Mathematical Sciences at the University of Liverpool

Stochastics

Jinglai Li works on in scientific computing, computational statistics, uncertainty quantification, data science and their applications in various scientific and engineering problems. Specific topics include:

  • Bayesian inferences, inverse problems
  • Monte Carlo simulations
  • Risk analysis and failure probability estimation
  • Data assimilation and filtering methods
  • Optimisation/decision-making under uncertainty
  • Machine learning.

Kai Liu works in stochastic analysis. Specific topics of interest include:

  • Stability of infinite dimensional stochastic systems - such topics as stochastic control, population biology and turbulence can give rise to such systems. Stability is often regarded as the first characteristic to study.
  • Large deviations of stochastic systems - the study of probabilities of very rare events, which may be interpreted as catastrophies.

Relevant mathematical techniques include stochastic calculus; large deviation theory; calculus (ODE, PDE); functional analysis; measure theory.

Alexei Piunovskiy and Zhang Yi work on optimal control, including analysis of communication networks. Specific topics of interest include

  • The convex analytic approach to controlled Markov jump processes and Markov decision processes - this approach, dual to the dynamic programming approach, is convenient for the study of constrained problems.
  • Optimal policies in multicriteria problems.
  • Communication networks – in particular, Active Queue Management models of TCP connection. With a varying input stream of jobs, given that one can control the input intensity, the goal is to find an optimal control policy.
  • Controlled Markov processes with local transitions - such processes appear in Queuing Theory, Reliability, Epidemiology, etc. In some cases, the underlying model can be approximated by a diffusion, or a deterministic ordinary differential equation. It is then of interest to evaluate the accuracy of such approximations and to elaborate optimal control policies.

Relevant mathematical techniques include probability and random processes; calculus (ODE, PDE); functional analysis; optimisation; computer programming (Matlab, Maple).

Kamila Żychaluk works on non-parametric statistics, Bayesian statistics, and applications in biology. Specific topics of interest include:

  • Kernel, wavelet and local polynomial estimation - traditional statistical techniques estimate an unknown function by assuming some parametric form, eg Linear Regression, Quadratic Regression, Logistic Regression. Nonparametric methods aim to fit a quite general (smooth) function to the observed data.
  • Applications in biology - eg coral reef development, psychometric response functions.

Relevant mathematical techniques include nonparametric statistics; generalized linear models; bootstrap; Bayesian statistics; statistical computing (R, Matlab). Software for model-free estimation of a psychometric function may be found at http://www.modelfree.liv.ac.uk/index.html