Joint UM-HKBU Workshop
on Mathematics and Statistics

In this talk, an adaptive and high order framework of finite volume solutions for steady Euler equations will be introduced. The algorithm consists of two main parts, i.e., a Newton iteration for the linearization of the Euler equations, and a geometrical multigrid iteration for the linear solver. Several numerical issues such as solution reconstruction, curved boundary, and regularization of the system will be introduced. To improve the efficiency, h-adaptive method is studied, and several kinds of error indicators are tested. Numerical results show the effectiveness of our method successfully.

When solving optimal control problems over a long time horizon, one can introduce additional parallelism in time by subdividing the time horizon into smaller, non-overlapping time intervals and by solving these subproblems in parallel. If the intermediate state and adjoint between time intervals are known exactly, this procedure yields the exact solution. Thus, the problem reduces to solving a nonlinear system in these intermediate states, which are related via certain propagation operators. In this talk, we present a parareal approach for solving this nonlinear system: here, the global problem is approximated by a simpler one using coarse propagators, while the fine propagation is performed in parallel over different time intervals. One then iterates until the intermediate states are consistent across time intervals. Unlike parareal for initial value problems, the coarse problem still contains a forward-backward coupling, but it is much cheaper to solve than the global fine problem. We analyze the convergence of the new method for a model linear problem and illustrate its behaviour numerically for nonlinear problems in which the control enters as an additive source term. (Joint work with Martin J. Gander and Julien Salomon)

Difference-based methods do not require estimating the mean function in nonparametric regression and are therefore popular in practice. In this paper, we propose a unified framework for variance estimation that combines the linear regression method with the higher-order difference estimators systematically. The unified framework has greatly enriched the existing literature on variance estimation that includes most existing estimators as special cases. More importantly, the unified framework has also provided an ingenious way to solve the challenging difference sequence selection problem that remains a long-standing controversial issue in nonparametric regression for several decades. Using both theory and simulations, we recommend to use the ordinary sequence in the unified framework, no matter if the sample size is small or if the signal-to-noise ratio is large. Finally, to cater for the demands of the application, we have developed a unified R package that integrates the existing difference-based estimators and the unified estimators in nonparametric regression and will make it freely available in the R statistical program http://cran.r-project.org/web/packages/.

We shall prove that the entropy production rate of a stochastic process in magnetic field satisfies a large deviation principle and its fluctuation satisfies Cohen-Gallavotti symmetry.

Nonlinear programming problems are a class of important optimization problems and there are no general/unified solvers for them. To efficiently solve a nonlinear programming problem, we generally need to consider the model meticulously in algorithmic design and analyze the theoretical issues such as the convergence carefully. I will present some of my own understandings of how to solve nonlinear programming problems, including how to understand the KKT conditions, how to understand some popular models via the variational inequality perspective, and how to develop application-tailored algorithms based on the well-known augmented Lagrangian method.

In this talk, I will briefly present the ideas of variational models for image recovery and segmentation.