In reliability analyis of an engineering structure, the combination of probabilistic and non-probabilistic methods has become an important issue. In particular, the uncertainty about the values concerning properties or parameters of an engineering structure can be modelled by a family of probability density functions parametrized by all t in a set T. The output of such a model typically consists of upper and lower probabilities, for example upper and lower failure probabilities.

The more interesting upper probability of failure is the solution of a (global) optimization problem

max {p(t): t in set T}

where p(t) is the failure probability for a fixed parameter value t. We estimate these function values p(t) using Monte Carlo simulation which means function evaluations (finite elements computations) for each of N sample points. This high computational effort has to be multiplied in addition by the number of function evaluations p(t) needed for solving the above optimization problem to find the optimal parameter t in T resulting in the upper probability.

For our numerical method we use importance sampling or reweighting techniques for two reasons:

(1) For computing the derivatives needed in the standard (global) optimization algorithms used.

(2) For reducing the number of parameter values t for which we need new samples in the optimization algorithm and in addition for reducing the sample sizes for these parameters. For this purpose we re-use the samples from parameters t of previous optimization steps taking the importance sampling ratios into account which leads to importance sampling on sets of a partition.

The efficiency of the method is analyzed by means of a moderate scale engineering structure.