M-PCM-OFFD: An effective output statistics estimation method for systems of high dimensional uncertainties subject to low-order parameter interactions

Abstract

The evaluation of output performance statistics for systems of high-dimensional uncertain input parameters is crucial for robust real-time decision-making tasks of large-scale complex systems that operate in an uncertain environment. We develop a framework that integrates Multivariate Probabilistic Collocation Method (M-PCM) and Orthogonal Fractional Factorial Design (OFFD) to achieve an effective and scalable output statistics estimation. In this paper, we prove that when the degree of each uncertain parameter does not exceed 3 and under the widely held assumption for high-dimensional systems that the interactions among uncertain input parameters are negligible beyond certain order, the integrated M-PCM–OFFD method breaks the curse of dimensionality for correct output mean estimation by maximally reducing the number of simulations from 22m to 2log2(m+1) for a system mapping of m uncertain input parameters. In addition, the resulting reduced-size simulation set is the most robust to numerical truncation errors of simulators among all subsets of the same size in the M-PCM simulation set. The analysis also provides new insightful formal interpretations of the optimality of OFFDs.