A sparse control approach to optimal sensor placement in PDE-constrained parameter estimation problems
A sparse control approach to optimal sensor placement in PDE-constrained parameter estimation problems
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DOI：
10.1007/s00211-019-01073-3
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发表时间：
2019-12-01
影响因子：
2.1
通讯作者：
Walter，Daniel
中科院分区：
文献类型：
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作者：
Neitzel，Ira;Pieper，Konstantin;Walter，Daniel
我们提出了一种系统的方法来优化有限多个传感器的位置，以便从相关参数依赖椭圆PDE的解的点评估中推断出有限维参数。根据测量传感器的分布，用渐近协方差矩阵的性质来量化相应最小二乘估计量的质量。我们制定了一个设计问题，其中我们最小化与相应置信区域的大小相关的函数，这些区域相对于点向测量的位置和数量。测量设置采用空间实验域上的正Borel度量建模，导致一个凸优化问题。对于算法解，导出了测度空间中的加速条件梯度方法，该方法利用了设计问题的结构特性来保证收敛到稀疏解。给出了该方法的收敛性，并通过数值实验对其结果进行了验证。
We present a systematic approach to the optimal placement of finitely many sensors in order to infer a finite-dimensional parameter from point evaluations of the solution of an associated parameter-dependent elliptic PDE. The quality of the corresponding least squares estimator is quantified by properties of the asymptotic covariance matrix depending on the distribution of the measurement sensors. We formulate a design problem where we minimize functionals related to the size of the corresponding confidence regions with respect to the position and number of pointwise measurements. The measurement setup is modeled by a positive Borel measure on the spatial experimental domain resulting in a convex optimization problem. For the algorithmic solution a class of accelerated conditional gradient methods in measure space is derived, which exploits the structural properties of the design problem to ensure convergence towards sparse solutions. Convergence properties are presented and the presented results are illustrated by numerical experiments.