标题： 初始条件在热力学模型中的不确定性传播 显示英文标题

标题： Uncertainty Propagation of Initial Conditions in Thermal Models

摘要： 机床的操作通常需要对刀具中心点（TCP）位置有高度精确的了解。 TCP随时间的位移可以从热模型中推断出来，这些模型包含一组几何耦合的热方程。 每个方程代表机床某部分的温度，并且通常是在复杂的几何结构上建立的。 TCP预测的准确性在很大程度上取决于模型参数的准确性，例如热交换参数和初始温度。 因此，确定这些参数对TCP位移预测的影响至关重要。 反过来，参数估计的准确性基本上由测量精度和传感器布置决定。 确定给定传感器配置的准确性是优化传感器布置的关键前提。 我们在这里开发了一个特定机床的热模型。 在此模型的基础上，我们提出了两种数值算法，以评估任何给定的热传感器配置在准确性方面的表现。 我们从后验协方差矩阵中计算出相对于不确定初始温度场的后验方差。 整个矩阵是密集的，且可能非常大，这取决于模型的大小。 因此，我们应用了一种低秩方法来近似相关条目，即其对角线上的方差。 我们首先介绍一种直接的方法来计算这种近似，这需要计算模型对初始值的敏感度。 此外，我们还介绍了一种低秩张量方法，该方法利用了底层系统结构。 我们比较了这两种算法在运行时间和内存需求方面的效率，并讨论了它们在最优传感器布置问题中的各自优势。 显示英文摘要

摘要： The operation of machine tools often demands a highly accurate knowledge of the tool center point's (TCP) position. The displacement of the TCP over time can be inferred from thermal models, which comprise a set of geometrically coupled heat equations. Each of these equations represents the temperature in part of the machine, and they are often formulated on complicated geometries. The accuracy of the TCP prediction depends highly on the accuracy of the model parameters, such as heat exchange parameters, and the initial temperature. Thus it is of utmost interest to determine the influence of these parameters on the TCP displacement prediction. In turn, the accuracy of the parameter estimate is essentially determined by the measurement accuracy and the sensor placement. Determining the accuracy of a given sensor configuration is a key prerequisite of optimal sensor placement. We develop here a thermal model for a particular machine tool. On top of this model we propose two numerical algorithms to evaluate any given thermal sensor configuration with respect to its accuracy. We compute the posterior variances from the posterior covariance matrix with respect to an uncertain initial temperature field. The full matrix is dense and potentially very large, depending on the model size. Thus, we apply a low-rank method to approximate relevant entries, i.e. the variances on its diagonal. We first present a straightforward way to compute this approximation which requires computation of the model sensitivities with with respect to the initial values. Additionally, we present a low-rank tensor method which exploits the underlying system structure. We compare the efficiency of both algorithms with respect to runtime and memory requirements and discuss their respective advantages with regard to optimal sensor placement problems.