Fri 01 January 2021
By Tobias Grafke

Samplepath Large Deviations: Theory and Numerical Tools

Summary: Rare but extreme events in complex systems can often efficiently be described by samplepath large deviations: In the limit of some smallness-parameter approaching zero (such as temperature for chemical reactions, inverse number of particles for thermodynamic limits, or inverse timescale separation for multiscale systems), probabilities and most likely pathways of occurence can are readily accessible. For large and strongly coupled stochastic systems, such as climate, atmosphere, or ocean, the corresponding computations pose a huge numerical challenge. These method borrow heavily from field theory, and represent the rare probability as a path integral, necessitating the computation of instantons and fluctuation determinants. In this project, we adress these challenges, including (1) how to compute the large deviation minimizer (instanton) for large systems, (2) how to compute next-order prefactor corrections, and (3) how to deal with heavy-tailed distributions

Relevant publications

  1. T. Grafke, T. Schäfer, and E. Vanden-Eijnden, "Sharp Asymptotic Estimates for Expectations, Probabilities, and Mean First Passage Times in Stochastic Systems with Small Noise", Communications on Pure and Applied Mathematics 77 (2024), 2268 (link)

  2. T. Schorlepp, S. Tong, T. Grafke, and G. Stadler, "Scalable Methods for Computing Sharp Extreme Event Probabilities in Infinite-Dimensional Stochastic Systems", Statistics and Computing 33 (2023), 137 (link)

  3. T. Schorlepp, T. Grafke, and R. Grauer, "Symmetries and Zero Modes in Sample Path Large Deviations", J Stat Phys 190 (2023), 50 (link)

  4. T. Schorlepp, T. Grafke, and R. Grauer, "Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems", J. Phys. A: Math. Theor. 54 (2021), 235003 (link)

  5. M. Alqahtani, and T. Grafke, "Instantons for rare events in heavy-tailed distributions", J. Phys. A: Math. Theor. 54 (2021), 175001 (link)

  6. G. Ferré and T. Grafke, "Approximate Optimal Controls via Instanton Expansion for Low Temperature Free Energy Computation", SIAM Multiscale Model. Simul. 19(3) (2021), 1310 (link)

  7. T. Grafke, and E. Vanden-Eijnden, "Numerical computation of rare events via large deviation theory", Chaos 29 (2019), 063118 (link)

  8. T. Grafke, "String Method for Generalized Gradient Flows: Computation of Rare Events in Reversible Stochastic Processes", J. Stat. Mech. 2019/4 (2019), 043206 (link)