Scalable methods for computing sharp extreme event probabilities in infinite-dimensional stochastic systems

• DOI: 10.1007/s11222-023-10307-2
• Publication Date: 2023-10-13T18:02:43Z
• Authors: Timo Schorlepp; Shanyin Tong; Tobias Grafke; Georg Stadler
• Keywords: FOS: Computer and information sciences; Optimization and Control (math.OC); Probability (math.PR); FOS: Mathematics; 0101 mathematics; Statistics - Computation; Mathematics - Optimization and Control; 01 natural sciences; Mathematics - Probability; Computation (stat.CO)

Abstract We introduce and compare computational techniques for sharp extreme event probability estimates in stochastic differential equations with small additive Gaussian noise. In particular, we focus on strategies that are scalable, i.e. their efficiency does not degrade upon temporal possibly spatial refinement. For purpose, extend algorithms based the Laplace method estimating of an to infinite dimensional path space. The limiting exponential scaling using a single realization random variable, large deviation minimizer. Finding this minimizer amounts solving optimization problem governed by equation. estimate becomes when it additionally includes prefactor information, which necessitates computing determinant second derivative operator evaluate integral around present approach dimensions Fredholm determinants, develop numerical compute these determinants efficiently high-dimensional systems arise discretization. also give interpretation process covariances transition tubes. An example model problem, provide open-source python implementation, is used throughout paper illustrate all methods discussed. To study performance methods, consider examples partial equations, including randomly forced incompressible three-dimensional Navier–Stokes equations.