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Event(s) on May 2009
- Friday, 8th May, 2009
Title: Asymptotics of MLE & LRT under boundary conditions Speaker: Professor Bimal Sinha, Presidential Research Professor, University of Maryland, USA Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversitAbstract: The standard asymptotic properties of the MLE and the LRT are well known under usual Cramer-type regularity conditions. In this talk details of such properties will be given when some parameters may lie on the boundaries. The key reference is a 1987 JASA paper by Self & Liang. - Tuesday, 12th May, 2009
Title: Two Millennium Problems: RH and BSDC Speaker: Prof. LIU, Jianya , School of Mathematical and System Sciences, , Shandong University,, China Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: In number theory, primes and Diophantine equations form two important research areas. From these two areas came two far-reaching conjectures: the Riemann Hypothesis (RH), and the Birch-Swinnerton-Dyer Conjecture (BSDC). The importance of RH and BSDC may be seen from the fact that they are among the seven millennium prize problems. The purpose of the talk is to explain the meaning of RH and BSDC, and discuss their consequences in the distribution of primes and in Diophantine equations, respectively. The talk is intended for general mathematical audience, and therefore its presentation is mainly from historical and philosophical perspective. - Tuesday, 19th May, 2009
Title: Improved Mean Estimation and Its Application to High-dimensional Data Classification Speaker: Prof. Tiejun Tong, Assistant Professor of Statistics, Department of Applied Mathematics, University of Colorado, USA Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: High-dimensional data such as microarrays have created new challenges to traditional statistical methods. In particular, the feature-specific estimates of means are usually unreliable when the number of samples is small. To address this problem, we propose a family of shrinkage estimators for means under the assumption of unequal and unknown variances. We show that the proposed estimators are minimax and thus dominate the sample means under the squared loss function. The proposed method is general and widely applicable, whereas we illustrate its usefulness in the framework of discriminant analysis. Specifically, we propose a shrinkage-based diagonal discriminant rule and demonstrate its improvement over the original rules through both simulation and real data analysis. - Friday, 22nd May, 2009
Title: JRIAM & ICM Colloquium: Fast Gradient Descent, Artificial Time Integration, and Applications to Image and Surface Reconstruction Speaker: Prof. Uri Ascher, Department of Computer Science, University of British Columbia, Canada Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: The integration to steady state of many initial value ODEs and PDEs using the forward Euler method can alternatively be considered as gradient descent for an associated minimization problem. Greedy algorithms such as steepest descent for determining the step size are as slow to reach steady state as is forward Euler integration with the best uniform step size. But other, much faster methods using bolder step size selection exist. Various alternatives are investigated from both theoretical and practical points of view. The steepest descent method is also known for the regularizing or smoothing effect that the first few steps have for certain inverse problems, amounting to a finite time regularization. We further investigate the retention of this property using the faster gradient descent variants in the contexts of deblurring and denoising of images, and of shape optimization involving data inversion of elliptic PDEs. When the combination of regularization and accuracy demands more than a dozen or so steepest descent steps, the alternatives offer an advantage, even though (indeed because) the absolute stability limit of forward Euler is carefully yet severely violated. - Tuesday, 26th May, 2009
Title: Anisotropic FEM and Applications Speaker: Prof.SHI Dong-yang, Department of Mathematics, Zhengzhou University, P. R. China Time/Place: 11:30 - 12:30
FSC1111, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: The classical finite element methods demand that the subdivisions should satisfy the regular condition or quasi-uniform requirement, i.e., there exists a constant C > 0, such that for all element $K$, $h_K/rho_Kleq C$ or $h/h_{min}leq C$, where $h=maxlimits_Kh_K ,h_{min}=minlimits_Kh_K$, $h_K$ and $rho_K$ are the diameter and the superior diameter of all balls contained in $K$, respectively. The conditions restrict the application of the finite element methods. In fact, when the domain concerned is very narrow, if we employ the regular partition, the computing cost will be very high. The obvious idea to overcome this difficulty is to use the anisotropic meshes with fewer degrees of freedom. In addition, the solutions of some elliptic problems may have anisotropic behavior in some parts of the solution domain. This means that the solution only vary significantly in certain directions. The better way to reflect this is anisotropy is to use anisotropic meshes with a finer mesh size in the direction of the rapid variation of the solution and a coarser mesh size in the perpendicular direction. Because the anisotropic elements $K$ are characterized by $frac{h_{K}}{rho_{K}}rightarrow infty$ when the limit is considered as $h rightarrow 0$, the famous Bramble-Hilbert lemma can not be used directly in the estimate of the interpolation error. At the same time, the consistency error estimate, the key of the nonconforming finite element analysis, will become very difficult to be dealt with, for there will appear a factor $frac{|F|}{|K|}rightarrow infty$ when the estimate is made on the longer sides $F$ of the element $K$. It means that the traditional finite element analysis techniques are no longer valid. In this report, we will introduce some new developments of anisotropic FEMs both in theoretical analysis and computations when they applied to the practical problems. At the same time, we propose some "open problems" for the further study on this aspect.