Sensitivity Analysis for Model with Dependent Inputs Using Sparse Polynomial Chaos Expansions

Abstract

Polynomial chaos expansions (PCE) meta-model has been wildly used and investigated in the last decades in sensitivity analysis (SA), which adopts a variety of orthogonal polynomials to approximate the system response and calculates sensitivity indices directly from the polynomial coefficients. The Sobol' index is one of prevalent sensitivity indices for model with independent inputs and can be easily obtained after constructing generalized polynomial chaos (gPC). But for dependent inputs, a typical approach is based on the procedure of transforming the dependent inputs into independent inputs according to the literature. This paper demonstrates a global sensitivity analysis (GSA) approach for dependent inputs, in which Gram-Schmidt orthogonalization (GSO) numerically computes the orthonormal polynomials for PCE. The especial procedure for dependent inputs to obtain sensitivity indices lies in the linearly independent polynomials basis for GSO must be in an intended order. Besides, to alleviate the curse of dimensionality, the sparse polynomial chaos (sPC) is built coupling with least angle regression (LAR) and a nested experimental design called weighted Leja sequences (wLS). Then cross validation (CV) determines the best truncated set for sPC with the suitable size of experimental design in use. In the end, this proposed approach is validated on a benchmark function with dependent inputs. The results reveal that the proposed approach performs well to calculate sensitivity indices for model with dependent inputs.

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