Colloquium/Seminar
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Event(s) on December 2014
- Wednesday, 3rd December, 2014
Title: Intersection Between Probability and Other Branches of Mathematics and Sciences Speaker: Prof. Zhi-Ming Ma, Chinese Academy of Sciences, China Time/Place: 17:00 - 18:00 (Preceded by Reception at 4:30pm)
WLB104, The Wing Lung Bank Building for Business Studies, Shaw Campus, HKBUAbstract: Probability and statistics have brought fruitful achievements intersecting with other branches of mathematics and sciences, as can be seen from the work of several recent Fields Medalists. In this lecture I shall briefly describe some research directions in this aspect, and introduce some of our recent results in this connection. - Friday, 5th December, 2014
Title: Application of Weak Galerkin Finite Element Methods Speaker: Prof. Ran Zhang, Department of Mathematics, Jilin University, China Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: The Weak Galerkin method is an extension of the standard Galerkin nite element method where classical derivatives were substituted by weakly defined derivatives on functions with discontinuity. Since the introduction of WG finite element methods, they have been applied successfully to the discretization of several classes of partial differential equations, cf., e.g., second order elliptic equations, the Biharmonic equations, the Stokes equations and the Brinkman equations. WG methods, by design, make use of discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons and polyhedrons. Weak functions and weak derivatives can be approximated by piecewise polynomials with various degrees. The flexibility of WG on these aspects of approximating polynomials makes it an excellent candidate for the numerical solu- tion of incompressible flow problems. - Friday, 5th December, 2014
Title: Extremum Problems for Laplacian Eigenvalues Speaker: Prof. Fanghua Lin, Courant Institute of Mathematical Sciences, New York University, USA Time/Place: 16:30 - 17:30 (Preceded by Reception at 4:00pm)
SPH, Ho Sin Hang Campus, Hong Kong Baptist UniversityAbstract: Eigenvalue Problems for Laplacians are among most well-known problems in classical analysis, partial differential equations, calculus of variations and mathematical physics. In this lecture I shall discuss a couple very basic extremum problems involving eigenvalues of the Laplacian. Such problems arise in shape optimizations, pattern formations and other studies in science, and solutions to these problems have been challenging for a very long time. In particular, we shall see how classical Rayleigh-Faber-Krahn inequalities and Weyl's asymptotic formula along with theorems concerning nodal domains coming into play in understanding of these problems. The lecture would be a brief and elementary survey of some of recent results. - Friday, 12th December, 2014
Title: Social Dynamics: Modeling, Analysis and Numerical Simulation Speaker: Prof. Eitan Tadmor, Department of Mathematics, University of Maryland, USA Time/Place: 11:00 - 12:00 (Preceded by Reception at 10:30am)
WLB104, Shaw Campus, Hong Kong Baptist UniversityAbstract: We discuss the dynamics of systems driven by the “social engagement” of its agents with their local neighbors through local gradients. Prototype examples include models for opinion dynamics in human networks, flocking, swarming and bacterial self-organization in biological organisms, or rendezvous in mobile systems. Two natural questions arise in this context: what is the large time behavior of such systems when the time T tends to infinity, and what is the effective dynamics of such large systems when the number of agents N tends to infinity. The underlying issue is how different rules of engagement influence the formation of clusters, and in particular, the tendency to form “consensus of opinions”. We analyze the flocking dynamics of agent-based models, present novel numerical methods which confirm the large time formation of Dirac masses at the kinetic level, and end up with critical threshold phenomena at the level of social hydrodynamics. - Monday, 15th December, 2014
Title: A sequential L_1-regularization approach to the L_p-regularization Speaker: Prof. Li Donghui , School of Mathematical Sciences, South China Normal University, China Time/Place: 11:00 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: The $L_p$-regularization problem with $pin (0,1)$ is non-convex, non-differentiable and non-Lipschitzian. The development of numerical methods for solving the problem is a challenging task. On the other hand, the $L_1$-regularization problem is relatively easy and can be solved by recently developed methods efficiently. In this talk, we propose an approach to the $L_p$-regularization problem via solving a sequence of $L_1$-regularization problems. The method is motivated by a sequential quadratic programming (SQP) method for solving a smooth constrained optimization reformulation to the problem. We show that the quadratic programming subproblem of the SQP method can be transformed into an unconstrained $L_1$-regularization problem. Moreover, its solution provides a descent direction for the objective function of the $L_p$-regularization problem. The next iterate is then obtained by the use of a backtracking line search process. Under certain conditions, we show the global convergence of the proposed method. - Tuesday, 16th December, 2014
Title: Lower Bounds of Dirichlet Eigenvalues for Degenerate Elliptic Operators Speaker: Prof. CHEN Hua, School of Mathematics and Statistics, Wuhan University, China Time/Place: 11:00 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Let Omega be a bounded open domain in R^n with smooth boundary and X=(X_1, X_2, …, X_m) be a system of real smooth vector fields defined on Omega. Under the Hörmander's condition, the sum of square operator triangle_X=sum_{j=1}^m X_j^2 is finite degenerate and denote here lambda_j is the j^{th} Dirichlet eigenvalue for -triangle_X on Omega. If the finitely degenerate elliptic operators satisfying the so-called Métivier's condition, we knew that lambda_k approx k^{2/nu}, as k -> +infty, where nu is the Métivier index. In this talk, for general finitely degenerate elliptic operators, we use the subelliptic estimate to get the lower bounds for the Dirichlet eigenvalues lambda_k. Next, for a class of Grushin type degenerate elliptic operators, we give the sharp lower bounds of lambda_k, which will be the extension of Métivier's asymptotic result. Finally, for a class of infinitely degenerate elliptic operators (i.e. the Hörmander's condition is not satisfied), which satisfying the logarithmic regularity estimate, we can also deduce the lower bounds of lambda_k which will be at least logarithmic increasing in k. - Tuesday, 16th December, 2014
Title: Projected Gradient Methods for Orthogonality Constrained Optimization and Application Speaker: Ms. ZHU Hong, Department of Mathematics, Hong Kong Baptist University, Hong Kong Time/Place: 14:30 - 15:15
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Optimization with orthogonal constraints plays an important role in many applications of science and engineering. This problem is difficult because the objective function is not necessary convex, and the constraints are non-convex, it also can lead to many local minimizers. We transform the problem to a series of optimization problem with convex constraints by adding a penalty style term on the objection function and relax the orthogonal constraints to convex constraints. Thus it can be handled by algorithms on convex set. We prove that if the parameter of penalty term is large enough (but not necessary goes to infinity), any local minimizer of relaxed problem is a local minimizer of the primal problem under the assume that the gradient of objective function is Lipschitz continuous. We also prove that the sequence generated for each non-convex subproblem converge to the stationary point under the suitable assumptions. Furthermore, we generalize this result to the case when ℓ1 - norm contained in the objective function. - Tuesday, 16th December, 2014
Title: Interior point continuous methods for convex programming Speaker: Mr. QIAN Xun, Department of Mathematics, Hong Kong Baptist University, Hong Kong Time/Place: 15:15 - 16:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: In this paper, two interior point based ordinary differential equation (ODE) systems are presented for solving convex programming with linear constraints. With the help of some potential functions, we are able to prove under a very mild condition, i.e. the existence of a finite optimal solution, that for any interior feasible point (i) the two continuous trajectories, which are the solutions of corresponding two ODE systems, are convergent; and (ii) the limit points are indeed the optimal solutions which are contained in the relative interior of the optimal solution set. Based on the attractive properties of these two continuous trajectories, a new algorithm for convex quadratic programming is proposed. Some preliminary numerical results are also provided. Then we will extend the method to the semidefinite programming. - Friday, 19th December, 2014
Title: Statistics for Big Data Speaker: Prof. Dennis K.J. Lin, Department of Statistics, The Pennsylvania State University, USA Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: After noting the relative absence of statisticians from the community of practice engaged with big data, we explain what big data is, how it's done, and who's working with it. Statisticians have much more to contribute in both the intellectual vitality and the practical utility of big data. At the same time, big data challenges statisticians to move out of some familiar habits to engage less structured problems, to become more comfortable with ambiguity, and to engage computer scientists in a more fruitful discussion of what the various parties can bring to this new mode of investigation. In this talk, we propose some potential directions for future research. This talk is mainly based upon a 2014 joint paper with Dr. John Jordan. - Friday, 19th December, 2014
Title: d-bar-Neumann Problem Speaker: Prof. Der-Chen Chang, Department of Mathematics and Statistics, Georgetown University, USA Time/Place: 15:00 - 16:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: {bar}{partial}-Neumann Problem A domain Omega subset C^{n+1} and its boundary partial{Omega} are said to be decoupled of finite type if there exists sub-harmonic, non-harmonic polynomials {P_j}_{j=1,…,n} with P_j(0) = 0 such that Omega = {(z_1, …, z_n, z_{n+1}) : Im(z_{n+1}) > sum^n_{j=1} P_j (z_j)} . We call the integer m_j = 2 + degree (triangle P_j) the degree of P_j. The ``type’’ of Omega is m=max{m_1, …, m_n}. In this talk, we use the method developed by Greiner-Stein and Chang-Nagel-Stein to construct the ``functional solution’’ N for the bar{partial}-Neumann problem: Given a (0,1)-form f=sum^{n+1}_{j=1} f_j bar{omega}_j, find a (0,1)-form u=N(f) such that (bar{partial}bar{partial}*+bar{partial}*bar{partial})u = f in Omega u_{n+1}=0 on partial{Omega} bar{Z}_{n+1}u_j|_{partial{Omega}} = 0 on partial{Omega} for j=1, …, n. Here bar{Z}_{n+1} = partial/partial{rho} + ipartial/partial{t} is the complex normal with t=Re(z_{n+1}) and rho = Im(z_{n+1}) - sum^n_{j=1}P_j(z_j) is the ``height function’’ defined on Omega. Then we will discuss possible sharp estimates for the operator N. - Friday, 19th December, 2014
Title: An intrinsic characterisation of weighted Kegel Sobolev spaces Speaker: Prof. Bert-Wolfgang Schulze, Institut für Mathematik, Universität Potsdam, Germany Time/Place: 16:00 - 17:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Operators on manifolds with higher singularities, defined as specific stratified spaces, are useful for numerous applications. Similarly as on a smooth manifold with boundary ellipticity is defined in terms of a principal symbolic hierarchy, here consisting of k components, according to the depth of the stratification. We first illustrate the way on how an operator acquires its symbols from the strata. If a stratum is of dimension >0, the corresponding edge symbols act in weighted Kegel spaces; those live on infinite cones transversal to the edge. Even in the edge case k=1 these spaces have remarkable properties in terms of their behaviour at the conical exit to infinity. Those originally refer to a Fourier-based description, but it seems more convenient to employ an approach in terms of specific Mellin operators with an edge-degenerate dependence on the axial variable r>0. We give an idea on how such structures are organised for higher singularities. The results are partly obtained in joint work with Lyu Xiaojing, Tianjin, China - Monday, 22nd December, 2014
Title: Disparity and Optical Flow Partitioning Using Extended Potts Priors Speaker: Dr. CAI Xiaohao, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK Time/Place: 11:00 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: This talk addresses the problems of disparity and optical flow partitioning based on the brightness invariance assumption. We investigate new variational approaches to these problems with Potts priors and possibly box constraints. For the optical flow partitioning, our model includes vector-valued data and an adapted Potts regularizer. Using the notation of asymptotically level stable functions we prove the existence of global minimizers of our functionals. We propose a modified alternating direction method of multipliers. This iterative algorithm requires the computation of global minimizers of classical univariate Potts problems which can be done efficiently by dynamic programming. We prove that the algorithm converges both for the constrained and unconstrained problems. Numerical examples demonstrate the very good performance of our partitioning method.