Sensitivity-enhanced generalized polynomial chaos for efficient uncertainty quantification

Abstract

We consider the Least Squares (LSQ) regression method for Uncertainty Quantification (UQ) using generalised polynomial chaos (gPC) and augment the linear system with the gradient of the Quantity of Interest (QoI) with respect to the stochastic variables. The gradient is computed very efficiently for all variables from the adjoint system of equations. To minimise the condition number of the augmented LSQ system, an effective sampling strategy of the stochastic space is required. We compare two strategies. In the first, we apply pivoted QR decomposition to the standard LSQ matrix and evaluate both the QoI and its gradient at the sample points identified. In the second strategy, we apply QR decomposition directly to the augmented matrix. We find that the first strategy is more efficient in terms of accuracy vs number of evaluations. We call the new approach sensitivity-enhanced generalised polynomial chaos, or se-gPC, and apply it to several test cases including an aerodynamic case with 40 stochastic parameters. The method can produce accurate estimations of the statistical moments using a small number of sampling points. The computational cost scales as ∼mp−1, instead of ∼mp of the standard LSQ formulation, where m is the number of stochastic variables and p the chaos order. The solution of the adjoint system of equations is implemented in many computational mechanics packages, thus the infrastructure exists for the application of the method to a wide variety of engineering problems.

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