Optimal Adaptive Designs with Inverse Ordinary Differential Equations
提出一种自适应构建实验设计的方法，利用A最优准则实时确定测量位置，在悬线高度测量、一维热传导和扩散问题中验证，统计模拟表明估计值随测量次数增加收敛到真值且方差和最小。
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.