A Theoretical Framework of the Scaled Gaussian Stochastic Process in Prediction and Calibration
A Theoretical Framework of the Scaled Gaussian Stochastic Process in Prediction and Calibration
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DOI：
10.1137/21m1409949
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发表时间：
2018-07
期刊：
影响因子：
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通讯作者：
Mengyang Gu;Fangzheng Xie;Long Wang
中科院分区：
文献类型：
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作者：
Mengyang Gu;Fangzheng Xie;Long Wang
模型标定或数据反演是不确定性量化的基本任务之一。在这项工作中，我们研究了比例高斯随机过程(S-GASP)的理论性质，以模拟现实和不完美数学模型之间的差异。通过正交级数表示，我们建立了高斯随机过程(GAP)与S-GAP之间的显式联系。在适当选择正则化和尺度参数的情况下，S-GAP校正模型的预测均值估计器以与GAP相同的速度收敛到实际。我们还表明，S-GAP定标中的定标数学模型收敛于使现实与数学模型之间的损失最小的数学模型，而具有其他广泛使用的协方差函数的GAP模型不具有这一性质。数值算例证实了该方法与最近几种方法相比具有很好的有限样本性能。
Model calibration or data inversion is one of fundamental tasks in uncertainty quantification. In this work, we study the theoretical properties of the scaled Gaussian stochastic process (S-GaSP), to model the discrepancy between reality and imperfect mathematical models. We establish the explicit connection between Gaussian stochastic process (GaSP) and S-GaSP through the orthogonal series representation. The predictive mean estimator in the S-GaSP calibration model converges to the reality at the same rate as the GaSP with a suitable choice of the regularization and scaling parameters. We also show the calibrated mathematical model in the S-GaSP calibration converges to the one that minimizes the $L_2$ loss between the reality and mathematical model, whereas the GaSP model with other widely used covariance functions does not have this property. Numerical examples confirm the excellent finite sample performance of our approaches compared to a few recent approaches.