会议名称：北京大学复旦大学厦门大学计算数学联合学术报告会

4月22日，海韵园实验楼108会议室

学术报告题目和摘要

Cheeger's Cut, Maxcut and Spectral Theory of 1-Laplacian on Graphs

邵嗣烘（北京大学）

This is primarily an expository talk surveying up-to-date known results on the spectral theory of 1-Laplacian on graphs and its applications to both Cheegers cut and maxcut problems. The structure of eigenspace, nodal domains, multiplicities of eigenvalues, and algorithms for graph cuts are collected.

关于Darcy-Stokes和CHDS问题的一些算法

陈文斌（复旦大学）

In this talk, Some numerical algorithms for Stokes-Darcy and  Cahn-Hilliard-Stokes-Darcy systems are introduced. Based on the idea of domain decomposition, the systems are decoupled, these schemes are very efficient and can be easily implemented using legacy codes. We establish the unconditional and uniform in time stability. High order schemes are also discussed.

HDG methods for the Maxwell equations

陈黄鑫（厦门大学）

In this talk we will first introduce a new HDG method for the steady state Maxwell equations based on a mixed curl-curl formulation.  We use a non-trivial subspace of polynomials of degree k+1 to approximate the numerical tangential trace of the electric field. If we assume the dual operator of the Maxwell equation has adequate regularity, we show that the $L^2$-norm error of the electric field is $O(h^{k+2})$. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this HDG method achieves superconvergence for the electric field without postprocessing. In particular, the convergence rate of the electric field is independent of the Lagrange multiplier when the HDG scheme is based on simplicial mesh. When we consider the Maxwell equations with low regularity of electric field, another HDG method and its optimal convergence rate will be shown, and the corresponding adaptive HDG method will be discussed.

Phase field modeling of Cell Polarity and Cell Delamination

张磊（北京大学）

Control of cellular behaviors plays a critical role in pattern formation, growth regulation and regeneration. Numerous developmental processes have been extensively studied from a mechanistic perspective, but only recently have serious efforts been directed toward systems biology approach. In this talk, I will present two biological systems to study pattern formation by using phase field model. First, we present a mathematical model that incorporates the interplays between Rac, filamentous actin (F-actin), and membrane tension for the formation of cell polarity. Second, I present a phase field approach to study the neuroblast delamination in Drosophila. Dynamics of cell ingression and role of actin-myosin network in apical constriction reveal that the myosin signaling drives neuroblast delaminiation in such rare event. The joint work with Feng Liu (PKU), Yan Yan (HKUST).

Filter based methods for statistical linear inverse problems

陆帅（复旦大学）

Ill-posed inverse problems are ubiquitous in applications. Understanding of algorithms for their solution has been greatly enhanced by a deep understanding of the linear inverse problem. In the applied communities ensemble-based filtering methods have recently been used to solve inverse problems by introducing an artificial dynamical system. This opens up the possibility of using a range of other filtering methods, such as 3DVAR and Kalman based methods, to solve inverse problems, again by introducing an artificial dynamical system. The aim of this talk is to analyze such methods in the context of the ill-posed linear inverse problem.

    Statistical linear inverse problems are studied in the sense that the observational noise is assumed to be derived via realization of a Gaussian random variable. We investigate the asymptotic behavior of filter based methods for these statistical linear inverse problems. Rigorous convergence rates are established for 3DVAR and for the Kalman filters, including minimax rates in some instances. Blowup of 3DVAR and its variant form is also presented, and optimality of the Kalman filter is discussed. These analyses reveal close connection between (iterative) regularization schemes in deterministic inverse problems and filter based methods in data assimilation. 

  It is a joint work with Dr. M. A. Iglesias (U. of Nottingham, UK), Dr. K. Lin (Fudan U., China) and Prof. A. M. Stuart (Caltech, USA).

A Spectral method for Maxwell equations in Cole-Cole dispersive media

黄灿（厦门大学）

In this talk, I shall consider time-dependent Maxwell equations in Cole-Cole dispersive media.  The Cole-Cole model are the standard Maxwell equations coupled with a fractional time derivative term,  which challenges the design and analysis of its numerical  algorithm.  By adopting the Matrix diagonalization  method,  the semidiscretization of the model can be broken into a set of ordinary integro-differential equations (OIDE) with weakly singular kernel.  Then, we propose an ansatz of solution for OIDEs. Using the ansatz,  together with mapped numerical quadrature technique and well-conditioned matrix technique, we are able to produce an accurate approximation for OIDEs and thus, the original model as well.