Integration of local and global surrogates for failure probability estimation

Abstract

This paper presents the development of an algorithm, termed the Global-Local Hybrid Surrogate (GLHS), designed to efficiently compute the probability of rare failure events in complex systems. The primary goal is to enhance the accuracy of reliability analysis while minimizing computational cost, particularly for high-dimensional problems where traditional methods, such as Monte Carlo simulations, become prohibitively expensive. The proposed GLHS builds upon the foundational work of Li et al., by integrating an adaptive strategy based on the General Domain Adaptive Strategy (Adcock et al.). The algorithm aims to approximate the failure domain of a given system, defined as the region in the input domain where the system transitions from safe to failure modes, described by a limit state surface. This failure domain is not explicitly known and must be learned iteratively during the analysis. The method employs a buffer zone, defined as the region surrounding the limit state surface. Within this buffer zone, Christoffel Adaptive Sampling is utilized to select new samples for constructing localized surrogate models, which are designed to refine the approximation in regions critical to failure probability estimation. The iterative process proceeds until convergence is reached. This results in a hybrid methodology that integrates a global surrogate to capture the overall trend with local surrogates that concentrate on critical regions near the limit state function. By adopting this strategy, the GLHS method balances computational efficiency with accuracy in estimating the failure probability.

Figures (13)

• Figure 1: Visualization for the iterative procedure in buffer zone domain learning and optimal sampling measure. Starts with computing a buffer zone using the surrogate. Next, we construct the hyperrectangle to resample in the next step, and finally, we construct a sampling measure to draw new samples.
• Figure 2: The first box represents the method described in the initialization \ref{['subsec:init']}. The second is the representation of the procedure that yields the first local surrogate $g_L^{(1)}(\bm{x})$ as described in \ref{['subsec:lscons']}. The last box represents the procedure to obtain the second local surrogate $g_L^{(2)}(\bm{x})$.
• Figure 3: Figure \ref{['fig:afailure']} is the representation for the optimum situation: the buffer zone $|g_S(\bm{x})|\leq 0$ fully encompasses the limit state surface, which is represented by the solid orange line. Figure \ref{['fig:bfailure']} represents the case where the buffer zone was not large enough and thus $g_T=0$ is only partially captured. In this situation, a bias will exist in the probability of failure estimation as the samples not captured by the buffer zone will not get updated.
• Figure 4: Figure \ref{['fig:cfailure']} represents the case where the buffer zone is not large enough and thus $g_T=0$ is not captured. In this situation, a bias will exist in the probability of failure estimation. Figure \ref{['fig:dfailure']} represents a scenario where the entire domain of failure is being updated by the local surrogate.
• Figure 5: Figure \ref{['fig:1D_S_T']} shows the ground truth function $g_T$ and the global surrogate $g_S$ for the 1D test case. The initial $m_0$ samples used to construct the global surrogate are marked with crosses. Figure \ref{['fig:1D_hyper']} illustrates the hyperplanes used to define the bounds for selecting the region of interest on the surrogate during resampling. The samples of $g_S$ that fall within the buffer zone are highlighted in orange.