The investigator develops efficient numerical schemes, software, and theoretical analysis of these schemes for the simulation of continuous-time discrete-state stochastic dynamic systems that exhibit "multiscale behaviour." Examples of such systems include chemically reacting molecular systems, traffic models, and markets. In principle such systems can be simulated exactly one "event" at a time. But the computational costs are often prohibitive for large systems. Continuous macroscopic models -- either stochastic or deterministic differential equaitons -- often prove to be valid approximations to such models and in that case several numerical methods exist for their efficient simulation. However, some systems exhibit "multiscale behaviour" in the sense that only parts of the system may be approximated by such macroscopic models while the remaining components need to be represented by a discrete and stochastic model. Motivating examples are found in intra-cellular gene transcription mechanisms, where the fluctuations of certain key molecular species present in very small numbers have a critical effect on the final state of the system. The investigator aims to develop numerical schemes and theory that target such multiscale systems. These schemes allow simulations to leap over several events while maintaining required level of accuracy. Computer simulation methods and their theory are well developed for systems, such as a space craft in motion, that follow a deterministic and continuous motion. However, in some complex dynamic systems, such as chemical reactions inside a cell, the behaviour of traders in a large market, and the internet traffic of data, the entities change in small jumps that occur randomly in time. The investigator develops important new tools for the efficient computer simulation of such dynamic phenomena. The impact of these tools is in the scientific computation of biological, social, and economic systems. This project also provides opportunities for graduate and undergraduate students to develop their skills in modeling and simulation of such phenomena, which are useful in financial, pharmaceutical, and biotechnology industries.
