# Numerical computation of rare events via large deviation theory

## TL;DR

This paper reviews numerical algorithms based on large deviation theory for computing rare event probabilities, discussing their implementation, challenges, and extensions across various stochastic systems.

## Contribution

It introduces new generalizations and improvements to minimum action methods for rare event computation and explores their integration with importance sampling techniques.

## Key findings

- Algorithms effectively compute rare event probabilities in diverse systems.
- Handling degenerate or multiplicative forcing presents specific challenges.
- Extensions to non-Gaussian noises and infinite-dimensional systems are feasible.

## Abstract

An overview of rare events algorithms based on large deviation theory (LDT) is presented. It covers a range of numerical schemes to compute the large deviation minimizer in various setups, and discusses best practices, common pitfalls, and implementation trade-offs. Generalizations, extensions, and improvements of the minimum action methods are proposed. These algorithms are tested on example problems which illustrate several common difficulties which arise e.g. when the forcing is degenerate or multiplicative, or the systems are infinite-dimensional. Generalizations to processes driven by non-Gaussian noises or random initial data and parameters are also discussed, along with the connection between the LDT-based approach reviewed here and other methods, such as stochastic field theory and optimal control. Finally, the integration of this approach in importance sampling methods using e.g. genealogical algorithms is explored.