Improvement to the discretized initial condition of the generalized density evolution equation

The probability density evolution method has been popularly used for various stochastic analysis of structural systems including the reliability analysis. The generalized density evolution equation is usually solved with its initial condition discretized as an impulse function. This, however, leads to oscillations close to the failure point of the structure in the final probability density function with large errors in the failure probability. A normal distribution is adopted in this paper to replace the conventional impulse function in modeling the initial condition. The assigned probability in the x-space direction is discretized at the initial time instant t = 0. The standard deviation, σ, in the normal distribution is optimized by the gradient descent method with Kullback-Leibler (KL) divergence as the objective function. The performance and accuracy of the proposed method are illustrated with four numerical examples, and the failure probabilities estimations are noted more accurate when the KL divergence of the probability density function is small.