Finite-element/progressive-lattice-sampling response surface methodology and application to benchmark probability quantification problems

Optimal response surface construction is being investigated as part of Sandia discretionary (LDRD) research into Analytic Nondeterministic Methods. The goal is to achieve an adequate representation of system behavior over the relevant parameter space of a problem with a minimum of computational and user effort. This is important in global optimization and in estimation of system probabilistic response, which are both made more viable by replacing large complex computer models with fast-running accurate and noiseless approximations. A Finite Element/Lattice Sampling (FE/LS) methodology for constructing progressively refined finite element response surfaces that reuse previous generations of samples is described here. Similar finite element implementations can be extended to N-dimensional problems and/or random fields and applied to other types of structured sampling paradigms, such as classical experimental design and Gauss, Lobatto, and Patterson sampling. Here the FE/LS model is applied in a ``decoupled`` Monte Carlo analysis of two sets of probability quantification test problems. The analytic test problems, spanning a large range of probabilities and very demanding failure region geometries, constitute a good testbed for comparing the performance of various nondeterministic analysis methods. In results here, FE/LS decoupled Monte Carlo analysis required orders of magnitude less computer time than direct Monte Carlo analysis, with no appreciable loss of accuracy. Thus, when arriving at probabilities or distributions by Monte Carlo, it appears to be more efficient to expend computer-model function evaluations on building a FE/LS response surface than to expend them in direct Monte Carlo sampling.