Timo Schorlepp (NYU Courant): Scalable extreme event estimation for stochastic differential equations via precise large deviation theory
Please note this event occurred in the past.
April 03, 2026 11:00 am - 12:00 pm ET
LGRT 1685
Abstract
Abstract:
Estimating the probability of extreme events is an important problem in many scientific and engineering disciplines, as they are often associated with rare system failures or catastrophes. Due to the lack of sufficient data, this is a difficult task, requiring specialized importance sampling or splitting methods. Asymptotically, rare event probabilities can also be estimated with a Laplace approximation, referred to as the second-order reliability method in engineering, or precise large deviation theory in mathematics. The approach involves (i) solving an optimization problem to identify the most likely realization of the random parameter that leads to a prescribed outcome, and (ii) calculating a determinant to account for Gaussian perturbations around the minimizer.
In this talk, I will discuss how to carry out both of these steps numerically in a scalable way for extreme events in stochastic differential equations (SDEs) with additive or multiplicative Brownian noise. This is an infinite-dimensional problem, with randomness induced by the Brownian motion, and commonly found in many applications. In particular, I will highlight the necessity to treat the determinant calculation in step (ii) correctly from an infinite-dimensional point of view to ensure scalability of the numerical method to high-dimensional state spaces. This leads to either a Fredholm or Carleman-Fredholm determinant computation, depending on whether the second variation of the noise-to-event map is trace-class or only Hilbert-Schmidt.
To illustrate these points, I will consider multiple examples of extreme event estimates for SDEs and stochastic partial differential equations, including the 1D viscous Burgers equation and the incompressible 3D Navier-Stokes equations with random forcing, as well as a 2D random advection-diffusion problem.
The talk is based on joint work with Shanyin Tong, Tobias Grafke, and Georg Stadler, published in Stat. Comput. 33(6), 137 (2023) and arXiv:2502.20114.
In this talk, I will discuss how to carry out both of these steps numerically in a scalable way for extreme events in stochastic differential equations (SDEs) with additive or multiplicative Brownian noise. This is an infinite-dimensional problem, with randomness induced by the Brownian motion, and commonly found in many applications. In particular, I will highlight the necessity to treat the determinant calculation in step (ii) correctly from an infinite-dimensional point of view to ensure scalability of the numerical method to high-dimensional state spaces. This leads to either a Fredholm or Carleman-Fredholm determinant computation, depending on whether the second variation of the noise-to-event map is trace-class or only Hilbert-Schmidt.
To illustrate these points, I will consider multiple examples of extreme event estimates for SDEs and stochastic partial differential equations, including the 1D viscous Burgers equation and the incompressible 3D Navier-Stokes equations with random forcing, as well as a 2D random advection-diffusion problem.
The talk is based on joint work with Shanyin Tong, Tobias Grafke, and Georg Stadler, published in Stat. Comput. 33(6), 137 (2023) and arXiv:2502.20114.