Continuously nested moment quadrature for uncertainty quantification of black-box models

This study presents a continuously nested moment quadrature method for uncertainty quantification of stochastic systems with arbitrary random input distributions. The method allows for continuous nesting and convergence testing simultaneously; therefore, existing model evaluation results can fully be reused to obtain a converged result at a minimum incremental computational demand. By incorporating a dynamic precision adjustment strategy and adopting criteria on the allowable number of negative weights, the proposed method overcomes the potential limitations of nesting only once under uniform distributions in the conventional Gauss-Kronrod formula, while achieving the highest possible algebraic precision in terms of polynomial degrees. The proposed method is applied to multiple classical and complex engineering and mathematical cases, including a computationally intensive 3D crack propagation problem. Results show that the proposed method requires less computational effort to achieve the same algebraic precision compared to the regular moment quadrature method and the Monte Carlo method. Notably, for problems with uniform random inputs, the computational demand can be reduced to one-fifth of that required by the regular moment quadrature method.