# A sparse control approach to optimal sensor placement in PDE-constrained parameter estimation problems

## TL;DR

This paper introduces a convex optimization framework using sparse control methods to optimally place sensors for PDE-based parameter estimation, improving estimation accuracy and efficiency.

## Contribution

It develops a novel measure-based convex optimization approach with accelerated algorithms for sparse sensor placement in PDE parameter estimation.

## Key findings

- Convex formulation for sensor placement problem.
- Accelerated conditional gradient algorithms ensure convergence.
- Numerical experiments demonstrate effectiveness of the approach.

## Abstract

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.