Sparse moment quadrature for uncertainty modeling and quantification

This study presents the Sparse Moment Quadrature (SMQ) method, a new uncertainty quantification technique for high-dimensional complex computational models. These models pose a challenge due to the long evaluation times and numerous random parameters. The SMQ method extends the existing moment quadrature method by incorporating the Smolyak rule to reduce the full tensor formula to a sparse tensor formula. The univariate Gauss quadrature rule is derived using the Hankel matrix of moments, allowing the rule to retain polynomial exactness under any distribution with bounded raw moments. Proper decompositions and transformations are used to handle multi-dimensional problems with correlated variables. The cost and accuracy of the method are analyzed and upper bounds are given. The SMQ method is demonstrated through examples involving 10-dimensional problems, dynamical oscillation, 20-, 100-, and 1000-dimensional nonlinear problems, and a practical membrane vibration problem. The proposed method yields nearly identical results to the conventional Monte Carlo method with thousands to millions of model evaluations. Overall, the SMQ method provides a practical solution to uncertainty quantification of high-dimensional problems involving complex computational models.