A new non-intrusive polynomial chaos using higher order sensitivities

- New non-intrusive method for Polynomial Chaos Expansion called Polynomial Chaos Decomposition with Differentiation.
- Higher-order sensitivities.
- Higher order sensitivities calculation using sample responses in multi-dimensions.
- Sensitivities with varying order of truncation error.
- Number of samples equal to the polynomial chaos terms.

This paper proposes a new non-intrusive method for uncertainty quantification called Polynomial Chaos Decomposition with Differentiation (PCDD) that uses higher-order sensitivities of the response. In PCDD, the polynomial chaos expansion (PCE) of the response is differentiated with respect to the basis random variables using multi-indices. This differentiation results in a system of linear equations which can then be solved to determine the expansion coefficients. Here, the higher accuracy, Modified Forward Finite Difference (ModFFD) that involves representation of the response using Taylor expansion of order equal to the chaos-order is used in combination with PCE. Therefore, the total number of samples required with this method is equal to the number of terms in the PCE. To verify the validity of this new technique, two analytical problems and two stochastic composite laminate problems were studied. The results of the analytical problems showed that the accuracy of PCDD using ModFFD is similar to that of PCDD using analytical sensitivities, which in addition is comparable to the exact results. For composite laminate problems, the PCDD displayed very high accuracy comparable to 50,000 Latin Hypercube Samples, which underlines the computational efficiency of this proposed method.