The soft Monte Carlo method

Abstract

This study introduces a novel probability concept, random probability density function (PDF), as an efficient alternative of the random sampling for probability/reliability analysis of multivariate problems. To this end, a solution is proposed for drawing 1-D random PDFs according to the joint PDF of random variables. It is shown that using this approach, few random PDFs could be used instead of millions of random samples for covering the sample space in probabilistic analysis. By drawing PDFs instead of samples, the expected value of multivariable functions can be represented as the expected value of several one-variable probability integrals and therefore, any capable 1-D integration technique can be henceforth used for accurate statistical moment estimation and reliability analysis of multivariable problems. The law of large numbers proves the correctness of the proposed approach and therefore, achieving correct solutions of complex probability and reliability problems will not be hereafter restricted to only using crude Monte Carlo simulation. Besides, a novel probability density simulator is proposed that can attain probability density and moments of multivariate functions in an efficient manner different from existing approaches. Accurate and efficient probability/reliability analyses of complex problems with small failure probabilities confirm that the proposed method can be a reliable and efficient alternative of the crude Monte Carlo method, especially for extremely rare-events and time-consuming problems. For such problems, the suggested approach can serve as a benchmark against which new methods can be compared.