Instanton based importance sampling for rare events in stochastic PDEs.

Area of Science:

• Nonlinear dynamics
• Statistical physics
• Computational methods

Background:

• Nonequilibrium systems governed by stochastic partial differential equations (SPDEs) often exhibit rare and large fluctuations.
• Accurately sampling these extreme events is computationally challenging for conventional methods.
• Understanding these fluctuations is crucial for predicting system behavior.

Purpose of the Study:

• To introduce a new, efficient method for sampling rare and large fluctuations in SPDEs.
• To accurately quantify the entire probability density function (PDF) of system observables, including extreme tails.
• To demonstrate the method's efficacy on a relevant physical model.

Main Methods:

• Utilized the instanton formalism, a saddle-point approximation within path integral formulations.
• Developed an alternative SPDE incorporating instanton solutions to constrain dynamics to extreme configurations.
• Applied a Girsanov theorem-based reweighting procedure to recover the full PDF.

Main Results:

• The instanton-based sampling method was successfully demonstrated on the one-dimensional Burgers equation.
• The proposed method significantly outperformed direct numerical simulations and Hybrid Monte Carlo methods.
• Accurate quantification of the velocity gradient PDF, from core to far tails, was achieved.

Conclusions:

• The instanton-based approach provides a powerful and efficient tool for studying rare events in SPDEs.
• This method offers a significant advantage over existing techniques for characterizing extreme fluctuations.
• The framework enables a comprehensive understanding of system dynamics, particularly in the tails of probability distributions.