Many industrial and engineering applications are built on the basis of differential equations. In some cases, parameters of these equations are not known and are estimated from measurements leading to an inverse problem. Unlike many other papers, we suggest to construct new designs in the adaptive fashion 'on the go' using the A-optimality criterion. This approach is demonstrated on determination of optimal locations of measurements and temperature sensors in several engineering applications: (1) determination of the optimal location to measure the height of a hanging wire in order to estimate the sagging parameter with minimum variance (toy example), (2) adaptive determination of optimal locations of temperature sensors in a one-dimensional inverse heat transfer problem and (3) adaptive design in the framework of a one-dimensional diffusion problem when the solution is found numerically using the finite difference approach. In all these problems, statistical criteria for parameter identification and optimal design of experiments are applied. Statistical simulations confirm that estimates derived from the adaptive optimal design converge to the true parameter values with minimum sum of variances when the number of measurements increases. We deliberately chose technically uncomplicated industrial problems to transparently introduce principal ideas of statistical adaptive design.

中文翻译：

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具有逆常微分方程的最优自适应设计

许多工业和工程应用都是建立在微分方程的基础上的。在某些情况下，这些方程的参数是未知的，并且是根据导致逆问题的测量值来估计的。与许多其他论文不同，我们建议使用 A 最优性标准以“随时随地”的自适应方式构建新设计。这种方法在几个工程应用中确定测量和温度传感器的最佳位置时得到了证明：(1) 确定测量悬挂线高度的最佳位置，以便估计具有最小方差的下垂参数（玩具示例），(2) 在一维逆传热问题中自适应确定温度传感器的最佳位置；(3) 在一维扩散问题的框架内自适应设计，当使用有限差分方法以数值方式找到解时。在所有这些问题中，都应用了参数识别和实验优化设计的统计标准。统计模拟证实，当测量次数增加时，从自适应优化设计得出的估计会收敛到具有最小方差总和的真实参数值。我们特意选择了技术上简单的工业问题来透明地介绍统计自适应设计的主要思想。应用参数识别和优化实验设计的统计标准。统计模拟证实，当测量次数增加时，从自适应优化设计得出的估计会收敛到具有最小方差总和的真实参数值。我们特意选择了技术上简单的工业问题来透明地介绍统计自适应设计的主要思想。应用参数识别和优化实验设计的统计标准。统计模拟证实，当测量次数增加时，从自适应优化设计得出的估计会收敛到具有最小方差总和的真实参数值。我们特意选择了技术上简单的工业问题来透明地介绍统计自适应设计的主要思想。