Adaptive Monte Carlo sampling
See here for details: “Adaptive Multilevel Monte Carlo for Probabilities”
Multilevel Path Branching
See here for details: “Multilevel Path Branching for Digital Options”.
Projects for prospective PhD students
Statistical Control of the Ecological Risks of Fisheries
Fishing competes with predators, such as birds and seals, for resources and might impact their populations. This PhD project will use state-of-the-art statistical methods to analyse the dynamics involved in fisheries and develop new models and tools to manage the ecological and financial impacts of fishing. The analysis of ecosystem dynamics will make use of extensive data including remote sensed environmental variables and time series of bird, mammal and fish population and performance estimates. The results of this analysis will then be used in a simulation framework to develop and test feedback methods for managing the fisheries to control the risks to marine ecosystems while maintaining economic benefits of fishing.
Hierarchical Methods for Stochastic Partial Differential Equations
Partial Differential Equations (PDEs) are important versatile tools for modelling various phenomena, like fluid dynamics, thermodynamics, nuclear waste, etc… Stochastic Partial Differential Equations (SPDEs) generalize PDEs by introducing random parameters or forcing. One is then interested in quantifying the uncertainty of outputs of such models through the computations of various statistics. Accurate computations of such statistics can be costly as it requires fine time- and space-discretization to satisfy accuracy requirements. Several hierarchical methods were developed to address such issues and applied successfully to Stochastic Differential Equations (SDEs) and in this project we will extend these works to deal with the more complicated SPDEs.
Efficient computation of Rare-risk measures
Certain rare events have high cost, both humanitarian and financial, which make them significant events that industries and governments must plan for. Taking measures to reduce or mitigate the risks of such events is the goal of risk management which requires accurate assessment of such risks. This project’s goal is to speed up computations of accurate risk measures of rare events to ensure effective risk management. The goal will be achieved by developing novel computational methods which utilize approximation properties of the underlying stochastic models and which are based on Monte Carlo and random sampling methods which are easily parallelizable and fully exploit the increased availability of computational resources.
Hierarchical Methods for Chaotic Systems
Chaotic systems appear in weather, ocean circulation and climate models, and incur enormous computational cost with currently available methods due to accuracy requirements imposing slow time-stepping and fine space-discretization. In this project, we will develop hierarchical methods to speed up uncertainty quantification (UQ) of such systems which will allow practitioners to conduct more thorough statistical studies that will ultimately result in better decision making.