Failure probability estimation in high dimensional spaces

ABSTRACT In high dimensional spaces standard methods for failure probability estimation as FORM/SORM run into problems. First it is difficult to find all relevant design points, second the estimation methods using the curvatures are becoming inaccurate in such circumstances. The subset simulation method — used for some years — is often described as very efficient for this task, but unfortunately it gives incorrect results for complex problems, since it cannot distinguish between local and global minimum distance points. But this is an essential prerequisite for algorithms calculating failure probabilities. So its use is problematic. Here an idea for approximating failure probabilities in such circumstances is outlined. Instead of trying to find immediately estimates, it tries to keep control of the limit state function and its behavior. This is done by minimizing it on centered hyperspheres. Since these are compact sets it is in general possible to locate the global minima there with reasonable numerical effort. So in a nutshell, instead of finding design points by solving the equation system for the Lagrangian u + λȉg(u) = 0, g(u) = 0, one solves the system: ȉg(u) + μu = 0, |u|² – γ² = 0 for increasing values of γ until one reaches the failure domain. Then further methods shall be applied for example dimension reduction and response surfaces around the found design points to find structural description of the limit state surface here.