Estimation of small failure probabilities based on thermodynamic integration and parallel tempering. (1st November 2019) - Xiao, Sinan; Reuschen, Sebastian; Köse, Gözde; Oladyshkin, Sergey; Nowak, Wolfgang
Highlights: A new efficient simulation method (TIPT) is proposed to estimate small failure probability. Multiple Markov chains are simulated simultaneously with collaboration. TIPT is suitable for high-dimensional problems and nonlinear problems with multiple failure domains. Test examples show the efficiency and stability of TIPT. Abstract: Estimating small failure probabilities of systems is one of the most challenging tasks in the reliability analysis of engineering systems. Classic Monte Carlo simulation estimates the expected value of a step function (indicator function for failure), which is computationally demanding. In this paper, a new simulation method based on thermodynamic integration and parallel tempering (TIPT) is proposed to estimate small failure probabilities. Through thermodynamic integration, we convert the problem into estimating expected values for a series of smooth functions, which results in a series of much simpler problems. To solve these simpler problems in parallel, we adopt a Markov chain Monte Carlo (MCMC) method called parallel tempering. Thus, each sub-problem has each own chain, and the chains communicate to assist each other. For high-dimensional problems (i.e. with a large number of uncertain variables), we use the preconditioned Crank-Nicolson version of MCMC in parallel tempering to increase efficiency. Three widely used analytic examples are used to test the efficiency of TIPT. Finally, TIPT is applied to estimate the failure probability in a benchmark problem for water distribution systems.