Michela Ottobre | Heriot-Watt University | United Kingdom
Quoting Niels Bohr, “it is hard to make predictions, especially about the future”; and one may add “even more if it is about the distant future”. We experience the difficulty of making long-term predictions in everyday life (think of weather forecasts, life expectancy for cancer patients, market behaviour etc). Beyond underlying modelling issues, our ability to make predictions relies either on the use of probabilistic/statistical approaches, which aim at quantifying the likelihood of possible scenarios (chance of rain/sun), or on numerical simulation methods. In the latter case one tries to “look into the future” by numerically approximating equations which are intended to model a given system (e.g. the spread of cancer cells or of disease in a population). Very often a combination of such approaches is used, with each of them having their strengths and demerits. In the case of numerical simulations, which are the focus of this minicourse, one of the big issues is that the numerical error typically increases in time, i.e. it increases if the simulation is run for longer. This makes long-term simulations (and predictions based on such simulations) less reliable. This minicourse will be structured in three parts:
* Part one. Introduction to numerical methods for Stochastic Differential Equation, including classical results on ergodicity (and loss of it) of numerical schemes. We will focus on numerical schemes for Langevin-type dynamics, which are also broadly used in the context of Markov Chain Monte Carlo (MCMC) algorithms .
* Part two. Following up from part one, we will discuss issues with simulating the long-time behaviour of SDEs and how some Markov Chain Monte Carlo Schemes can be seen as ameliorating some of such issues. (Time allowing, we will show how choosing appropriate numerical integrators can ameliorate the so-called curse of dimensionality in MCMC schemes.)
* Part three. Research perspectives: can we build numerical schemes that approximate a given dynamics, uniformly in time ?