A PDE approach for the invariant measure of stochastic oscillators with hysteresis

2026-01-11

AI Summary

This study addresses the computational inefficiency of conventional Monte Carlo simulations in evaluating the invariant measure of stochastic bilinear oscillators with hysteresis. To overcome this limitation, the authors propose an alternative approach based on partial differential equations (PDEs). The method extends the stochastic elastoplastic framework from a two-dimensional perfectly plastic model to a three-dimensional bilinear model, thereby more accurately capturing complex hysteretic behaviors such as the Bauschinger effect. By constructing a Lyapunov function and leveraging tools from stochastic dynamical systems theory, the proposed PDE-based formulation efficiently computes the invariant measure. This enables accurate and efficient estimation of threshold-crossing rates—replacing the classical Rice formula—and serviceability probabilities, achieving significant gains in computational efficiency without sacrificing accuracy.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper presents a PDE approach as an alternative to Monte Carlo simulations for computing the invariant measure of a white-noise-driven bilinear oscillator with hysteresis. This model is widely used in engineering to represent highly nonlinear dynamics, such as the Bauschinger effect. The study extends the stochastic elasto-plastic framework of Bensoussan et al. [SIAM J. Numer. Anal. 47 (2009), pp. 3374--3396] from the two-dimensional elasto-perfectly-plastic oscillator to the three-dimensional bilinear elasto-plastic oscillator. By constructing an appropriate Lyapunov function, the existence of an invariant measure is established. This extension thus enables the modelling of richer hysteretic behavior and broadens the scope of PDE alternatives to Monte Carlo methods. Two applications demonstrate the method's efficiency: calculating the oscillator's threshold crossing frequency (providing an alternative to Rice's formula) and probability of serviceability.

Problem

• invariant measure
• stochastic oscillator
• hysteresis
• bilinear elasto-plastic
• white noise

Innovation

• PDE approach
• invariant measure
• stochastic oscillator
• hysteresis
• bilinear elasto-plastic