Sensitivity-enhanced generalized polynomial chaos for efficient uncertainty quantification
Sensitivity-enhanced generalized polynomial chaos for efficient uncertainty quantification
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DOI：
10.1016/j.jcp.2023.112377
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发表时间：
2022-06
期刊：
影响因子：
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通讯作者：
Kyriakos D. Kantarakias;G. Papadakis
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文献类型：
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作者：
Kyriakos D. Kantarakias;G. Papadakis
摘要：本文研究了利用广义多项式混沌（gPC）的最小二乘（LSQ）回归方法求解不确定性量化（UQ）问题，并用兴趣量（qi）相对于随机变量的梯度对线性系统进行扩充。对于伴随方程组的所有变量，梯度计算非常有效。为了使增广LSQ系统的条件个数最小化，需要一种有效的随机空间采样策略。我们比较两种策略。首先，我们将旋转QR分解应用于标准LSQ矩阵，并在识别的样本点处评估qi及其梯度。在第二种策略中，我们将QR分解直接应用于增广矩阵。我们发现，就准确性而言，第一种策略比评估次数更有效。我们将新方法称为灵敏度增强广义多项式混沌，或se-gPC，并将其应用于几个测试案例，包括具有40个随机参数的空气动力学案例。该方法可以用少量的采样点对统计矩进行准确的估计。计算代价尺度为~ mp−1，而不是标准LSQ公式的~ mp，其中m是随机变量的数量，p是混沌顺序。伴随方程组的解在许多计算力学包中实现，因此存在将该方法应用于各种工程问题的基础设施。
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∼ m p− 1, instead of∼ 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.