Stochastic differential equations (SDEs) are widely used in science and engineering in order to model systems where random effects play a significant role. They often require sampling a corresponding invariant distribution, that is, the computation of averages with respect to a defined probability density that is a function of many variables. Such sampling problems arise in many application areas, including molecular dynamics and Bayesian sampling techniques used in emerging machine learning applications. However, the design of efficient and accurate numerical algorithms for such problems is highly nontrivial, since they are often in high dimensional spaces (or large-scale dataset in data science applications) and some of them are even rare events, moreover the largest timestep usable is often limited in order to maintain numerical stability. This is a challenging but active field of research.
Based on the idea of splitting the vector field of the stochastic system in such a way that each subsystem can be solved exactly in terms of distributions, recent progress has been made in the setting of (adaptive) Langevin dynamics and related dissipative particle dynamics (DPD), a coarse-graining technique introduced for simulating complex hydrodynamic behaviour (e.g., DNA and blood flow) at a mesoscopic level that is not accessible by conventional molecular dynamics. The aim of this project is to further explore the optimal design of numerical algorithms in more challenging settings (e.g., the generalized Langevin dynamics that incorporates memory effects and other systems with a configuration/parameter-dependent damping term) as well as in large-scale machine learning applications (e.g., stochastic gradient Markov chain Monte Carlo methods). Moreover, it would be very interesting to explore whether or not there exists a general framework to construct numerical algorithms with desired properties.
The project will involve rigorous mathematical analysis (providing comprehensive training in numerical analysis, scientific computing, and Bayesian sampling techniques) and implementation of the developed algorithms as well as extensive numerical experimentation, thus equipping the student with highly desirable skills for working in either industry or academia.
We are looking for an enthusiastic and highly-motivated graduate with
- a first class degree in Mathematics or a closely related discipline with strong mathematical component (Master’s level or equivalent);
- a solid background in numerical methods/analysis of SDEs;
- excellent programming skills;
- good communication skills (oral and written).
Good knowledge of molecular dynamics, statistical mechanics as well as basic understanding of machine learning techniques and software will be advantageous.
The application procedure and the deadlines for scholarship applications are advertised at https://www.birmingham.ac.uk/schools/mathematics/phd/phd.aspx
Informal inquiries should be directed to Dr Xiaocheng Shang, email: [email protected]
Funding Notes for Chinese candidates:
The China Scholarship Council (CSC) Scholarship: https://www.csc.edu.cn/chuguo
China Scholarship Council (CSC) PhD Scholarships Programme at the University of Birmingham: https://www.birmingham.ac.uk/postgraduate/funding/china-scholarship-council-phd.aspx
PhD Placements and Supervisor Mobility Grants China-UK: https://www.britishcouncil.cn/en/programmes/education/higher/opportunities/phd