Application of high-credible statistical results calculation scheme based on least squares Quasi-Monte Carlo method in multimodal stochastic problems

Abstract

Solving stochastic problems based on multimodal distributions with high accuracy and efficiency is the focus of current research in the stochastic field. However, since exact statistical results in practical engineering problems are unknown, how to efficiently obtain statistical results with high credibility is of great importance. Therefore, a high-credible statistical results calculation scheme is proposed in this paper. First, the multimodal model is transformed into a weighted sum form of multiple unimodal models by Gaussian mixture model. Then, a high-credible statistical results calculation scheme is designed based on the reusability of low-discrepancy sequences in the quasi-Monte Carlo (QMC) method. The scheme calculates the coefficient of variation for the random response statistical results of the unimodal model across various sample ranges. When the coefficient of variation is less than the tolerance error, which indicates low fluctuation, the statistical results are deemed to have converged to the exact solution. In order to further accelerate the calculation efficiency of the proposed scheme, this paper proposes a least squares quasi-Monte Carlo (LSQMC) method to yield high accuracy statistical moments. Compared with the averageness weights in the QMC method, the non-averageness weights obtained by LSQMC method can more effectively reflect the numerical Characteristics of the random variables and the location distribution of the sampling points in the probability space. Finally, numerical examples have demonstrated that the high-credible statistical results calculation scheme based on the LSQMC method can obtain the high accuracy statistical moments with high efficiency in multimodal stochastic problems.