This paper proposes a novel fixed inducing points online Bayesian calibration (FIPO-BC) algorithm to efficiently learn the model parameters using a benchmark database. The standard Bayesian calibration (STD-BC) algorithm provides a statistical method to calibrate the parameters of computationally expensive models. However, the STD-BC algorithm does not scale well with regard to the number of data points and also it lacks an online learning capability. The proposed FIPO-BC algorithm greatly improves the computational efficiency of the algorithm and, in addition, enables online calibration to be performed by executing the calibration on a set of predefined inducing points.

To demonstrate the procedure of the FIPO-BC algorithm, two tests are performed, finding the optimal value and exploring the posterior distribution of 1) the parameter in a simple function, and 2) the high-wave number damping factor in a scale-resolving turbulence model (scale adaptive simulation shear-stress transport model/SAS-SST). The results (such as the calibrated model parameter and its posterior distribution) of FIPO-BC with different inducing points are compared to those of STD-BC. It is found that FIPO-BC and STD-BC can provide very similar results, once the predefined set of inducing points in FIPO-BC is sufficiently fine. Given that fewer datapoints are needed in the proposed FIPO-BC algorithm, compared to the STD-BC algorithm, it will be a more computational efficient algorithm. In our demonstration test cases, the proposed FIPO-BC algorithm is at least ten times faster than the STD-BC algorithm. Meanwhile, the online feature of the FIPO-BC allows continuous updating of the calibration outputs and potentially reduces the workload on generating the database.

本文提出了一种新颖的不动点在线贝叶斯定标（FIPO-BC）算法，以使用基准数据库有效地学习模型参数。标准贝叶斯校准（STD-BC）算法提供了一种统计方法来校准计算上昂贵的模型的参数。但是，STD-BC算法在数据点数量方面不能很好地扩展，并且也缺乏在线学习能力。提出的FIPO-BC算法大大提高了该算法的计算效率，此外，通过对一组预定义的诱导点执行校准，可以执行在线校准。

为了演示FIPO-BC算法的过程，进行了两项测试，找到最佳值并探索后验分布：1）简单函数中的参数，以及2）尺度解析中的高波数阻尼因子湍流模型（比例自适应模拟切应力传输模型/ SAS-SST）。将具有不同诱导点的FIPO-BC的结果（例如校准的模型参数及其后验分布）与STD-BC进行比较。发现一旦FIPO-BC中预定义的诱导点集足够精细，FIPO-BC和STD-BC可以提供非常相似的结果。与STD-BC算法相比，由于所提出的FIPO-BC算法需要较少的数据点，因此它将是一种计算效率更高的算法。在我们的演示测试案例中，提出的FIPO-BC算法至少比STD-BC算法快十倍。同时，FIPO-BC的在线功能允许连续更新校准输出，并有可能减少生成数据库的工作量。