﻿ 厦门大学数学科学学院

北京大学-厦门大学计算数学联合学术交流会

2015117日，星期六，海韵实验楼105报告厅）

 时间 报告人 单位 报告题目 主持人 8:45-9:00 开幕式 9:00-9:30 陈  龙 北京大学 Multigrid Methods for Fractional Laplacian Equations 汤华中 教授 9:30-10:00 蔡云峰 北京大学 A matrix polynomial spectral approach for general joint block diagonalization 10:00-10:30 白正简 厦门大学 A Riemannian Fletcher-Reeves Conjugate Gradient Method for Doubly Stochastic Inverse Eigenvalue Problems 10:30-11:00 休息、合影 11:00-11:30 董  彬 北京大学 Wavelet Frame Transforms and Differential Operators: Bridging Discrete and Continuum for Image Restoration 邱建贤 教授 11:30-12:00 邵嗣烘 北京大学 Graph Cut and Spectrum of the 1-Laplacian 12:00-12:30 熊  涛 厦门大学 Hybrid asymptotic preserving high order nodal DG-IMEX schemes for the BGK equation

Multigrid Methods for Fractional Laplacian Equations

(北京大学)

Abstract: In this talk, we will present fast multilevel methods for the approximate solution of the discrete problems that arise from the discretization of fractional Laplacian. To localize the nonlocal fractional Laplacian operator, we solve a Dirichlet to Neumann-type operator via an extension problem. However, this comes at the expense of incorporating one more dimension to the problem, thus motivates our study of adaptive multilevel methods.

Because of the singularity of the solution of the extended equation, anisotropic elements in the extended variable are needed in order to obtain quasi-optimal error estimates. For this reason, we consider a multigrid method with a line smoother and obtain nearly uniform convergence rates using the multilevel framework developed by Xu and Zikatanov.

A matrix polynomial spectral approach for general joint block diagonalization

Abstract: Joint block diagonalization (JBD) of a given Hermitian matrix set { ,i=0,1,…,p} is tofind a nonsingular matrix W such that  for i=0,1,…,p are all block diagonal matriceswith the same prescribed block diagonal structure. General JBD (GJBD) attempts to solve JBD without knowing the resulting block diagonal structure in advance, which is more difficult than JBD.

In this talk, we show that GJBD of { ,i=0,…,p} is strongly connected with the spectral informationof the corresponding matrix polynomial P( ) = . Under some conditions, the solutions to GJBD are characterized by the spectral information, and a necessary and sufficient condition is given for the existence of nontrivial solutions to GJBD.

In addition, based on the established theory, two feasible numerical methods are proposed to solve GJBD.

A Riemannian Fletcher-Reeves Conjugate Gradient Method for Doubly Stochastic Inverse Eigenvalue Problems

Abstract: We consider the inverse eigenvalue problem of reconstructing a doubly stochastic matrix from the given spectrum data. We reformulate this inverse problem as a constrained nonlinear least squares problem over several matrix manifolds, which minimizes the distance between isospectral matrices and doubly stochastic matrices. Then a Riemannian Fletcher-Reeves  conjugate gradient method is proposed for solving the constrained nonlinear least squares problem and its global convergence is established. An extra gain is that a new Riemannian isospectral flow method is obtained. Our method is also extended to the case of prescribed entries. Finally, some numerical tests are reported to illustrate the efficiency of the proposed method.

Wavelet Frame Transforms and Differential Operators: Bridging Discrete and Continuum for Image Restoration

彬（北京大学）

Abstract: My talk is mainly based on a series of three papers ([1-3] below). In [1], we established connections between wavelet frame transforms and differential operators in variational framework. In [2], we established their connections for nonlinear evolution PDEs. Based on [1,2], we proposed a new piecewise smooth image restoration model based on wavelet frames in [3], and linked it with a brand new variational model, a special case of which resembles, but is superior to, the well-known Mumford-Shah model. The connections established in [1-3] provide us with new insights and inspiring interpretations of both wavelet frame and differential operator based approaches, which enable us to create new models and algorithms for image restoration that combine the merits of both approaches.  The significance of our findings is beyond what it may appear. In fact, our analysis and discussions in [1-3] already indicate that wavelet frame based approach is a new and useful tool in numerical analysis to discretize and solve variational and PDE models in general, which enriches the existing theory and applications of numerical PDEs, variational techniques, wavelet frames, etc.

Graph Cut and Spectrum of the 1-Laplacian

ABSTRACTGraph cut, partitioning the vertices of a graph into two or more disjoint subsets, is a fundamental problem in graph theory and has become a very powerful tool in data clustering. In this talk, we first try to connect the Cheegercut  problem with a continuous function optimization problem through the graph 1-Laplacian. That is, the original discrete combination optimization problem can be equivalently transformed into the continuous function optimization problem and then the nonlinear eigenvalue problem. Second, we discuss theory and algorithms for such nonlinear eigenvalue problem which is usually not only nonconvex but also nondifferentiable. Finally, some preliminary numerical results and future directions are provided.

Hybrid asymptotic preserving high order nodal DG-IMEX schemes for the BGK equation

涛（厦门大学）

Abstract: In the asymptotic limit, the BGK equation in the hyperbolic scaling will lead to compressible Euler or Navier-Stokes (NS) equations. We have developed a class of high-order asymptotic preserving (AP) nodal DG-IMEX schemes, which formally could be demonstrated that the scheme,

when the Knudsen number is small, becomes a local DG discretization of the NS equations, and when the Knudsen number vanishes, becomes a DG discretization of the Euler equations. However, it seems that when the Knudsen number is small or zero, it would be better to solve the NS equations or Euler equations directly, which could save a lot of computational cost. We now develop a hybrid nodal DG-IMEX scheme based on a moment realization creteria. We will show the good performance of the scheme from some numerical expriments.