Research at FAST

17 Department of Applied Mathematics Numerical Analysis and Scientific Computing Dr Zhonghua Qiao has significantly contributed to numerical analysis and scientific computing, especially numerical methods for partial differential equations. He has undertaken a systematical study of the numerical approximation for the Cahn-Hilliard type equation, a key component of phase-field modeling. He has designed and analyzed semi-implicit unconditionally energy stable numerical methods for solving the Cahn-Hilliard equation in phase field models, for which the stability proof does not rely on assumptions of the nonlinearity or a priori bounds of the numerical solution. He has also significantly contributed to the design of efficient adaptive methods. In 2018, Dr Qiao received Hong Kong Mathematical Society Award for Young Scholars. Mathematical Analysis of Chemotaxis Models: New Ideas on Traveling Waves and Well-posedness The (nonlinear) stability of traveling wave solutions of the Keller-Segel model, describing chemotactic movement, is a long-standing open question. Dr Zhian Wang, with his collaborators, successfully established the stability of traveling wave solutions of the Keller-Segel model by the Cole-Hopf transformation when the consumption rate is linear. This is the first and only stability result for the stability of traveling wave solutions of the singular Keller-Segel system. Our Cole-Hopf transformation stimulated a great deal of follow-up research work and our research results have been cited in many papers. Dr Wang received the 2019 Hong Kong Mathematical Society Award for Young Scholars. Our Achievements Regarding the minimal Keller-Segel system, Dr Wang showed that a solution with subcritical mass will asymptotically converge to the unique constant equilibrium if the domain is radially symmetric. This partially answers an open question for the asymptotic behaviors of minimal Keller-Segel model solutions with subcritical mass within radially symmetric domains. By introducing skillful variable transformations, the PolyU team obtained the first results on the well-posedness of the Attraction-Repulsion Keller-Segel system and found the competition index determining the global boundedness, or blow-up, of solutions. This competition index number provides a basic tool to exploit the Attraction-Repulsion type Keller-Segel system and subsequently its variants. Our team has also derived a new Gagliardo-Nirenberg type inequality and developed the idea of weighted energy estimates to handle degeneracy in density-suppressed motility. This offers an approach to study other degenerate models with density-suppressed motility like prey-taxis models. Stochastic Optimal Control with Partial Information and Related Maximum Principle This work belongs to the fundamentally theoretical research on the optimal control theory of stochastic systems and its application. The major strategic requirements for risk management and stochastic control closely focus on theoretical problems that restrict the accurate measurement and effective control of financial risks. Partially observable information in the field of stochastic system optimization control has been a long-standing yet challenging theoretical problem, especially the forward-backward stochastic system. AMA has made a series of important and fundamental breakthroughs in theoretical problems such as optimal control, differential game, and optimal filtering of stochastic systems with related topics. Dr Jianhui Huang and Dr Xun Li received one of the 2018 Ministry of Education Higher Education Outstanding Scientific Research Output Awards, the Natural Science Award (2 nd Class) based on their studies, with their research collaborators (Dr Guangcheng Wang and Dr Jingtao Shi from Shandong University). In particular, this award is administrated by Ministry of Education of the People’s Republic of China.