Multifidelity sensor placement in Bayesian state estimation problems

TL;DR

This work addresses the problem of placing sensors with heterogeneous costs and fidelities under a budget to optimize Bayesian state estimation. It reframes multifidelity sensor placement as a budgeted D-optimal design, linking it to column subset selection and deploying a greedy, cost-normalized strategy along with Sherman–Morrison-based updates, plus an iterative fidelity-alternation method. The authors prove monotonicity and partial submodularity of the objective, show that greedy lacks a universal approximation guarantee, and demonstrate through SST and cylinder-flow benchmarks that the iterative approach reliably improves information gain and reconstruction accuracy over random designs. The findings highlight practical pathways for scalable, cost-aware sensor design in complex, high-dimensional inverse problems, with potential for accelerated, large-scale deployment.

Abstract

We study optimal sensor placement for Bayesian state estimation problems in which sensors vary in cost and fidelity, resulting in a budget-constrained multifidelity optimal experimental design problem. Sensor placement optimality is quantified using the D-optimality criterion, and the problem is approached by leveraging connections with the column subset selection problem in numerical linear algebra. We implement a greedy approach for this problem, whose computational efficiency we improve using rank-one updates via the Sherman-Morrison formula. We additionally present an iterative algorithm that, for each feasible allocation of sensors, greedily optimizes over each sensor fidelity subject to previous sensor choices, repeating this process until a termination criterion is satisfied. To the best of our knowledge, these algorithms are novel in the context of cost constrained multifidelity sensor placement. We evaluate our methods on several benchmark state estimation problems, including reconstructions of sea surface temperature and flow around a cylinder, and empirically demonstrate improved performance over random designs.

Figures (5)

• Figure 1: MAP reconstructions of sea surface temperature using sensors selected by the greedy (top) and the iterative (bottom) algorithms. Both methods employ the same settings, yielding comparable relative errors of about $11\%$. For the iterative method, the candidate set contained $|\mathcal{K}| = 11$ allocations, and the algorithm performed a total of $t=2$ greedy refinement iterations across all candidates. Markers indicate sensor locations (stars denote expensive sensors), and zoomed insets highlight regions of higher sensor density.
• Figure 2: MAP reconstructions of flow past a cylinder obtained using greedy (top) and iterative (bottom) sensor selection. Both approaches achieve comparable relative errors of approximately $9\%$. For the iterative method, the candidate set consists of $|\mathcal{K}|=11$ sensor allocations, and the algorithm performed a total of $t=9$ greedy refinement iterations across all candidates. The zoomed insets highlight sensor placements concentrated near regions of high flow variability.
• Figure 3: Greedy sensor selection results for different costs and measurement noise standard deviations.
• Figure 4: Sensor placement results for the greedy (left) and iterative (right) algorithms. Rows correspond to increasing budgets $b = 50, 100, 200$ from top to bottom. The corresponding $(k_{\mathrm{ch}}, k_{\mathrm{exp}})$ values are indicated below each image, with cheap sensors shown in red and expensive sensors shown in blue.
• Figure 5: D-optimality comparison of random, greedy, and iterative sensor selection at $b=500$. Both greedy and iterative selection outperform random selection.

Theorems & Definitions (13)

• Lemma 2.1: Sherman-Morrison Formula golub2013matrix
• Lemma 2.2: Matrix Determinant Lemma HornJohnson2012
• Lemma 2.3: Minkowski Determinant Theorem Marcus1965ASO
• Corollary 2.4
• proof
• Proposition 3.1
• proof
• Proposition 3.2: Monotonicity and Submodularity
• proof
• Proposition 3.3: Expensive sensors are more informative