This project will focus on the development and analysis of numerical methods to probe the complex energy landscapes related to various interface and defect problems, such as microstructures in materials undergoing phase transitions, quantized vortices in geometrically frustrated configurations, and so on. The associated stochastic dynamics and hydrodynamics, as well as their numerical approximations will also be investigated. He will also study robust and adaptive algorithms that are useful in the statistical analysis of the underlying structures and important features. These research issues, on one hand, are driven by practical applications through collaborations with other scientists, and on the other hand, also motivate new studies of mathematical subjects ranging from geometry and topology to numerical and stochastic analysis. The research to be carried out is of interdisciplinary nature, encompassing subjects like computational mathematics, physics, information, materials and biological sciences. There are many mathematical and numerical challenges involved in the research such as the understanding of the collective behaviors of families and paths of solutions of nonlinear PDEs and stochastic dynamics, and the exploration of the hidden structures and statistics in the simulated results.
The research on the algorithmic development and numerical simulations can help enhancing the capability as well as the predictive power of scientific computations, which has potentially significant scientific, social and economic impact and is one of the top research priorities internationally. Meanwhile, interfaces and defects are also ubiquitous in nature which play fundamental roles in many aspects of physical and biological systems. A better mathematical and computational understanding of interfaces and defects, especially in a stochastic setting, will enrich the scientific knowledge base, which in turn may aid the efforts by physicists, materials scientists and engineers in discovering new materials with desirable properties and in developing new scientific devices and commercial instruments. This project will also provide valuable interdisciplinary research opportunities for the future generation of workforce and researchers. With an emphasis on the TEAMS (Training in Experiments, Analysis, Modeling and Simulations) spirit, young students can be better prepared to conduct interdisciplinary research in their career.
This project is focused on the development and analysis of numerical methods to probe the complex energy landscapes related to various interfaces and defects, such as microstructures in materials undergoing phase transitions and so on. The associated stochastic dynamics and hydrodynamics, as well as their numerical approximations will also be investigated. We also study robust and adaptive algorithms that are useful in the analysis of the underlying structures and important features. Much progress have been made in the last three years, resulting in over two dozens of publications. Among our major research findings, a particular breakthrough is on the mathematical understanding and numerical analysis of a new transition state (saddle point) search algorithm (we call it shrinking dimer dynamics) and its variants. The new algorithms are closely connected to a number of other more popular methods used in practice. The effectiveness of our methods has been demonstrated for a number of applications including critica nucleations. Our theory provided natural analogy between saddle point search and equilibrium computation which will open up new research directions in this field. We also published works, including a paper in SIAM Review, to develop rigorous mathematical foundation to study nonlocal models which are getting more and more important, especially for problems involving multiphysics and multiscales. The new noolocal calculus framework we have proposed also provides new language to formulate nonlocal models and leads to new approaches to analyze and solve nonlocal problems and to do multiscale coupling. Our research findings have a broad impact beyond mathematics. For example, exploring energy landscape and modeling rare events are very important in many scientific fields. Our development of shrinking dimer dynamics may have a broad impact in such domains. Also, our paper on computing minimal surfaces appeared in SIGRAPH 2012, one of the premier conference in computer graphics, which may provide new approaches to geometric modeling. We have engaged young students and scholars in our research activities. Four mathematics ph.d students have graduated. We have also supervised a post-doc and a couple of visiting graduate students. We also incorporated research findings in classroom teaching and lectures at summer schools for graduate students such as the 2013 IMA summer school on geometric flows and deformation in physiological processes.
