Event(s) on August 2019

• Thursday, 8th August, 2019

  Title:	PhD Oral Exam: Nonlinear Optimized Schwarz Preconditioning for Heterogeneous Elliptic Problems
  Speaker:	Mr GU Yaguang, Department of Mathematics, Hong Kong Baptist University, HKSAR
  Time/Place:	14:30  -  16:30 FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist University
  Abstract:	In the oral defense, we study problems with heterogeneities using the zeroth order optimized Schwarz preconditioning. There are three main parts. In the first part, we propose an Optimized Restricted Additive Schwarz Preconditioned Exact Newton approach (ORASPEN) for nonlinear diffusion problems. In this approach, we use the Robin condition to communicate subdomain errors. We will see that for problems with large heterogeneities, the Robin parameter has a significant impact on the convergence behavior when subdomain boundaries cut through the discontinuities. In the second main part, therefore, we perform an algebraic analysis for a linear diffusion model problem. In this analysis, we will carefully discuss two possible choices of Robin parameters on the artificial interfaces and derive asymptotic expressions of both the optimal Robin parameter and the convergence rate for each choice. Finally, in the third main part, we will study a time-dependent nonequilibrium Richards equation (NERE), which can be used to model preferential flow in physics. We semi-discretize the NERE in time, and then study how the ORASPEN approach performs for the resulting elliptic problems.

  
  

• Tuesday, 13th August, 2019

  Title:	Strong Approximation of Monotone Stochastic Partial Different Equations Driven by Multiplicative Noise
  Speaker:	Dr LIU Zhihui, Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong
  Time/Place:	11:00  -  12:00 FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist University
  Abstract:	We establish a general theory of strong error estimation for numerical approximations of a second order parabolic stochastic partial differential equation with monotone drift driven by a multiplicative infinite dimensional Wiener process. The equation is spatially discretized by Galerkin finite element method and temporally discretized by drift-implicit Euler or Milstein schemes. By the monotone assumption, we use both the variational and semigroup approaches to derive a spatial Sobolev regularity under the LωpLt∞H˙x1+γ-norm and a temporal H¨older regularity under the LωpLx2-norm for the solution of the proposed equation with an H˙x1+γ-valued initial datum for γ ∈ [0, 1]. In the second step, we introduce an auxiliary process and show that both this process and the discrete solutions are uniform unconditionally stable. Finally, we make full use of the monotonicity of the equation and tools from stochastic calculus to derive the sharp strong convergence rates O(h1+γ + τ1/2) and O(h1+γ + τ(1+γ)/2) for the Galerkin-based Euler and Milstein schemes, respectively.

  
  

• Friday, 16th August, 2019

  Title:	A Versatile Estimation Procedure without Estimating the Nonignorable Missingness Mechanism
  Speaker:	Prof Yanyuan Ma, Department of Statistics, Penn State University, Pennsylvania, USA
  Time/Place:	11:00  -  12:00 FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist University
  Abstract:	We consider the estimation problem in a regression setting where the outcome variable is subject to nonignorable missingness and identifiability is ensured by the shadow variable approach. We propose a versatile estimation procedure where modeling of missingness mechanism is completely bypassed. We show that our estimator is easy to implement and we derive the asymptotic theory of the proposed estimator. We also investigate some alternative estimators under different scenarios. Comprehensive simulation studies are conducted to demonstrate the finite sample performance of the method. We apply the estimator to a children's mental health study to illustrate its usefulness.

  
  

• Tuesday, 20th August, 2019

  Title:	PhD Oral Exam: Statistical Methods of MendelianRandomization Using GWAS Summary Data
  Speaker:	Ms HU Xianghong, Department of Mathematics, Hong Kong Baptist University, HKSAR
  Time/Place:	11:00  -  13:00 FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist University
  Abstract:	Mendelian Randomization (MR) is a powerful tool for accessing causality of exposure on an outcome using GWAS data, However, the accuracy of the MR causal effect estimates could be challenged in case of the MR assumptions are violated. The source of biases could attribute to the weak effects arising because of polygenicity, the presentence of horizontal pleiotropy and other biases, e.g., selection bias. In this thesis, we firstly propose a Bayesian weighted Mendelian randomization for causal inference, which takes into account weak effects and violation of MR assumptions due to pleiotropy. Based on the framework of BWMR, we further develop a method for correction of selection bias in MR analysis. We evaluate the performance of our methods through comprehensive simulations and real data analysis, demonstrating advantages over competitors. With the increasing availability of GWAS summary, our methods are believed to be of great practical value.

  
  

• Wednesday, 21st August, 2019

  Title:	Super-resolution in Imaging High Contrast Targets from the Perspective of Scattering Coefficients
  Speaker:	Prof Yat Tin CHOW, Department of Mathematics, University of California, USA
  Time/Place:	11:00  -  12:00 FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist University
  Abstract:	"In this talk, we are concerned with an acoustic / transverse electric (TE) / transverse magnetic (TM) inverse scattering problem, and mathematically analyze the experimentally observed phenomenon of super-resolution in imaging high-contrast targets based on the concept of scattering coefficient. We first introduce the notion of scattering coefficients for heterogeneous media and analyse this entity to help us understand the exponential ill-posedness of the problem at a fixed frequncy. Based on this novel concept of scattering coefficients, sensitivity and resolution analysis are performed to mathematically assess the reconstruction quality and justify the super-resolution phenomenon in imaging some high contrast targets. We illustrate our main findings with a variety of numerical examples. These findings may help in developing resonant structures for resolution enhancement. This talk is based on joint works with Habib Ammari (ETH) and Jun Zou (CUHK)."

  
  

• Wednesday, 28th August, 2019

  Title:	PhD Oral Exam: Some New Developments in Data Transformation and Meta-analysis with Small Number of Studies
  Speaker:	Mr LIN Enxuan, Department of Mathematics, Hong Kong Baptist University, HKSAR
  Time/Place:	10:30  -  12:30 FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist University
  Abstract:	In this presentation, we focus on the three critical issues for the statistical parts in meta-analysis. The first issue is how to convert OR to RR in the case-control study. In view of this, we establish a new formula for this transformation to fulfill the gap. The performance of the new method will be examined through simulations and real data analysis. Our method and formulas can not only handle meta-analyses with different effect sizes, but also offer some insights for medical researchers to further understand the meaning of OR and RR in both cohort and case-control studies. Another issue is the model selection in meta-analyses with few studies. we propose to further improve the estimation accuracy of the average effect in the fixed-effects model by assigning different weight for each study as well as fully utilizing the information in the within-study variances. Through theory and simulation, we demonstrate that the fixed-effects model can serve as the most convincing model for meta-analysis with few studies. And most importantly, with a total of three candidate models, we expect that meta-analysis can be conducted more flexibly, more meaningfully, and more accurately. The third issue is that most existing methods for the heterogeneity measurement were derived under the assumption of known within-study variances. In practice, however, a direct use of the reported within-study variance estimates may largely reduce the power of the tests and also lower the accuracy of the estimates, especially when the sample sizes in some studies are not sufficiently large. To overcome this problem, we propose a family of shrinkage estimators for the within-study variances that are able to borrow information across the studies, and derive the optimal shrinkage parameters under the Stein loss function. We then apply the new estimates of the within-study variances to some well known methods for measuring heterogeneity. Simulation studies and real data examples show that our shrinkage estimators can dramatically reduce the estimation bias and hence improve the exiting literature.

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