Direct Numerical Solutions to Stochastic Differential Equations with Multiplicative Noise

Area of Science:

• Computational physics
• Numerical analysis
• Nonlinear dynamics

Background:

• Classical stochastic differential equations (SDEs) with multiplicative noise present significant computational challenges.
• Traditional methods often rely on trajectory averaging, which can be computationally expensive and may not accurately capture critical phenomena like bifurcations.
• Path integral solutions from quantum mechanics offer alternative theoretical frameworks.

Purpose of the Study:

• To develop a novel numerical method for solving classical SDEs with multiplicative noise.
• To avoid the computational burden and potential inaccuracies of trajectory-averaging methods.
• To accurately simulate systems approaching bifurcation regimes.

Main Methods:

• Inspired by quantum relaxation path integral solutions.
• Development of a trajectory-averaging-free numerical method for classical SDEs.
• Simulation of a classical oscillator coupled to non-Markovian noise.
• Acceleration of the method using tensor factorization techniques.

Main Results:

• The developed method accurately estimates the transition into the bifurcation regime of a classical oscillator.
• The method significantly outperforms trajectory-averaging simulations in accuracy.
• The computational cost is orders of magnitude lower compared to traditional methods.

Conclusions:

• The novel numerical method provides an efficient and accurate approach for simulating classical SDEs with multiplicative noise.
• This method is particularly advantageous for studying systems near critical points, such as bifurcations.
• Tensor factorization techniques offer substantial computational speedups for these simulations.