FORM/SORM, SS and MCMC: A Mathematical Analysis of Methods for Calculating Failure Probabilities

A basic problem in structural reliability is the calculation of failure probabilities in high dimensional spaces. FORM/SORM concepts are based on the Laplace method for the pdf of the failure domain at its modes. With increasing dimensions the quality of SORM decreases considerably. The straightforward solution would have been to improve the SORM approximations. However, instead of this, a new approach, subset simulation (SS) was championed by many researchers. By the proponents of SS it is maintained that SS does not suffer from the deficiencies of SORM and can solve high-dimensional reliability problems for very small probabilities easily. However by the author in numerous examples the shortcomings of SS were outlined and it was finally shown that SS is in fact a disguised Monte Carlo copy of asymptotic SORM. The points computed by SS are converging towards the beta points as seen for example in the diagrams in many SS papers. One way to improve FORM/SORM one runs, starting near the modes i.e. beta points, MCMC’s which move through the failure domain \(F=\{\mathbf {x} ; g(\mathbf {x})< 0\}\) with \(g(\mathbf {x})\) the LSF. With MCMC one can calculate integrals over F with the pdf \(\phi (\mathbf {x})\), but not the normalizing constant P(F). However, a little artifice helps. Comparing the failure domain with another having a known probability content; not P(F) has to be estimated, but the quotient of these two probabilities. A good choice for this is \(F_L=\{x ; g_L(\mathbf {x})< 0\}\) given by the linearized LSF \(g_L(\mathbf {x})\), so \(P(F_L)= \Phi (-|\mathbf {x}^*|)\) with \(\mathbf {x}^*\) the beta point. Running two MCMC’s, one on F and one on \(F_L\)  by comparing them it is possible to obtain an estimate for the failure probability P(F). Another way is to use a modified line sampling method. For each design point for a random set of points on the tangential plane the distance of the plane to the limit state surface on the ray normal to the tangential space is determined and the corresponding normal line integral. Improving FORM/SORM by MCMC adds the advantages of analytic methods to the flexibility of the Monte Carlo approach.