉间断解；另一方面由于此耦合系统是一个多区域耦合的界面问题，要求数值方法易于处理在界面处网格剖分不匹配的情形。基于以上两个方面的考虑，我们选用具有精确捕捉间断解、允许悬挂点、高精度、允许任意多边形网格剖分、局部质量守恒等性质的间断Galerkin有限元方法求解此耦合问题。
Immersed finite Element Methods: Development, Analysis, and Applications
林涛
（弗吉尼亚理工大学）
This presentation is a brief introduction to the development, analysis, and applications of immersed finite-element (IFE) methods for solving interface problems of partial differential equations with interface-independent meshes. An IFE method uses standard polynomial finite element functions on non-interface elements, but it employs macro polynomials designed according to interface jump conditions on interface elements. We will describe a unified framework for constructing a group of IFE spaces based on lower degree polynomials such as linear, or bilinear, or rotated-Q1 polynomials. We will discuss the approximation capability of these IFE spaces. A partially penalized IFE (PPIFE) scheme will presented for solving the typical elliptic interface problems. Then, we will consider some applications of this PPIFE discretization to time dependent interface problems and some inverse problems.
等离子体物理中非线性色散方程的有限元数值解法
梅立泉
（西安交通大学）
宇宙中超过99%的重子物质由等离子体组成，等离子体物理在空间科学研究和空间工程应用中具有非常重要的地位。非线性色散方程广泛应用于从流体动力学、等离子体物理到非线性光学的物理科学，在化学和生物等学科中也有广泛的应用。
本文主要针对等离子体中的三类非线性波动现象：孤立波、激波、怪波，研究等离子体物理中非线性色散方程的有限元数值解法。首先，对于RLW方程，当使用标准Crank-Nicolson公式进行时间离散时，在每一个时间层上需要求解一个非线性常微分方程组，而这个方程组需要用迭代法进行数值求解。首先，对高维RLW和SRLW方程，研究其求解的有限元数值格式，对格式进行数值分析并对孤立波的传播、两个孤立波的碰撞进行了模拟。
其次，对(2+1)-维Schrӧdinger方程，建立其求解的显式多步有限元求解格式，并对孤立波的传播及Bose-Einstwein凝聚进行模拟。
然后，对分数阶Schrӧdinger方程，建立其求解的二阶的能量稳定的数值格式，对格式进行数值分析，并通过算例模拟等离子体物理中的反常扩散现象。
Numerical analysis of the stabilized characteristic mixed IPDG method for the incompressible Navier-Stokes equations
王淑琴
（西北工业大学）
Here a stabilized second order characteristic mixed interior penalty discontinuous Galerkin (IPDG) method is introduced for the incompressible Navier-Stokes equations in $\mathbb{R}^2$. It is shown that the numerical approximation of velocity is bounded in $W^{1,\infty}$-norm when the time step satisfies $\tau\leq Ch$ where $C$ is a constant. With the boundedness, the optimal error estimates of velocity in $L^2$-norm and in DG norm $|||\cdot|||_{1,h}$ are established by using Stokes projection method. In addition, the suboptimal error estimates of pressure in $L^2$-norm is proven. Some numerical experiments are given to validate the theoretical results.
Numerical methods for solving linear complementarity problems
汪祥
（南昌大学）
In this talk, we will introduce two numerical methods to solve a class of linear complementarity problems. Convergence analysis will show these two new methods will converge under certain conditions. Numerical experiments further show that the proposed methods are superior to the existing methods in actual implementation.
Toeplitz矩阵重建的几种算法
温瑞萍
（太原师范学院）
矩阵重建问题主要衍生于近几年非常流行的压缩感知技术, 主要分为矩阵填充和矩阵恢复问题, 在图像与信号处理、计算机视觉、推荐系统等方面发挥着重要的作用.
Toeplitz矩阵作为一种特殊的矩阵, 在图像与信号处理中有着广泛的应用, 其奇异值分解的算法复杂度仅为O(n^2logn), 目前大部分的矩阵重构算法都是基于奇异值分解的. 而在解决Toeplitz矩阵的矩阵重构问题时, 现有的算法存在计算量大、速度慢等问题. 这里主要报告以下两方面工作：
针对Toeplitz矩阵填充问题, 首先提出了Toeplitz矩阵的保结构算法, 该算法中利用二次规划技术及均值技术寻找最优填充矩阵并保持Toeplitz结构, 从而利用其结构特点降低奇异值分解时间. 之后又提出了Toeplitz矩阵的奇异值阈值算法和修正的增广Lagrange算法. 理论上证明了算法的收敛性. 数值实验表明新算法的高效性.
针对Toeplitz矩阵恢复问题, 提出了Toeplitz矩阵恢复的阈值算法, 算法分别利用均值和中值使得迭代矩阵保持Toeplitz矩阵结构, 同时利用其快速奇异值分解算法降低奇异值分解及CPU时间. 随后提出了增广Lagrange算法. 与原算法对比, 当数据或图像污染很严重时, 新的算法更有效.
Randomized GLRAM-type algorithms for high dimensionality reduction and image reconstruction
吴钢
（中国矿业大学）
High-dimensionality reduction techniques are very important tools in machine learning and data mining. The method of generalized low rank approximations of matrices (GLRAM) and its variations are popular for dimensionality reduction and image reconstruction, which are based on native two-dimensional matrix patterns. However, they often suffer from heavily computational overhead in practice, especially for data with high dimensionality. In order to reduce the computational complexities of these type of algorithms, we apply randomized singular value decomposition (RSVD) on them and propose three randomized GLRAM-type algorithms. Theoretical results are established to show the validity and rationality of our proposed algorithms.
First, we discuss the decaying property of singular values of the matrices during iterations of the GLRAM algorithm, and provide the target rank required in the RSVD process from a theoretical point of view. Second, we show the relationships between the reconstruction errors generated by the original GLRAM-type algorithms and the randomized GLRAM-type algorithms. Third, we shed light on the convergence of the randomized GLRAM algorithm.
Numerical experiments on some real-world data sets illustrate the superiority of our proposed algorithms over their original counterparts and some state-of-the-art algorithms, for image reconstruction and face recognition.
Minimum residual HSS iteration method for non-Hermitian positive definite complex linear systems
伍渝江
（兰州大学）
This talk will present a non-stationary iteration method, or a minimum residual Hermitian and skew-Hermitian (MRHSS) iteration method for solving non-Hermitian positive definite complex linear systems. Convergence analysis and numerical results will be also given to illustrate the efficiency of the MRHSS method.
Analysis of symmetric schemes and robust preconditioners for incompressible MHD system
张国栋
（烟台大学）
In this talk, we consider efficient schemes for solving incompressible MHD (magnetohydrodynamics) system and design robust preconditioners for them. We propose the first order and second order symmetric schemes with augmented symmetric terms in magnetic equations that are introduced in consideration of designing uniformly robust preconditioners. We also carry out the optimal error estimates of the proposed scheme. Furthermore, we design diagonal block preconditioners for the schemes, and rigorously prove that the condition number of preconditioned system is uniformly bounded by a constant that only depends on the computational domain. Finally, some numerical experiments, including accuracy tests and physical benchmark problems, are presented to verify the uniform robustness of the preconditioner and test the convergence orders of the schemes.
