Introductory description

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 climate, finance, epidemiology, risk
quantification and safety design, etc.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. Introduction: Metastability and Extreme Events
2. Preliminaries in Stochastic Processes
   • Markov processes, SDEs
   • Algorithms: Stochastic Integrators
3. Transition State Theory and Transition Path Theory
   • Algorithms: Computing committor functions
   • Applications: Molecular Dynamics, Transition to Turbulence
4. Large Deviation Theory, Freidlin-Wentzell Theory
   • Algorithms: String Method, Minimum Action Method
   • Applications: Thermodynamics, Coulomb Gas, Rogue Waves
5. Monte-Carlo methods, Rare event simulation
   • Algorithms: Importance Sampling, Splitting methods
   • Applications: Option pricing, Epidemiology, Climate

Learning outcomes

By the end of the module, students should be able to:

• By the end of the module, students should be able to: - identify the occurrence of rare events for stochastic problems in applied maths - distinguish between different rare event sampling techniques for complex stochastic systems - demonstrate understanding of the probabilistic formulation of rare events - understand the theoretical foundation of transition path theory - understand the theoretical foundation of large deviation theory - demonstrate ability to implement rare event samplers for simple stochastic models - apply rare event techniques to complex stochastic systems

Indicative reading list

"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

Subject specific skills

By the end of the module, students should be able to:

• identify the occurrence of rare events for stochastic problems in
  applied maths
• distinguish between different rare event sampling techniques for
  complex stochastic systems
• demonstrate understanding of the probabilistic formulation of rare events
• understand the theoretical foundation of transition path theory
• understand the theoretical foundation of large deviation theory
• demonstrate ability to implement rare event samplers for
  simple stochastic models
• apply rare event techniques to complex stochastic systems

Transferable skills

• sourcing research material
• prioritising and summarising relevant information
• absorbing and organizing information
• presentation skills (both oral and written)