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The probability density evolution method (PDEM) provides a feasible approach for the dynamic response analysis of nonlinear stochastic structures. The key step in this regard is to solve a generalized density evolution equation (GDEE) in order to establish the probability density function (PDF). Previously, a finite difference method (FDM) has often been resorted to solve the GDEE. However, one may encounter the problem of mesh sensitivity in the application of FDM to the PDEM. To this end, a novel difference-wavelet method that can improve the finite difference result by means of a nonlinear wavelet density estimation method is proposed in the present paper. By exploiting the multi-resolution property of wavelet functions and by choosing the optimal scale at each instant, it is expected that the bothering mesh sensitivity issue in finite difference method can be overcome to some extent and a better probability density result can be obtained. In order to verify the proposed method, a single-degree-of-freedom (SDOF) oscillator and an 8-story frame structure are investigated in detail. The results show the notable superiority of the proposed method to finite difference method. [ABSTRACT FROM AUTHOR]

The key issue of the probability density evolution method (PDEM) is to solve a generalized density evolution equation (GDEE). Previously, the GDEE was solved in the framework of the point evolution method which is essentially a zero-order ensemble evolution method. In this paper, a first-order ensemble evolution method is proposed aiming at increasing the accuracy and robustness of the PDEM. The main idea of the proposed method is to incorporate information of standard deviation of each probability subdomain into the probability density evolution equation (PDEE) by introducing an ensemble velocity term. Compared with the point evolution method, the proposed method can truly reflect the fluctuation of a stochastic dynamic system. In order to estimate the ensemble velocity term accurately, a piecewise quadratic polynomial fitting method is also proposed. In addition, a GF-discrepancy based point selection method and a finite difference scheme that is total variation diminishing are adopted to solve the new PDEE. A single-degree-of-freedom oscillator, a Riccati equation and a 2-span 8-storey frame structure are investigated in detail to demonstrate the advantage of the proposed method over the original one. [ABSTRACT FROM AUTHOR]

The probability density evolution method (PDEM) synthesizing the change of probability measure (COM) provides a compatible framework for quantifying simultaneous aleatory and epistemic uncertainties. Nonetheless, when the effective support for the change of measure becomes narrow, the quantification error of the PDEM-COM may increase, and the PDEM-COM may even be invalid if the effective support is an empty set. To address this issue, in the present paper, the error analysis of the PDEM-COM is firstly explored, and the generalized F-discrepancy via the COM is utilized to measure the similarity of two different distributions. Then, an augmenting approach is proposed to improve the performance of the PDEM-COM by adding a number of computational simulations. The results indicate that, the PDEM-COM can still be relatively accurate as long as the effective support is sufficiently large, even when there is a significant shift in the probability measure. However, if the effective support turns to be extremely narrow, the error of hybrid uncertainty quantification via the PDEM-COM may be unsatisfied, but it can be significantly reduced by the suggested augmenting approach. Numerical examples are investigated to demonstrate the accuracy and efficiency of the proposed method. Some issues to be further studied are also outlined. • Error analysis of the PDEM-COM is studied. • Numerical illustration of the PDEM-COM for different changes of probability measure. • An augmenting approach is proposed to improve the PDEM-COM. • The accuracy and efficiency of the proposed method is validated by three engineering applications. [ABSTRACT FROM AUTHOR]