Abstract: We formally derive the path-kernel formula for the linear response, or the parameter-derivative of averaged statistics, of SDEs. The parameter controls the diffusion coefficients. Our formula combines and extends the path-perturbation and the kernel method (also called likelihood ratio or Cameron-Martin-Girsanov) by extending Bismut-Elworthy-Li's compensation idea to perturbation on dynamics. It tempers the unstableness by gradually moving the path-perturbation to hit the probability kernel. It does not assume hyperbolicity. We prove by direct comparison of bundles of paths across different parameters. Based on the new formula, we derive a pathwise sampling algorithm for linear responses and demonstrate it on the 40-dimensional Lorenz 96 system with noise; this numerical example is difficult for all other algorithms.

Bio: I compute the parameter-derivatives of averaged observable of in a dynamical system, which is typically chaotic / high-dimensional / small-noise. I gave the fast response method, which enables sampling linear responses for hyperbolic systems, so people can finally work in high-dimensions. I gave the path-kernel method, which enables sampling linear responses for unstable SDEs with multiplicative noise. More specifically, the tools we developed

1. Fast response formula, equivariant divergence formula, and algorithms.
2. Ergodic and foliated kernel differentiation (or likelihood ratio) method.
3. Adjoint theories and algorithms, backpropagation under gradient explosion.
4. Non-intrusive shadowing algorithms.
5. Path-Kernel method.
I am also interested in dynamical system and probability and their interaction with all fields, such as fluids, geophysics, inference, data assimilation, and machine learning.

Organizer: 邱凌云