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Global optimality conditions for sensor placement, with extensions to binary low-rank A-optimal designs

Christian Aarset

TL;DR

This work develops explicit global-optimality conditions for the 1-relaxed sensor placement problem within Bayesian linear inverse problems, enabling exact characterisation of dominant and redundant sensors via a non-smooth first-order criterion. It introduces a p-continuation algorithm that builds binary, near-optimal designs by fixing dominant/redundant sensors and progressively solving with smaller p, underpinned by a computationally efficient low-rank FEM formulation of the A-optimal objective. The approach yields scalable gradient/Hessian computations that avoid per-sensor traces and exploits a frozen QR-based factorisation to handle large-scale discretisations, demonstrated on a Helmholtz inverse-source problem with hundreds of candidate sensors. These results provide a path to high-quality, resource-aware experimental designs with provable optimality structure and practical computational speedups for stochastic inverse problems.

Abstract

The \emph{sensor placement problem} for stochastic linear inverse problems consists of determining the optimal manner in which sensors can be employed to collect data. Specifically, one wishes to place a limited number of sensors over a large number of candidate locations, quantifying and optimising over the effect this data collection strategy has on the solution of the inverse problem. In this article, we provide a global optimality condition for the sensor placement problem via a subgradient argument, obtaining sufficient and necessary conditions for optimality\revix{, and marking certain sensors as \emph{dominant} or \emph{redundant}, i.e.~always on or always off}. We demonstrate how to take advantage of this optimality criterion to find approximately optimal binary designs, i.e.~designs where no fractions of sensors are placed. Leveraging our optimality criteria, we derive a powerful low-rank formulation of the A-optimal design objective for finite element-discretised function space settings, demonstrating its high computational efficiency, particularly in terms of derivatives, and study globally optimal designs for a Helmholtz-type source problem and extensions towards optimal binary designs.

Global optimality conditions for sensor placement, with extensions to binary low-rank A-optimal designs

TL;DR

This work develops explicit global-optimality conditions for the 1-relaxed sensor placement problem within Bayesian linear inverse problems, enabling exact characterisation of dominant and redundant sensors via a non-smooth first-order criterion. It introduces a p-continuation algorithm that builds binary, near-optimal designs by fixing dominant/redundant sensors and progressively solving with smaller p, underpinned by a computationally efficient low-rank FEM formulation of the A-optimal objective. The approach yields scalable gradient/Hessian computations that avoid per-sensor traces and exploits a frozen QR-based factorisation to handle large-scale discretisations, demonstrated on a Helmholtz inverse-source problem with hundreds of candidate sensors. These results provide a path to high-quality, resource-aware experimental designs with provable optimality structure and practical computational speedups for stochastic inverse problems.

Abstract

The \emph{sensor placement problem} for stochastic linear inverse problems consists of determining the optimal manner in which sensors can be employed to collect data. Specifically, one wishes to place a limited number of sensors over a large number of candidate locations, quantifying and optimising over the effect this data collection strategy has on the solution of the inverse problem. In this article, we provide a global optimality condition for the sensor placement problem via a subgradient argument, obtaining sufficient and necessary conditions for optimality\revix{, and marking certain sensors as \emph{dominant} or \emph{redundant}, i.e.~always on or always off}. We demonstrate how to take advantage of this optimality criterion to find approximately optimal binary designs, i.e.~designs where no fractions of sensors are placed. Leveraging our optimality criteria, we derive a powerful low-rank formulation of the A-optimal design objective for finite element-discretised function space settings, demonstrating its high computational efficiency, particularly in terms of derivatives, and study globally optimal designs for a Helmholtz-type source problem and extensions towards optimal binary designs.

Paper Structure

This paper contains 34 sections, 6 theorems, 63 equations, 10 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. The sensor placement problem and the A-optimal objective functional
  4. Penalty functional in the sensor placement problem
  5. Penalty functional in the best $m_0$ sensors placement problem
  6. State of the art
  7. Contribution
  8. Optimality in the sensor placement problem
  9. Low-rank redundant-dominant techniques for binary A-optimal designs
  10. The $p$-continuation algorithm
  11. The A-optimal objective in finite element discretisation
  12. Low-rank calculation of the A-optimal objective
  13. Computational complexity
  14. Precomputation
  15. Inverse $L_{\bm{w}}^{-1}$
  16. ...and 19 more sections

Key Result

Theorem 3

Given Assumption ass:standing and $m_0\in\mathbb{N}$, $m_0\leq m$, fix any ${\bm{w}}^*\in\consK[1]$. By reordering, if necessary, assume moreover that the gradient $\nabla\mathcal{J}({\bm{w}}^*)\in\mathbb{R}^m$ satisfies the ordering We distinguish between two possible cases:

Figures (10)

  • Figure 1: Feasible set $\consK[1][1]$ in $\mathbb{R}^3$
  • Figure 2: Domain. Inner circle: Source domain $\Omega$ with example source $f$\ref{['eq:f_ground']}. Outer circle: Measurement domain $\mathbb{M}$ with wave field $u:=\mathcal{S}_{50}f$ (rescaled to fit same color scale).
  • Figure 3: Outer circle: Circular grid of all candidate sensor locations. Inner circle: Pointwise variance field corresponding to placing every sensor.
  • Figure 4: Left: $1$-relaxed optimal design ${\bm{w}}^*$ using $m_0=36$ out of $m=334$ sensors. Right: corresponding gradient. Dominant indices (${\bm{w}}^*_k=1$, resp. large negative gradient) as green squares, redundant indices (${\bm{w}}^*_k=0$, resp. small negative gradient) as cyan triangles, free indices red.
  • Figure 5: Comparison of the A-optimal objective of outputs of Algorithm \ref{['alg:p_cont']} (green) vs. $10^3$ random designs (red) vs. the globally optimal non-binary designs ${\bm{w}}^*$ (blue) for $7\leq m_0\leq 36$.
  • ...and 5 more figures

Theorems & Definitions (19)

  • Definition 1
  • Theorem 3: Redundant-dominant classification of the optimum
  • proof
  • Example 4
  • Corollary 5
  • Remark 6
  • Remark 7
  • Theorem 8: A-optimal objective via QR
  • proof
  • Remark 9
  • ...and 9 more