In this paper, we develop a mixed covariance function Kriging (MCF-Kriging) model for uncertainty quantification. The mixed covariance function is a linear combination of a traditional stationary covariance function and a nonsta-tionary covariance function constructed by the inner product of orthonormal polynomial basis functions. We use a weight matrix to control the contribution of each polynomial basis to the whole model representation, and a trade-off parameter is used to balance the contribution of the two different covariance functions. The optimal values of these model hyperparameters are obtained through an iterative algorithm derived by maximum likelihood estimation (MLE), and sparse representation is achieved automatically in the MLE step by removing the basis functions with small contribution. Additionally, the hyperparameters of stationary covariance function are tuned by minimizing the leave-one-out cross-validation error of the surrogate model. For validation, we investigate three benchmark test functions with different dimensionalities, and compare the accuracy and efficiency with the state-of-art sequential PC-Kriging and optimal PC-Kriging models. The results show that the MCF-Kriging model provides comparable performance compared to the two PC-Kriging models for nonlinear problems, that are moderate and even high-dimensional. Finally, we apply our model to a heat conduction problem to demonstrate its effectiveness in engineering application.

中文翻译：

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不确定性量化的混合协方差函数克里金模型

在本文中，我们开发了一种用于不确定性量化的混合协方差函数克里金 (MCF-Kriging) 模型。混合协方差函数是传统的平稳协方差函数和由正交多项式基函数的内积构造的非平稳协方差函数的线性组合。我们使用权重矩阵来控制每个多项式基对整个模型表示的贡献，并使用权衡参数来平衡两个不同协方差函数的贡献。这些模型超参数的最优值是通过最大似然估计（MLE）派生的迭代算法获得的，并且在MLE步骤中通过去除贡献较小的基函数来自动实现稀疏表示。此外，通过最小化代理模型的留一法交叉验证误差来调整平稳协方差函数的超参数。为了验证，我们研究了三个不同维度的基准测试函数，并将准确性和效率与最先进的顺序 PC-Kriging 和最佳 PC-Kriging 模型进行比较。结果表明，与两个 PC-Kriging 模型相比，MCF-Kriging 模型提供了相当的性能，用于处理中等甚至高维的非线性问题。最后，我们将我们的模型应用于热传导问题，以证明其在工程应用中的有效性。我们研究了三个不同维度的基准测试函数，并将准确性和效率与最先进的顺序 PC-Kriging 和最优 PC-Kriging 模型进行了比较。结果表明，与两个 PC-Kriging 模型相比，MCF-Kriging 模型提供了相当的性能，用于处理中等甚至高维的非线性问题。最后，我们将我们的模型应用于热传导问题，以证明其在工程应用中的有效性。我们研究了三个不同维度的基准测试函数，并将准确性和效率与最先进的顺序 PC-Kriging 和最优 PC-Kriging 模型进行了比较。结果表明，与两个 PC-Kriging 模型相比，MCF-Kriging 模型提供了相当的性能，用于处理中等甚至高维的非线性问题。最后，我们将我们的模型应用于热传导问题，以证明其在工程应用中的有效性。