Optimally tensor-structured quadrature rule for uncertainty quantification

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Non-intrusive polynomial chaos (NIPC) is an efficient method for solving forward uncertainty quantification (UQ) problems. If using the integration-based approach, NIPC results in multidimensional integration problems that are often solved using the numerical quadrature approach. For the numerical quadrature rule, the full-grid Gauss quadrature method maintains a fully tensorial structure of univariate Gauss quadrature rule, and the number of quadrature points increases exponentially with the number of dimensions, while for the designed quadrature method, the quadrature rule does not maintain a tensorial structure and the number of quadrature points increases at a slower rate than the Gauss quadrature method. Recently, a new method called accelerated tensor-grid evaluations (ATE) was proposed to reduce the cost of model evaluations on the tensor-grid inputs. The ATE method modifies the computational graph of the model in the middle-end of a three-stage compiler so that repeated evaluations on the operation level are eliminated. With ATE, the full-grid NIPC method can be the most efficient UQ method on problems involving sparse computational graph structures. However, for many high-dimensional problems, there may exist a partially tensorial structure for the quadrature rule that is optimal to use with ATE. In this paper, we propose a new method called optimally tensor-structured quadrature rule. This method is based on the designed quadrature idea but finds the quadrature rule that has the optimal tensorial structure so that the cost of model evaluations is minimized while achieving a specific level of accuracy with the integration method. The proposed method also generates the same quadrature rule as full-grid Gauss quadrature or designed quadrature method when the optimal tensorial structure is a fully tensorial or non-tensorial structure, respectively. This method has been tested on a 4D uncertainty quantification problem that involves a low-fidelity multi-disciplinary model of a laser-beam powered UAV. On this problem, the NIPC method using the optimally tensor-structured quadrature points with ATE is at least 50% more efficient than the existing UQ methods we compared with, while achieving the same level of accuracy.