论文标题
灵敏度增强的广义多项式混乱，以有效的不确定性定量
Sensitivity-enhanced generalized polynomial chaos for efficient uncertainty quantification
论文作者
论文摘要
我们使用广义多项式混乱（GPC）提出了最小二乘（LSQ）回归方法的富集公式（LSQ）回归方法。更具体地说，我们用相对于随机变量的利益量的梯度（或灵敏度）的其他方程丰富了线性系统。通过在随机空间的每个采样点求解一个方程式，可以非常有效地计算出所有变量的灵敏度。相关的计算成本类似于直接问题的一种解决方案。对于采样点的选择，我们应用了基于测量矩阵的旋转QR分解的贪婪算法。我们称新方法敏感性增强的广义多项式混乱或SE-GPC。我们将该方法应用于几个测试用例，以通过增加的混乱顺序测试准确性和收敛性，包括$ 40 $随机参数的空气动力学案例。发现该方法使用最小数量的采样点对统计矩进行准确的估计。计算成本比例为$ \ sim m^{p-1} $，而不是标准LSQ公式的$ \ sim m^p $，其中$ m $是随机变量的数量，而混乱订单的$ p $。方程式的伴随系统的解决方案在许多计算机械包中实现，因此将方法应用于各种工程问题的基础架构存在。
We present an enriched formulation of the Least Squares (LSQ) regression method for Uncertainty Quantification (UQ) using generalised polynomial chaos (gPC). More specifically, we enrich the linear system with additional equations for the gradient (or sensitivity) of the Quantity of Interest with respect to the stochastic variables. This sensitivity is computed very efficiently for all variables by solving an adjoint system of equations at each sampling point of the stochastic space. The associated computational cost is similar to one solution of the direct problem. For the selection of the sampling points, we apply a greedy algorithm which is based on the pivoted QR decomposition of the measurement matrix. We call the new approach sensitivity-enhanced generalised polynomial chaos, or se-gPC. We apply the method to several test cases to test accuracy and convergence with increasing chaos order, including an aerodynamic case with $40$ stochastic parameters. The method is found to produce accurate estimations of the statistical moments using the minimum number of sampling points. The computational cost scales as $\sim m^{p-1}$, instead of $\sim m^p$ 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.
