Asymptotic Analysis and Rare Event Simulation for Failure Probabilities in Discrete Random Media

The problem of material failure is of considerable importance in several applications. We will analyze a discrete atom chain model as a means of studying a material failure problem in a random medium. For different assumptions on the atomistic interaction potential, we determine the conditions necessary for material failure, and conclude failure may only occur in the event of a large deviation in the random model parameters. This observation is then used to derive asymptotic bounds on the probability of failure. Furthermore, we use our theoretical results to motivate the development of an importance sampling algorithm to calculate rare failure probabilities with greater efficiency than standard Monte Carlo methods.