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A sparse control approach to optimal sensor placement in PDE-constrained parameter estimation problems. (English) Zbl 1423.35452

Summary: 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.

MSC:

35R30 Inverse problems for PDEs
49K20 Optimality conditions for problems involving partial differential equations
62K05 Optimal statistical designs
65K05 Numerical mathematical programming methods
49M25 Discrete approximations in optimal control

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