We propose an algorithm for an optimal adaptive selection of points from the design domain of input random variables that are needed for an accurate estimation of failure probability and the determination of the boundary between safe and failure domains. The method is particularly useful when each evaluation of the performance function g(x) is very expensive and the function can be characterized as either highly nonlinear, noisy, or even discrete-state (e.g., binary). In such cases, only a limited number of calls is feasible, and gradients of g(x) cannot be used. The input design domain is progressively segmented by expanding and adaptively refining mesh-like lock-free geometrical structure. The proposed triangulation-based approach effectively combines the features of simulation and approximation methods. The algorithm performs two independent tasks: (i) the estimation of probabilities through an ingenious combination of deterministic cubature rules and the application of the divergence theorem and (ii) the sequential extension of the experimental design with new points. The sequential selection of points from the design domain for future evaluation of g(x) is carried out through a new learning function, which maximizes instantaneous information gain in terms of the probability classification that corresponds to the local region. The extension may be halted at any time, e.g., when sufficiently accurate estimations are obtained. Due to the use of the exact geometric representation in the input domain, the algorithm is most effective for problems of a low dimension, not exceeding eight. The method can handle random vectors with correlated non-Gaussian marginals. The estimation accuracy can be improved by employing a smooth surrogate model. Finally, we define new factors of global sensitivity to failure based on the entire failure surface weighted by the density of the input random vector.
翻译:我们提出了一种算法,用于从输入随机变量的设计域中自适应选择点,以便精确估计故障概率并确定安全区域和故障区域之间的边界。当性能函数 g(x) 的每次评估非常昂贵,并且该函数可被描述为高度非线性、嘈杂或甚至离散状态(例如二进制)时,该方法特别有用。在这种情况下,只有有限数量的调用是可行的,且 g(x) 的梯度无法使用。输入设计域通过扩展和自适应细化网格状无锁几何结构来逐步分割。所提出的基于三角化的方法有效地结合了模拟和逼近方法的特点。该算法执行两个独立的任务:(i) 通过确定性续积分规则与散度定理的妙用,通过计算每个局部区域内的故障概率估计是准确的,(ii) 序列扩展实验设计以将新点加入到域中,扩宽执行。从设计域中选择未来评估 g(x) 的点的顺序是通过新的学习函数完成的,该函数通过最大化与局部区域相对应的概率分类的瞬时信息增益来完成。延伸可以在任何时候停止,例如获得足够准确的估计时。由于在输入域中使用精确的几何表示,因此该算法对最多八个维度的问题最为有效。该方法可以处理随机向量具有相关的非高斯边缘。通过使用平滑代理模型可以提高估计精度。最后,我们基于整个故障面和输入随机向量密度加权定义新的全局敏感性因子。