學術委員會：
主席：江松（北京應用物理與計算數學研究所）
委員（按姓氏拼音順序）：
白中治（中國科學院）
韓渭敏（愛荷華大學）
何曉明（密蘇裡科技大學）
何銀年（西安交通大學）
侯延仁（西安交通大學）
黃艾香（西安交通大學）
李開泰（西安交通大學）
林濤（維吉尼亞理工大學）
梅立泉（西安交通大學）
聶玉峰（西北工業大學）
溫瑞萍（太原師範學院）
伍渝江（蘭州大學）
Session 1
地點：西北大學賓館1100會議室
Session 2
地點：西安交通大學數學樓2-1會議室
Session 3
地點：西北大學非線性科學研究中心學術報告廳
白中治
（中國科學院）
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.
韓渭敏
（愛荷華大學）
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.
何曉明
（密蘇裡科技大學）
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.
何銀年
（西安交通大學）
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.
侯延仁
（西安交通大學）
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.
黃艾香
（西安交通大學）
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.
黃玉梅
（蘭州大學）
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.
李開泰
（西安交通大學）
血管裡的血液流動和血管壁的彈性殼體的耦合系統的球單元有限元逼近，血管壁視為三維彈性殼體，服從三維彈性方程，血液流動視為不可壓縮的牛頓流體，服從Navier_stokes方程，連接邊界服從速度和法向應力連續條件。應用有限元方法求解，有限元單元為三維球單元。血管壁作為彈性殼體，在厚度方向只用一個球單元，在血液流內，元素的幾何都是三維球，但幾何尺寸
是不均勻的。尚未進行數值分析。
李瑞
（陝西師範大學）
自由流和多孔介質流耦合問題在工程中有著重要的應用，例如地表水和地下水的交互、地下水在喀斯特巖溶含水層的流動、血液在血管和器官之間的流動、工業過濾、汙染物的運移、石油開採、地熱能的研究、二氧化碳的封存、海綿城市的建設等。而在實際多孔介質區域, 地質結構複雜多變, 基質中通常含有裂縫。裂縫的幾何區域狹長, 分布具有隨機性, 大小跨越多個數量級, 且填充嚴重, 既可作為高導流通道, 也可成為流動屏障. 自由流和裂縫多孔介質區域耦合問題流體流動通道尺度差異大，多種流動形式共存，不僅存在滲流，還存在自由流，以及三個區域之間的流體交換，同時，裂縫的存在加劇了孔隙介質的非均勻性，各向異性和不連續性，從而導致基質中流體的速度和壓力不連續。本文針對不可壓縮單相流體在自由流區域和裂縫多孔介質區域中的耦合流動問題，結合數學建模、理論分析和數值模擬展開研究。我們擬利用區域耦合的建模框架結合數學上的理論推導，建立描述自由流和裂縫多孔介質流區域耦合問題的可計算數學模型。為了準確模擬自由流和裂縫多孔介質流區域耦合問題的流體流動，一方面要求數值方法可以精準捕捉間斷解；另一方面由於此耦合系統是一個多區域耦合的界面問題，要求數值方法易於處理在界面處網格剖分不匹配的情形。基於以上兩個方面的考慮，我們選用具有精確捕捉間斷解、允許懸掛點、高精度、允許任意多邊形網格剖分、局部質量守恆等性質的間斷Galerkin有限元方法求解此耦合問題。
林濤
（維吉尼亞理工大學）
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.
梅立泉
（西安交通大學）
宇宙中超過99%的重子物質由等離子體組成，等離子體物理在空間科學研究和空間工程應用中具有非常重要的地位。非線性色散方程廣泛應用於從流體動力學、等離子體物理到非線性光學的物理科學，在化學和生物等學科中也有廣泛的應用。
本文主要針對等離子體中的三類非線性波動現象：孤立波、激波、怪波，研究等離子體物理中非線性色散方程的有限元數值解法。首先，對於RLW方程，當使用標準Crank-Nicolson公式進行時間離散時，在每一個時間層上需要求解一個非線性常微分方程組，而這個方程組需要用迭代法進行數值求解。首先，對高維RLW和SRLW方程，研究其求解的有限元數值格式，對格式進行數值分析並對孤立波的傳播、兩個孤立波的碰撞進行了模擬。
其次，對(2+1)-維Schrӧdinger方程，建立其求解的顯式多步有限元求解格式，並對孤立波的傳播及Bose-Einstwein凝聚進行模擬。
然後，對分數階Schrӧdinger方程，建立其求解的二階的能量穩定的數值格式，對格式進行數值分析，並通過算例模擬等離子體物理中的反常擴散現象。
王淑琴
（西北工業大學）
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.
汪祥
（南昌大學）
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.
溫瑞萍
（太原師範學院）
矩陣重建問題主要衍生於近幾年非常流行的壓縮感知技術, 主要分為矩陣填充和矩陣恢復問題, 在圖像與信號處理、計算機視覺、推薦系統等方面發揮著重要的作用.
Toeplitz矩陣作為一種特殊的矩陣, 在圖像與信號處理中有著廣泛的應用, 其奇異值分解的算法複雜度僅為O(n^2logn), 目前大部分的矩陣重構算法都是基於奇異值分解的. 而在解決Toeplitz矩陣的矩陣重構問題時, 現有的算法存在計算量大、速度慢等問題. 這裡主要報告以下兩方面工作：
針對Toeplitz矩陣填充問題, 首先提出了Toeplitz矩陣的保結構算法, 該算法中利用二次規劃技術及均值技術尋找最優填充矩陣並保持Toeplitz結構, 從而利用其結構特點降低奇異值分解時間. 之後又提出了Toeplitz矩陣的奇異值閾值算法和修正的增廣Lagrange算法. 理論上證明了算法的收斂性. 數值實驗表明新算法的高效性.
針對Toeplitz矩陣恢復問題, 提出了Toeplitz矩陣恢復的閾值算法, 算法分別利用均值和中值使得迭代矩陣保持Toeplitz矩陣結構, 同時利用其快速奇異值分解算法降低奇異值分解及CPU時間. 隨後提出了增廣Lagrange算法. 與原算法對比, 當數據或圖像汙染很嚴重時, 新的算法更有效.
吳鋼
（中國礦業大學）
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.
伍渝江
（蘭州大學）
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.
張國棟
（煙臺大學）
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.
國家天元數學西北中心
國家天元數學西北中心是國家自然科學基金委員會天元數學基金為推動中國數學率先趕上世界先進水平、推動中國數學區域、領域均衡發展而設立的數學研究機構（平臺）。中心依託交大、立足西北、面向全國、放眼世界，將建設成為數學工作者與其它學科領域學者深度交叉融合的學術交流中心和數學與數學技術研究中心。