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Mathematics Institute

MA9M5 - Topics in Rare Events

Lecturer: Tobias GrafkeLink opens in a new window

Timetable:  

Mon 14:00–16:00, C1.06
Thu 12:00–13:00, B3.02

in Term 2. Lectures are held in-person and online on Teams, with the links given below.

Lecture Notes:

  1. Introduction: (pdf, booklet)
  2. Importance Sampling: (pdf, booklet)
  3. Importance Splitting: (pdf, booklet)
  4. Transition Path Theory (pdf, booklet)
  5. Large Deviation Theory (pdf, booklet)
  6. Advanced Topics (pdf, booklet)
  7. Appendix: (pdf, booklet)

All lecture notes so far: (pdf)

Lecture Dates and Notes:

The lectures will be held at the below dates in C1.06 (Mon) and B3.02 (Thu) in the Zeeman building and online.

We will discuss: Rare event sampling techniques, Monte Carlo estimates, Importance Sampling, Optimal Biasing, Importance Splitting and Cloning, Transition State and Transition Path Theory, Large Deviation Theory and its relation to Statistical Mechanis. In the lectures, these methods will be explained and applied to problems in engineering, chemistry, physics, climate, and others. The module is suitable and relevant for postgraduate students in applied mathematics, statistics, theoretical/computational physics, or theoretical/computational chemistry.

Assessment: Oral Examination 100%

Prerequisities

Assumed: undergraduate-level probability theory, stochastic processes
Useful: some background in programming (python, matlab, ...)

Content

Rare events are often relevant despite their low probability: Either because they have a disproportionately large impact (such as for earth quakes, stock market crashes, heat waves or floods), or because they are rare merely on their intrinsic scale, but ubiquitous in our everyday life (such as chemical reactions as overcoming an energy barrier, bitflips in communication networks or magnetic reversal in harddrives). Direct observations of rare events in complex systems, both by experiments or modelling, quickly become prohibitive if the events are too rare and are thus practically not analysable or observable. This prevents us from understanding their causes or even quantifying their probability.

There is a multitude of mathematical techniques that deal with exactly this situation, starting from large deviation theory in probability, over transition state theory from dynamical systems and path integral techniques from quantum field theory, to numerical rare event sampling techniques to get quantitative answers. These strategies are applied in fields as diverse as molecular dynamics, climate, finance, epidemiology, risk quantification and safety design, etc.

The aim of this module is to give an overview of rare event techniques from an applied mathematicians standpoint, connecting the theoretical foundations to concrete applications and algorithms. The module starts from the basics of stochastic processes to introduce transition path theory, large deviations theory, and importance sampling Monte Carlo methods. The explicit goal is however to demonstrate these techniques in practical problems. To this end, each theoretical area will be introduced in conjunction with numerical algorithms directly applied to areas of applied maths.

References

"Stochastic Methods", Gardiner, Springer (Berlin), ISBN 978-3-540-70712-7
"Stochastic Simulation. Algorithms and Analysis", Asmussen, Glynn, Springer (New York), ISBN 978-0-387-30679-7
"Stochastic Differential Equations", Oksendal, Springer (Heidelberg), ISBN 978-3-540-04758-2
"Random Perturbations of Dynamical Systems", Freidlin, Wentzell, Springer (Heidelberg) ISBN 978-3-642-25846-6
"Rare Event Simulation using Monte Carlo Methods", Rubino, Tuffin (Eds), Wiley (Chichester), ISBN 978-0-470-77269-0
"Introduction to Rare Event Simulation", Bucklew, Springer (New York) ISBN 0-387-20078-9
"Large Deviations Techniques and Applications", Dembo, Zeitouni, Springer (Heidelberg), ISBN 978-3-652-03310-0
"Large Deviations for Stochastic Processes", Feng, Kurtz, American Mathematical Society, ISBN 978-0-8218-4145-7