Colloquium/Seminar
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Event(s) on October 2018
- Thursday, 4th October, 2018
Title: Mathematical program with geometric constraints: enhanced optimality, weaker constraint qualifications, error bound and sensitivity Speaker: Dr Zhang Jin Time/Place: 10:30 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: We study the mathematical program with geometric constraints (MPGC) such that the image of a mapping from a Banach space is included in a nonempty and closed subset of a finite dimensional space. We obtain the nonsmooth enhanced Fritz John necessary optimality conditions in terms of the approximate subdifferential. In the case where the Banach space is a weakly compactly generated Asplund space, the optimality condition obtained can be expressed in terms of the limiting subdifferential while in the general case it can be expressed in terms of the Clarke subdifferential. One of the technical difficulties in obtaining such a result in an infinite dimensional space is that no compactness result can be used to show the existence of local minimizers of a perturbed problem. In this paper we employ the celebrated Ekeland's variational principle to obtain the results instead. The enhanced Fritz John condition allows us to obtain the enhanced Karush-Kuhn-Tucker condition under the pseudonormality and the quasinormality conditions which are weaker than the classical normality conditions. We then prove that the quasinomality is a sufficient condition for the existence of local error bounds of the constraint system. Finally we obtain a tighter upper estimate for the subdifferentials of the value function of the perturbed problem in terms of the enhanced multipliers. - Friday, 5th October, 2018
Title: Crank-Nicolson ADI Method for Space-Fractional Diffusion Equations with Non-separable Coefficients Speaker: Mr Xuelei Lin, Department of Mathematics, Hong Kong Baptist University Time/Place: 10:30 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: In this paper, we study Crank-Nicolson ADI method for two-dimensional Riesz space-fractional diffusion equations with non-separable coefficients. Existing ADI methods are only shown to be unconditional stable when coefficients are some special separable functions. The main contribution of this paper is to show under mild assumptions the unconditional stability of the proposed Crank-Nicolson ADI method in discrete $ell^2$ norm and the consistency of cross perturbation terms arising from Crank-Nicolson ADI method. Also, we demonstrate that several consistent spatial discretization schemes satisfy the required assumptions. Numerical results are presented to examine the accuracy and the efficiency of the proposed ADI methods. - Thursday, 11th October, 2018
Title: Some new optimization theory for convergence analysis of first-order algorithms Speaker: Dr Zhang Jin, Department of Mathematics, Hong Kong Baptist University Time/Place: 10:30 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: We present some new theoretical results for the topics of optimality conditions, constraint qualifications and error bounds in optimization, and show how these theoretical results can be used for analyzing the convergence of some very popular first-order algorithms which have been finding wide applications in data science domains. Some fundamental models such as the LASSO and grouped LASSO are studied, and it is shown that the linear convergence can be obtained if some algorithms are implemented for these models such as the proximal gradient method, the proximal alternating linearized minimization algorithm and the randomized block coordinate proximal gradient method. We provide a novel analytic framework based on variational analysis techniques (e.g., error bound, calmness, metric subregularity) for the convergence analysis of first-order algorithms. By this new analytic framework, we significantly improve some convergence rate results in the literature and obtain some new results. - Monday, 15th October, 2018
Title: Mixture Model in Deep Learning and Its Applications Speaker: Prof Liu Jun, Department of Mathematics, Beijing Normal University, China Time/Place: 15:30 - 16:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Mixture model is a statistical method which can be used to classification. We will integrate the mixture model and expectation maximum (EM) method into the popular deep learning technique and propose a mixture model based learning method for mixture noise removal and a mixture model based generative adversarial network (GAN). - Thursday, 25th October, 2018
Title: Partial error bound conditions and the linear convergence rate of ADMM Speaker: Dr Zhang Jin, Department of Mathematics, Hong Kong Baptist University Time/Place: 10:30 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: In the literature, error bound conditions have been widely used to study the linear convergence rates of various first-order algorithms. Most of the literature focuses on how to ensure these error bound conditions, usually posing numerous assumptions or special structures on the model under discussion. In this paper, we focus on the alternating direction method of multipliers (ADMM), and show that the known error bound conditions for studying the ADMM's linear convergence rate can indeed be further weakened if the error bound is studied over the specific iterative sequence it generates. revise {An error bound condition based on ADMM's iterations} is thus proposed, and linear convergence under this condition is proved. Furthermore, taking advantage of a specific feature of ADMM's iterative scheme by which part of the perturbation is automatically zero, we propose the so-called partial error bound condition, which is weaker than known error bound conditions in the literature, and we derive the linear convergence rate of ADMM. We further show that this {partial error bound} condition is useful for interpreting the difference if the two primal variables are updated in different orders when implementing the ADMM. This has been empirically observed in the literature, yet no theory is known. - Thursday, 25th October, 2018
Title: Smallest Eigenvalue of Large Hankel Matrices at Critical Point: Comparing a Conjecture with parallelised computation Speaker: Prof Yang CHEN, Faculty of Science and Technology, University of Macau, Macau, China Time/Place: 14:30 - 15:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: "Hankel matrices are matrices of moments, (see Heine, Handbuch der Kugelfunc-tionen, 1878), plays a fundamental role in approximation theory. It transpired that, the logarithm of the n×n Hankel determinants which may depend on parameters in the weight, plays an important role in finite n aspect of integrable systems. The early pioneers are Jimbo-Miwa-Ueno-Okamoto-McCoy-Tracy-Widom. Since Hankel matrices are moments of positive continuous weight function, it form positive definite quadratic form. From which one may seek to find the smallest eigenvalue, with the aid of polynomials orthogonal with respect to the weight. The talk will focus on the weight characterized by a parameter (> 0), w(x) = e−x , > 0, 0 x < 1. and (HN)i,j = Z 1 0 xi+jw(x)dx, 0 i, j N − 1.. The “critical point” occurs at = 1/2. Joint work, Yang Chen (University of Macau), Jakub Sikirowski (Computing Lab. in Cam-bridge, UK)), Mengkun Zhu (Qilu University, Shandong, China)" - Monday, 29th October, 2018
Title: Mathematical study on elastic cloaking, geometric body generation and geomagnetic anomalies detection Speaker: Ms Tsui Wing Yan, Department of Mathematics, Hong Kong Baptist University Time/Place: 10:30 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: The study on inverse problems has played a pivotal role to various disciplines of science and technology. In the view of their promising applications, we investigate the potentials for three inverse scattering approaches for (i) the enhancement of nearly elastic invisibility cloak for the three-dimensional inclusion embedded in isotropic medium by using scattering vanishing structures, (ii) the geometric body generation by customizing characteristic parameters of a specific geometric body in machine learning perspective and (iii) the identification of electromagnetic anomalies beneath the Earth generated by seismic waves using geomagnetic monitoring. Both inverse problems to above approaches for active measurements and for passive observations are interested. - Tuesday, 30th October, 2018
Title: Research on the separation of multicomponent non-stationary signals: A comparison between SST and SSO Speaker: Prof LI Lin, School of Electronic and Engineering, Xidian University, Xi’an, Shanaxi, Ch Time/Place: 11:00 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Separation of multicomponent non-stationary signals has broad applications in many engineering fields, such as seismic signal analysis, vibration fault diagnosis, biomedical signal analysis, speech enhancement, sonar signal processing, etc. This work mainly deals with the separation methods based on synchrosqueezing transform (SST) and signal separation operator (SSO). Firstly, we will introduce the principle of synchrosqueezing, the ridge extraction and sub-signal reconstruction. Moreover, we will introduce some new developments of SST, such as second-order SST, higher-order SST, and adaptive SST, etc. Secondly, we will introduce an algorithm with which the SSO can be used to process the sampled signals. Then the SSO is extends to an adaptive case with time-varying Gaussian windows. Finally, we use various numerical data, including simulated signal and real signals to demonstrate the separation performance of SST and SSO. We will also compare to other separation methods.