Optimal sensor placement under model uncertainty in the weak-constraint 4D-Var framework

Alen Alexanderian; Hugo Díaz; Vishwas Rao; Arvind K. Saibaba

doi:10.3934/fods.2026002

Article Contents

2026, Volume 11: 1-28. Doi: 10.3934/fods.2026002

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Optimal sensor placement under model uncertainty in the weak-constraint 4D-Var framework

1.
Department of Mathematics, North Carolina State University, USA
2.
Argonne National Laboratory, USA

^*Corresponding author: Alen Alexanderian
^*Corresponding author: Alen Alexanderian

Received: January 2025

Revised: September 16, 2025

Early access: December 18, 2025

Published online: December 18, 2025

The work of AA was supported in part by the US National Science Foundation (NSF) grant DMS-2111044. HD and AKS were supported by the Department of Energy, Office of Science, Advanced Scientific Computing Research (ASCR) Program through the award DE-SC0023188. AKS was also supported by the NSF, in part, through the award DMS-1845406. VR was supported by the U.S. Department of Energy, Office of Science, ASCR Program under contracts DE-AC02-06CH11357 and DE-SC0023188.

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Abstract

In data assimilation, the model may be subject to uncertainties and errors. The weak-constraint data assimilation framework enables incorporating model uncertainty in the dynamics of the governing equations. We propose a new framework for near-optimal sensor placement in the weak-constrained setting. This is achieved by first deriving a design criterion based on the expected information gain, which involves the Kullback-Leibler divergence from the forecast prior to the posterior distribution. An explicit formula for this criterion is provided, assuming that the model error and background are independent and Gaussian and the dynamics are linear. We discuss algorithmic approaches to efficiently evaluate this criterion through randomized approximations. To provide further insight and flexibility in computations, we also provide alternative expressions for the criteria. We provide an algorithm to find near-optimal experimental designs using column subset selection, including a randomized algorithm that avoids computing the adjoint of the forward operator. Through numerical experiments in one and two spatial dimensions, we show the effectiveness of our proposed methods.

Keywords:

Mathematics Subject Classification: Primary: 65M32, 62K05; Secondary: 65F40, 65F60.

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Figure 1. The sensor locations determined by RAF-OED for different values of $ k $

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Figure 2. Comparison of different sensor placement methods. The histogram represents the criterion values for random designs

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Figure 3. Figures illustrating the candidate sensor locations (left), initial condition $ c_0 $ (middle), and velocity field $ \boldsymbol v $ (right)

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Figure 4. Snapshots of $ c_h $

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Figure 5. Comparison between the RAF-OED (dashed black) and the random designs (histogram)

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Figure 6. The sensor locations determined by RAF-OED for different $ k $ values

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Table 1. A summary of matrices involved for all the different formulations for computing the WC4D-Var criterion $ \phi_{\mathrm{EIG}} $

Formulation/Matrix	$ {\bf{\Gamma }}_\mathit{\boldsymbol{R}}^{-1} $	$ {\bf{\Gamma }}_{{\rm{mod}}}^{-1} $	$ {\bf{\Gamma }}_{{\rm{mod}}}^{\frac 12} $	$ {\bf{L}}^{-1} / {\bf{L}}^{\top} $	Matrix Size
Unpreconditioned eq. 25	$\checkmark$	$\checkmark$	✘	✘	$ N_d $
Preconditioned (22)	$\checkmark$	✘	$\checkmark$	$\checkmark$	$ N_d $
SP-I (26)	✘	✘	✘	✘	$ N_d+N_m $
SP-II (27)	$\checkmark$	✘	✘	✘	$ 2N_d $

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Table 2. Computational complexity of XNysTrace and Stochastic Lanczos Quadrature algorithms. Here, $N$ denotes the number of random probes, and $T_\boldsymbol {f({\bf{E}})}$ represents the computational cost of a single application of $f({\bf{E}})$ via Lanczos, where $n_{\rm{iter}}$ is the number of Lanczos iterations. The cost $T_{{\bf{E}}}$ will depend of each formulation; see Table 1

Algorithm	Operations (flops)	Reference
SLQ	$N\cdot T_\boldsymbol {f({\bf{E}})}=N(n_{\rm{iter}}T_{{\bf{E}}}+d\cdot n^2_{\rm{iter}} )$	[48, Section 3]
XNysTrace+Lanczos	$N\cdot T_\boldsymbol {f({\bf{E}})}+\mathcal{O}(N^2 d)$	[18, Section 2.2]

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Table 3. Results for Log-Determinant Approximation using SLQ with $ N = 2^3 $

Formulation	Matrix size $ d $	Lanczos Iters	Rel. Error
Unpreconditioned (25)	$ 4010 $	533	$ 7.9 \times 10^{-4} $
Preconditioned (22)	$ 4010 $	37	$ 1.5 \times 10^{-4} $
SP-I (26)	$ 8300 $	350	$ 1.1 \times 10^{-4} $
SP-II (27)	$ 8020 $	550	$ 9.0 \times 10^{-2} $

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Table 4. Number of samples, average Lanczos iterations, average relative errors, and standard deviations for Hutchinson's and XNysTrace estimators over 100 trials

Samples	Lanczos	SLQ		XNysTrace+Lanczos
$ (N) $	Iters.	Mean	Std. Dev.	Mean	Std. Dev.
2	73.5	$ 2.99 \times 10^{-4} $	$ 2.49 \times 10^{-4} $	$ 4.17 \times 10^{-4} $	$ 2.99 \times 10^{-4} $
4	148.2	$ 2.08 \times 10^{-4} $	$ 1.62 \times 10^{-4} $	$ 2.40 \times 10^{-4} $	$ 1.85 \times 10^{-4} $
8	296.3	$ 1.48 \times 10^{-4} $	$ 1.28 \times 10^{-4} $	$ 1.80 \times 10^{-4} $	$ 1.48 \times 10^{-4} $
16	592.1	$ 1.01 \times 10^{-4} $	$ 8.08 \times 10^{-5} $	$ 1.11 \times 10^{-4} $	$ 7.85 \times 10^{-5} $
32	1186.9	$ 8.31 \times 10^{-5} $	$ 5.48 \times 10^{-5} $	$ 5.59 \times 10^{-5} $	$ 4.11 \times 10^{-5} $
64	2369.5	$ 4.85 \times 10^{-5} $	$ 4.58 \times 10^{-5} $	$ 4.01 \times 10^{-6} $	$ 3.18 \times 10^{-6} $
128	4741.9	$ 3.75 \times 10^{-5} $	$ 2.90 \times 10^{-5} $	$ 1.05 \times 10^{-7} $	$ 8.24 \times 10^{-8} $

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Table 5. EIG Values from Different Methods

Method	$ \phi_{\mathrm{EIG}}({\bf{S}}) $	Method	$ \phi_{\mathrm{EIG}}({\bf{S}}) $
Best ($ {\bf{S}}_ \rm{max} $)	106.30	Best ($ {\bf{S}}_ \rm{max} $)	unknown
Greedy ($ {\bf{S}}={\bf{S}}_\text{G} $)	95.76	Greedy ($ {\bf{S}}={\bf{S}}_\text{G} $)	127.31
Algorithm 1	97.71	Algorithm 1	125.89
RAF-OED	97.71	RAF-OED	127.22
k = 5		k = 10

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Table 6. Comparison of SLQ and XNysTrace over 100 trials

$ N $	SLQ			XNysTrace
$ N $	Mean	RSD	SD	Mean	RSD	SD
2	$ 4.1612 \times 10^{5} $	$ 8.83 \times 10^{-2} $%	368	$ 3.9789 \times 10^{5} $	$ 2.96 \times 10^{-2} $%	118
4	$ 4.1612 \times 10^{5} $	$ 6.75 \times 10^{-2} $%	281	$ 3.9790 \times 10^{5} $	$ 2.29 \times 10^{-2} $ %	91
8	$ 4.1609 \times 10^{5} $	$ 5.15 \times 10^{-2} $%	$ \mathit{\boldsymbol{214}} $	$ 3.9788 \times 10^{5} $	$ 1.56 \times 10^{-2} $%	$ \mathit{\boldsymbol{62}} $

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Table 7. SC-based EIG values evaluated on sensor sets selected for the WC formulation. $ \mathbf{S}^\text{RGKS}_\text{WC} $ denotes the WC-selection obtained using the RAF-OED algorithm

$ \phi_{\mathrm{EIG}}^\text{SC}(\mathbf{S}^\text{RGKS}_\text{WC}) $
$ \alpha = 5\% $	$ 179.65 $
$ \alpha = 3\% $	$ 178.26 $
$ \alpha = 2\% $	$ 175.39 $
$ \alpha = 1\% $	$ 170.54 $

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