A control variate method for threshold crossing probabilities of plastic deformation driven by transient coloured noise

TL;DR

The paper addresses the challenge of computing rare threshold-crossing probabilities for plastic deformation under transient coloured noise. It introduces a hybrid framework that couples non-standard boundary PDEs for white-noise statistics with Monte Carlo simulations for coloured-noise systems, using a control variate to reduce variance. The main contributions are the derivation of PDEs with non-standard boundaries, the development of simple and practical optimal control variate estimators, and numerical demonstrations showing substantial variance reduction and computational efficiency across PSDs. The approach is particularly relevant for reliability assessment in earthquake engineering and can be extended to non-stationary excitation and higher-dimensional models.

Abstract

We propose a hybrid method combining partial differential equation (PDE) and Monte Carlo (MC) techniques to obtain efficient estimates of statistics for plastic deformation related to kinematic hardening models driven by transient coloured noise. Our approach employs a control variate strategy inspired by [CPAM, 75 (3), 455-492, 2022] and relies on a class of PDEs with non-standard boundary conditions, which we derive here. The solutions of those PDEs represent the statistics of models driven by transient white noise and are significantly easier to solve than the coloured noise version. Our approach uses a coupling between the white-noise-driven process and the coloured-noise-driven process, yielding a variance-reduced estimator through control variate techniques. We apply our method to threshold-crossing probabilities, which are used as failure criteria known as ultimate and serviceability limit states under non-stationary excitation. Our contribution provides solid grounds for such calculations and is significantly more computationally efficient in terms of variance reduction compared to standard MC simulations.

Figures (7)

• Figure A1: Description of the kinematic hardening bilinear elastoplastic model.(a) In the rheological model, $x(t)$ is the total displacement, that is the sum of the elastic $z(t)$ and plastic $\Delta(t)$ ones. The friction component prevents the elastic line from exceeding yield strength $S_\text{y}$. The damping term $\mathfrak{b}$ resists the total velocity $y$. (b) For the archetypal evolution of $\mathbb{F}(t)$ and $x(t)$ when $\mathfrak{b}=0$, starting from a neutral position, the behaviour starts being elastic (first elastic phase). Once $x$ exceeds $z_\text{max}$ with a positive velocity $y$, $z$ is frozen and $y$ becomes the rate of change of the plastic displacement $\Delta$ (plastic phase). Then, when the velocity becomes negative, $\Delta$ is frozen and $y$ becomes the rate of change of the elastic displacement (second elastic phase).
• Figure A2: Probabilities $P_1^0$ (left) and $P_2^0$ (right) as functions of the displacement threshold $X_f$ and plastic deformation threshold $\Delta_f$, respectively, taken from the numerical solution of the PDEs and from the Monte Carlo Simulations, with elasto-plastic stiffness ratios $a=0$, $a=0.5$, and $a=1$.
• Figure A3: Monte Carlo estimation of $P^\epsilon_1$ (for $a=0.5$ and $X_f=2$) using $\hat{I}_N^\varepsilon$ (standard MC), $\hat{J}_N^\varepsilon$ (simple control variate) and $K_N^\varepsilon$ (practical optimal control variate). For $\hat{J}_N^\varepsilon$ and $K_N^\varepsilon$, the expectation of the control variate $P_1^0$ is obtained by solving the PDE \ref{['eq:pdeBEPO1']}. This is done numerically by finite differences shown in Section \ref{['sec:fdm_pde']}. The driving coloured noise is related to PSD1 for the left column (a)-(b)-(c), and PSD2 for the right column (a')-(b')-(c'). The time discretization of the KBEPO dynamics together with the driving noise (coloured or white) is given in \ref{['eq:OU_EM_BEPO']}-\ref{['eq:colorednoise_EM_BEPO']}-\ref{['eq:whitenoise_EM_BEPO']} in the appendix.
• Figure A4: Monte Carlo estimation of $P^\epsilon_2$ (for $a=0.5$ and $\Delta_f=1$) using $\hat{I}_N^\varepsilon$ (standard MC), $\hat{J}_N^\varepsilon$ (simple control variate) and $K_N^\varepsilon$ (practical optimal control variate). For $\hat{J}_N^\varepsilon$ and $K_N^\varepsilon$, the expectation of the control variate $P^0_2$ is obtained by solving the PDE \ref{['eq:pdeBEPO2']}. This is done numerically by finite differences shown in Section \ref{['sec:fdm_pde']}. The driving coloured noise is related to PSD1 for the left column (a)-(b)-(c), and PSD2 for the right column (a')-(b')-(c'). The time discretization of the KBEPO dynamics together with the driving noise (coloured or white) is given in \ref{['eq:OU_EM_BEPO']}-\ref{['eq:colorednoise_EM_BEPO']}-\ref{['eq:whitenoise_EM_BEPO']} in the appendix.
• Figure A5: Monte Carlo estimation of $P^\epsilon_1$ (for $a=0$ and $X_f=2$) using $\hat{I}_N^\varepsilon$ (standard MC), $\hat{J}_N^\varepsilon$ (simple control variate) and $K_N^\varepsilon$ (practical optimal control variate). Results have been obtained with the same methodology as in Figure \ref{['fig:P1_PSD1_PSD2']}. The left column (a)-(b)-(c) corresponds to PSD1, and the right column (a')-(b')-(c') corresponds to PSD2.

Theorems & Definitions (5)

• Remark 1: Choice of PSDs
• Remark 2
• Remark 3
• Remark 4
• Remark 5