A greedy Galerkin method to efficiently select sensors for linear dynamical systems

A key challenge in inverse problems is the selection of sensors to gather the most effective data. In this paper, we consider the problem of inferring the initial condition to a linear dynamical system and develop an efficient control-theoretical approach for greedily selecting sensors. Our method e...

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Bibliographic Details
Published in:Linear algebra and its applications 2023-12, Vol.679, p.275-304
Main Authors: Kouri, Drew P., Hua, Zuhao, Udell, Madeleine
Format: Article
Language:English
Subjects:
Galerkin method
Gramian
Greedy algorithm
Inverse problems
Linear dynamics
Lyapunov equation
MATHEMATICS AND COMPUTING
Observability
Optimal control
Preconditioning
Sensor placement
Submodularity
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Summary:A key challenge in inverse problems is the selection of sensors to gather the most effective data. In this paper, we consider the problem of inferring the initial condition to a linear dynamical system and develop an efficient control-theoretical approach for greedily selecting sensors. Our method employs a Galerkin projection to reduce the size of the inverse problem, resulting in a computationally efficient algorithm for sensor selection. As a byproduct of our algorithm, we obtain a preconditioner for the inverse problem that enables the rapid recovery of the initial condition. We analyze the theoretical performance of our greedy sensor selection algorithm as well as the performance of the associated preconditioner. Finally, we verify our theoretical results on various inverse problems involving partial differential equations.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2023.09.003