Abstract
Numerical simulations are nowadays used on a daily basis in order to model physical, biological, chemical or financial systems. The question of how to quantify the uncertainty on the output of these simulations is crucial in order to use these models with confidence. The objective of these lectures, at the interface between applied probability, statistics and numerical analysis, is to introduce mathematical methods to model, characterize and analyse the uncertainty on the results. Various deterministic and stochastic sampling techniques will be introduced. We will also discuss metamodels to build surface response. These metamodels are useful for example to analyze the sensitivity of the output of a numerical model. Finally, rare event sampling techniques will also be presented.
Here is a preliminary schedule:
– Introduction and basics about Monte Carlo methods.
– Parameter estimation: Maximum Likelihood Estimator, Bayesian approaches.
– Building metamodels using regression, Gaussian process regression.
– Sensitivity analysis.
– Reduced basis techniques and proper orthogonal decomposition.
– Low rank approximation and greedy algorithms, proper generalized decomposition.
– Risk analysis: FORM/SORM methods, quantile estimation, extreme value theory.
– Monte Carlo methods for rare events: importance sampling and splitting methods.
Exercise sheets will be provided to practice the concepts introduced in the lectures. The assessment will be done through projects, with numerical experiments to be conducted on simple examples.
Schedule
Lectures will be held in the CDT room 402 between 11am-1pm on the following dates in 2020:
- Monday 13th Jan
- Monday 20th Jan
- Monday 27th Jan
- Monday 3rd Feb
- Monday 10th Feb
- Monday 24th Feb
- Monday 2nd March
- Monday 9th March (Nb. this lecture will take place between 4pm-6pm)
- Monday 16th March
- Monday 23rd March
Three Research Lectures will be held between 3pm-4pm on the following dates in 2020:
- Monday 03 Feb – CDT Room 401
- Monday 10 Feb – CDT Room 401
- Monday 24 Feb – Huxley Room 139
Speaker
Professor Tony Lelièvre is a world leader in the mathematical analysis of stochastic numerical methods, and their applications to molecular dynamics simulations, in particular. Among his achievements are new mathematical frameworks and algorithms for sampling multimodal measures, for sampling metastable stochastic trajectories, and for coarse-graining high dimensional problems. In terms of mathematical advances, his contributions lie at the interface between probability theory and the analysis of partial differential equations.