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Sparse Sensor Allocation for Inverse Problems of Detecting Sparse Leaking Emission Sources

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The central claim: sensor placement for sparse leak detection can be solved as a bilevel optimization whose outer objective is the wind-averaged mean squared estimation error of a nonnegative elastic-net inverse solver.

desk verdict A promising sensor-placement framework with a real gap in the hypergradient derivation: the SBA update likely descends a partial derivative that ignores how sampled data depend on sensor locations. read the letter →

arxiv 2509.05559 v1 pith:VJPHYYRU submitted 2025-09-06 stat.AP cs.ITmath.ITstat.ME

Xinchao Liu , Youngdeok Hwang , Dzung Phan , Levente Klein , Xiao Liu , Kyongmin Yeo This is my paper · ORCID
classification stat.APcs.ITmath.ITstat.ME MSC 62K0590C2690C3062F15
keywords optimalsensorplacementbileveloptimizationinverseproblemselasticnetintegratedmeansquarederrorsampleaverageapproximationstochasticgradientdescentmethaneleakdetection
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Leaking emission sources are sparse and nonnegative, wind is uncertain, and classical linear-Gaussian Bayesian designs cannot handle these features. This paper treats sensor placement as a bilevel optimization: the outer problem chooses sensor positions to minimize the wind-averaged integrated mean squared error of the estimated emission rates, while the inner problem estimates those rates with a nonnegative elastic-net inversion. The paper derives closed-form hypergradients for both levels, provides convergence bounds for a repeated sample-average approximation and a stochastic-gradient bilevel algorithm, and shows numerically that the resulting placements reduce estimation error relative to random, space-filling, and A-optimal initial designs. If correct, it turns sensor deployment for industrial leak monitoring into a computationally tractable design step rather than a heuristic.

What carries the argument

The load-bearing object is the bilevel formulation (11) with the nonnegative elastic-net inverse problem (12) as its lower level and the wind-averaged IMSE as its upper objective. Its mechanism is the implicit-differentiation formula (17)-(18) for the hypergradient: differentiate the KKT conditions of the lower-level QP with respect to sensor locations, keeping only the active constraints (positions pinned at zero), to obtain the gradient that drives sensor updates. Around this sits an augmented primal-dual algorithm for solving each inner QP and a stochastic-approximation outer loop that re-samples wind and emission scenarios each step.

What would settle it

In a small case with a degenerate lower-level solution (a true zero coefficient whose optimal Lagrange multiplier is also zero), compute the hypergradient from (17)-(18) and compare it with a central finite-difference gradient of the IMSE; a mismatch beyond numerical tolerance would show the descent direction is not the true gradient.

Watch

Extended reading notes

Core claim

The paper's central claim is that the sensor layout itself should be chosen as the solution of the bilevel program (11): minimize the Monte Carlo estimate of $\mathbb{E}_{\theta,\beta,\Phi}\lVert \hat\theta(\Phi,\beta,s)-\theta\rVert_2^2$, where each $\hat\theta$ is the solution of a convex quadratic program imposing $\theta\ge 0$ with elastic-net penalties. Because the lower-level solver is differentiable through its KKT conditions under strict complementarity, the paper obtains the gradient of the upper-level objective with respect to sensor positions and uses it in two algorithms: rSAA, which repeats small sample-average problems and averages their outputs, and SBA, a stochastic projected-gradient method that re-samples wind and emissions each iteration. The convergence analysis shows the SBA iterate reaches a stationary point as the inner solver accuracy and batch size grow, and the numerical experiments report that in the 10-sensor/20-source test the mean absolute percentage error drops from 69.06% (random), to 50.79% (A-optimal start), to 29.94% (bilevel-optimized design).

Load-bearing premise

The hypergradient formula that drives every SBA sensor update assumes strict complementarity in the lower-level QP: every source whose estimated rate is pinned at zero must have a strictly positive dual multiplier, which can fail for elastic-net penalties when zero sits inside the L1 subdifferential.

Editorial extensions

If this is right

  • Sensor layouts become functions of the wind distribution: with prevailing north-to-south wind, the algorithm places sensors along the downwind boundary, and reversing the wind rose would mirror the layout.
  • The same machinery carries physical constraints other than nonnegativity (e.g., bounds, couplings) as long as the lower-level estimator remains a differentiable QP.
  • Because the design objective is the estimation error of the actual sparse estimator rather than a posterior covariance proxy, the method can be applied whenever the inverse solver is regularized.
  • The MAPE reduction reported in the 10-sensor/20-source validation brings the optimized design near the ~30% uncertainty ceiling used in methane quantification guidelines with fewer sensors than naive designs.
  • The rSAA stochastic upper bound gives operators a computable certificate on how far a candidate placement is from the true optimum.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to replace the strict-complementarity assumption with an epsilon-active-set or proximal differentiation rule; if the current hypergradient is biased on degenerate elastic-net solutions, such a fix would restore the descent property without changing the bilevel structure.
  • The formulation should extend to other uncertain forward parameters (source height, diffusivity, atmospheric stability) by adding them to the sampling distribution, since the IMSE averaging does not depend on the parameter being wind.
  • Because the paper's final layout is sensitive to initialization in nonconvex experiments, a practical deployment rule would be to run SBA from several starting designs (A-optimal, support points, random) and select by the rSAA-style confidence bound.
  • For nonlinear dispersion models, the same bilevel objective could be paired with a learned differentiable surrogate of the inverse map, which is the paper's stated future direction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper considers the sparse placement of sensors for estimating sparse, nonnegative emission sources under uncertain wind conditions. It formulates the problem as a bilevel optimization: the upper level minimizes the wind-averaged integrated mean squared error (IMSE) of the emission-rate estimates over sensor locations, while the lower level solves a nonnegative elastic-net-regularized inverse problem for each Monte Carlo scenario. Two algorithms are proposed: a repeated sample average approximation (rSAA) using a global solver on small deterministic problems, and a stochastic gradient-based bilevel approximation (SBA) that differentiates through the lower-level KKT system. Convergence bounds are stated for both methods, and numerical experiments with Gaussian-plume synthetic data compare the resulting designs against random, A-optimal, support-point, and other initial designs. Open-source GPU-accelerated code is provided.

Significance. If the central derivation were correct, the paper would offer a practical, scalable design tool that extends classical A-optimal experimental design to non-Gaussian, sparsity-promoting, physically constrained inverse problems with uncertain forward-model parameters. The formulation is a natural and useful extension, the numerical study covers a range of problem sizes, and the availability of reproducible code with GPU support is a concrete strength. However, the paper's main algorithmic claim for SBA is undercut by an apparently missing term in the hypergradient derivation, and the convergence guarantee rests on an unverified strict-complementarity assumption. The framework itself remains viable and the identified flaw is local and correctable, so the contribution is potentially significant but not yet established as stated.

major comments (3)
  1. [Section 3.2, Eq. (18) and Appendix A.1] The implicit differentiation of the KKT conditions treats the observations Phi^(i) as independent of s, but in Eqs. (5)-(7) the data are generated as Phi^(i)(s) = F(beta^(i),s) theta^(i) + epsilon^(i), so nabla_s Phi^(i) = (nabla_s F) theta^(i) is generally nonzero. The derivative of d^(i)(s) = lambda_2 1 - sigma_epsilon^{-2} F^*(beta^(i),s) Phi^(i)(s) therefore contains the additional term -sigma_epsilon^{-2} F^*(beta^(i),s) (nabla_s F(beta^(i),s)) theta^(i), which is absent from (18). Appendix B derives only derivatives of the plume kernel A_j(s_i), not the derivative of Phi with respect to s. Consequently, the SBA update (19) is not an unbiased stochastic gradient of the objective in (11a)/(6) unless this term is included, and Theorem 1's convergence statement applies to a different functional. This issue is load-bearing for the claim that SBA minimizes the IMSE; the rSAA algorithm may be unaffected, but the paper does not specify whether its sample generation re-evaluates Phi at each candidate sensor location s.
  2. [Section 3.2, paragraph after Proposition 2] The hypergradient formula requires strict complementarity, namely positive dual multipliers bar{eta} > 0 on all active constraints, but the paper neither proves this for the nonnegative elastic-net QP (12) nor verifies it numerically. For L1-type regularized problems, an active coordinate can have a zero dual multiplier when the subgradient at zero contains zero, in which case the active set is not differentiable and the implicit-function argument behind (18) breaks down. Since Lemma 1 and Theorem 1 are built on this differentiation, the convergence guarantee is conditional on an unverified assumption. The authors should either prove that strict complementarity holds generically for the Gaussian-plume forward model with lambda_1, lambda_2 > 0, or report numerical checks (e.g., the minimum of bar{eta} over active constraints across all iterations and Monte Carlo samples).
  3. [Section 4.4, Figure 19] The headline MAPE values (69.06%, 50.79%, and 29.94% for random, A-optimal initial, and proposed designs) are reported as single point estimates without standard errors, confidence intervals, or multiple independent replications. Because the objective (6) is stochastic in theta, beta, and Phi, and because Figure 12 shows that SBA converges to different local optima from different initializations, the Section 5 claim that the proposed approach 'can significantly improve the accuracy of the inverse estimation over some of the existing designs' is not fully substantiated. The authors should report means and dispersions over at least several independent runs, and state whether the same random seeds or Monte Carlo samples were used across the compared designs.
minor comments (5)
  1. [Abstract and Section 4.2] The abstract says the approach reduces IMSE 'starting from various initial sensor deployment generated by existing approaches,' but in Example II only the Support Points and A-optimal designs are actually used to initialize SBA; the random, K-means, GP, and SSPOR designs appear only in the qualitative comparison of Figure 7. The final validation in Section 4.4 uses random, A-optimal, and the proposed design. The wording should be made consistent with the experiments.
  2. [Appendix C.2] The proof of Theorem 1 invokes 'Assumption 7,' but only Assumptions 1-6 are stated anywhere in the paper. In the proof of Lemma 1 in Appendix C.1, the bound labeled 'according to Assumption 6' appears to require Assumption 5, the Lipschitz condition on D(s,hat{theta}(xi),hat{eta}(xi)). The assumptions should be renumbered and cross-referenced consistently.
  3. [Algorithm 1 and Section 3.3] The paper refers to 'existing global solvers' for the deterministic bilevel problems in rSAA but does not name the solver, the reformulation used, or the software. Since the reproducibility of the rSAA results depends on this, a concrete reference or implementation detail would be helpful.
  4. [Notation throughout] The symbol hat{Psi} is used both for the Monte Carlo sample-average objective (7) and for the true IMSE (6); in Theorem 1 and its proof, expressions such as hat{Psi}(s_m; theta*) and Psi(s_0; theta*) mix the two without definition. Please disambiguate these, for example by writing Psi for the true objective and hat{Psi}_N for the sample average.
  5. [Section 4.3] The sentence 'the iteration number M is chosen according to the computing budget' gives no criterion or sensitivity analysis for M. Since Figures 10-16 show trajectories and final designs for varying M, a brief statement on convergence monitoring or stopping rules would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bilevel objective and the lower-level estimator are independently defined, and the numerical validation is benchmarked against external designs.

full rationale

The paper's derivation chain is self-contained and does not reduce to its own inputs. The upper-level objective (6) is an independent Monte Carlo expectation of the squared estimation error over stated priors for wind, emission rates, and observation noise, while the lower-level estimator (11b) is defined by an elastic-net-regularized, nonnegativity-constrained quadratic program. The sensor-allocation problem is not defined in terms of its own solution: the estimator is fixed first, and the sensor locations are then chosen to minimize the resulting IMSE. The hypergradient in Proposition 2 is obtained by implicit differentiation of the lower-level KKT conditions (Appendix A.1), and the convergence analysis in Theorem 1 is a standard stochastic-gradient argument rather than a restatement of the objective. The numerical validation re-evaluates the IMSE with large independent Monte Carlo samples and compares against external initial designs including random, K-means, Support Points, GP, SSPOR, and A-optimal placements, so the claimed improvement is an empirical benchmark result rather than a construction. Self-citations to Liu et al. (2022) for the global solver and to Liu and Yeo (2023) and Yeo et al. (2019) for the absence of closed-form solutions are supporting references for standard facts or implementation tools, and they do not carry the central derivation. A possible omission of the derivative of the sampled data with respect to sensor locations in the hypergradient expression (18) would be a correctness issue, not a circularity issue, because it does not make the output equivalent to the input by definition.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced. The free parameters are user-chosen regularization strengths, and the assumptions are standard regularity and modeling conditions, with strict complementarity being the most fragile.

free parameters (2)
  • lambda1 (ridge penalty) = 0.0001, 0.01, 0.001 in examples
    Regularization hyperparameter in the lower-level elastic net; controls prior width for emission rates and affects sensor placement.
  • lambda2 (lasso penalty) = 0.0001, 0.01, 0.001 in examples
    Sparsity regularization hyperparameter in the lower-level elastic net; encourages zero emission rates.
assumptions (7)
  • domain assumption Strict complementarity for the lower-level QP active set
    Required for the implicit hypergradient in Proposition 2; can fail for L1-regularized problems, and the paper does not verify it numerically.
  • standard math Lipschitz smoothness of hypergradient (Assumption 1)
    Standard smoothness assumption used to derive SGD convergence bounds; not verified but plausible.
  • domain assumption Lower-level optimality gap bounded by delta (Assumption 2)
    Assumes the approximate lower-level solver is close to the true solution; paper asserts it is small but provides no measured delta.
  • standard math Bounded gradient constants (Assumptions 3 to 6)
    Regularity conditions on norms of gradients and inverses needed for the constants in Theorem 1.
  • domain assumption Gaussian plume forward model
    The linear dispersion model (27) is taken as the physical truth; results are conditional on this model.
  • domain assumption Known prior distributions p(theta) and p(beta)
    The design objective assumes these distributions are specified by the user from data or expert judgment.
  • domain assumption Background emission rate set to zero
    Section 2 sets mu=0 without loss of generality, which is only WLOG under a linear model with known background.

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Pith. "Pith review of Sparse Sensor Allocation for Inverse Problems of Detecting Sparse Leaking Emission Sources." pith.science (2026). https://pith.science/paper/VJPHYYRU

@misc{pith2026250905559,
  author       = {Pith},
  title        = {Pith review of: Sparse Sensor Allocation for Inverse Problems of Detecting Sparse Leaking Emission Sources},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VJPHYYRU}},
  note         = {Machine review of arXiv:2509.05559}
}
read the original abstract

This paper investigates the sparse optimal allocation of sensors for detecting sparse leaking emission sources. Because of the non-negativity of emission rates, uncertainty associated with parameters in the forward model, and sparsity of leaking emission sources, the classical linear Gaussian Bayesian inversion setup is limited and no closed-form solutions are available. By incorporating the non-negativity constraints on emission rates, relaxing the Gaussian distributional assumption, and considering the parameter uncertainties associated with the forward model, this paper provides comprehensive investigations, technical details, in-depth discussions and implementation of the optimal sensor allocation problem leveraging a bilevel optimization framework. The upper-level problem determines the optimal sensor locations by minimizing the Integrated Mean Squared Error (IMSE) of the estimated emission rates over uncertain wind conditions, while the lower-level problem solves an inverse problem that estimates the emission rates. Two algorithms, including the repeated Sample Average Approximation (rSAA) and the Stochastic Gradient Descent based bilevel approximation (SBA), are thoroughly investigated. It is shown that the proposed approach can further reduce the IMSE of the estimated emission rates starting from various initial sensor deployment generated by existing approaches. Convergence analysis is performed to obtain the performance guarantee, and numerical investigations show that the proposed approach can allocate sensors according to the parameters and output of the forward model. Computationally efficient code with GPU acceleration is available on GitHub so that the approach readily applicable.

Figures

Figures reproduced from arXiv: 2509.05559 by the authors.

Figure 1
Figure 1. ∼1,800 point “super-emitters” in Permian Basin, Texas with a simulated concen￾tration field using the Gaussian Plume model for a small region with 20 sources (the geo￾referenced sources and their emission strengths are obtained from Cusworth et al. (2021)). Very often, such problems share three important characteristics. (i) sparsity in sensor allocation: the number of sensors that can be placed is far less than the… view at source ↗
Figure 2
Figure 2. Placement of sensors (green stars) on the straight line (blue line). [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Results from Example I.                     [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (24 more)
Figure 4
Figure 4. Figure 4: The stochastic upper bound defined in (21), [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Trajectories using different initial values of [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Trajectories using different initial values of [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Initial sensor locations (indicated by “ [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: The proposed approach significantly reduces the values of the objective function by moving the initial sensor locations (“▲”) to the final sensor locations (“⋆”) [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Wind rose plot for Example II. The locations of sensors and the corresponding objective values along the iterations are shown Figures 10 and 11. In these figures, the objective value is re-evaluated with large Monte Carlo samples (i.e., 100,000 samples) for each iterat…
Figure 10
Figure 10. Figure 10: Deployment of 5 sensors for 10 emission sources (initial location: [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Deployment of 6 sensors for 10 emission sources (initial location: [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: Allocation of 7 sensors for 10 emission sources with different initial guesses. [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: Allocation of 10 sensors for 20 emission sources (initial location: [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Allocation of 10 sensors for 50 sources (initial location: [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]
Figure 15
Figure 15. Figure 15: Sensor placement for 50 emission sources (initial location: [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
Figure 16
Figure 16. Figure 16: Placement of 50 sensors for 100 emission sources with [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 17
Figure 17. Figure 17: Allocation of 10 sensors (S1-S10) for 20 sources (E1-E20). [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]
Figure 18
Figure 18. Figure 18: Effect of sensor noise and emission uncertainty on estimation error. [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]
Figure 19
Figure 19. Figure 19: Comparison of the estimated emission rates based on different sensor allocations. [PITH_FULL_IMAGE:figures/full_fig_p024_19.png]
Figure 20
Figure 20. Figure 20: The screenshot of the GUI of the code that implements the proposed approach. 5 Conclusions This paper provided comprehensive investigations, technical details, in-depth discussions and implementation of the optimal sensor placement problem for linear dispersion proces…
Figure 21
Figure 21. Figure 21: Allocation of 8 sensors for 10 emission sources ( [PITH_FULL_IMAGE:figures/full_fig_p041_21.png]
Figure 22
Figure 22. Figure 22: Allocation of 9 sensors for 10 emission sources ( [PITH_FULL_IMAGE:figures/full_fig_p041_22.png]
Figure 23
Figure 23. Figure 23: Allocation of 7 sensors for 10 emission sources ( [PITH_FULL_IMAGE:figures/full_fig_p041_23.png]
Figure 24
Figure 24. Figure 24: Allocation of 9 sensors for 10 emission sources ( [PITH_FULL_IMAGE:figures/full_fig_p042_24.png]
Figure 25
Figure 25. Figure 25: Comparison of final designs between different hyperparameters (10 emission [PITH_FULL_IMAGE:figures/full_fig_p042_25.png]
Figure 26
Figure 26. Figure 26: Initial sensor locations by random guess (10 emission sources and 7 sensors) [PITH_FULL_IMAGE:figures/full_fig_p043_26.png]
Figure 27
Figure 27. Figure 27: 10 sampled scenarios using the Gaussian Plume model and parameters in the [PITH_FULL_IMAGE:figures/full_fig_p043_27.png]

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Works this paper leans on

56 extracted references · 52 canonical work pages

  1. [1]

    Petra, G

    Alexanderian, A., N. Petra, G. Stadler, and O. Ghattas (2014). A-optimal design of experiments for infinite-dimensional bayesian linear inverse problems with regularized \_0-sparsification. SIAM Journal on Scientific Computing\/ 36\/ (5), A2122--A2148

    work page 2014

  2. [2]

    Antil, H., Z. W. Di, and R. Khatri (2020). Bilevel optimization, deep learning and fractional laplacian regularization with applications in tomography. Inverse Problems\/ 36\/ (6), 064001

    work page 2020

  3. [3]

    Leyffer, and T

    Attia, A., S. Leyffer, and T. Munson (2023). Robust a-optimal experimental design for bayesian inverse problems. arXiv preprint arXiv:2305.03855v1\/

    arXiv 2023

  4. [4]

    Brunton, B. W., S. L. Brunton, J. L. Proctor, and J. N. Kutz (2016). Sparse sensor placement optimization for classification. SIAM Journal on Applied Mathematics\/ 76\/ (5), 2099--2122

    work page 2016

  5. [5]

    Chen, Q., M. Modi, G. McGaughey, Y. Kimura, E. McDonald-Buller, and D. T. Allen (2022). Simulated methane emission detection capabilities of continuous monitoring networks in an oil and gas production region. Atmosphere\/ 13\/ (4)

    work page 2022

  6. [6]

    Chepuri, S. P. and G. Leus (2014). Continuous sensor placement. IEEE signal processing letters\/ 22\/ (5), 544--548

    work page 2014

  7. [7]

    Chow, F. K., B. Kosovi \'c , and S. Chan (2008). Source inversion for contaminant plume dispersion in urban environments using building-resolving simulations. Journal of applied meteorology and climatology\/ 47\/ (6), 1553--1572

    work page 2008

  8. [8]

    Cusworth, D. H., R. M. Duren, A. K. Thorpe, W. Olson-Duvall, J. Heckler, J. W. Chapman, M. L. Eastwood, M. C. Helmlinger, R. O. Green, G. P. Asner, et al. (2021). Intermittency of large methane emitters in the permian basin. Environmental Science & Technology Letters\/ 8\/ (7), 567--573

    work page 2021

Show all 56 references
  1. [9]

    Karpatne, K

    Daw, A., A. Karpatne, K. Yeo, and L. Klein (2022). Source identification and field reconstruction of advection-diffusion process from sparse sensor measurements. Conference on Neural Information Processing Systems

    2022

  2. [10]

    de Silva, B. M., K. Manohar, E. Clark, B. W. Brunton, S. L. Brunton, and J. N. Kutz (2021). Pysensors: A python package for sparse sensor placement. arXiv preprint arXiv:2102.13476\/

    2021 arXiv

  3. [11]

    Jain, and U

    Ganguly, A., S. Jain, and U. Watchareeruetai (2023). Amortized variational inference: A systematic review. Journal of Artificial Intelligence Research\/ 78 , 167--215

    2023

  4. [12]

    Kent, and L

    Giovannelli, T., G. Kent, and L. N. Vicente (2021). Inexact bilevel stochastic gradient methods for constrained and unconstrained lower-level problems. arXiv preprint arXiv:2110.00604\/

    2021 arXiv

  5. [13]

    Golub, G. H., P. C. Hansen, and D. P. O'Leary (1999). Tikhonov regularization and total least squares. SIAM journal on matrix analysis and applications\/ 21\/ (1), 185--194

    1999

  6. [14]

    Guidelines, F. (2022). Assessing methane emissions from orphaned wells to meet reporting requirements of the 2021 infrastructure investment and jobs act (bil): Federal program guidelines. Available online, https://www.doi.gov/sites/ doi.gov/files/federal-orphaned-wells-methane...

    2022

  7. [15]

    Horesh, and L

    Haber, E., L. Horesh, and L. Tenorio (2009). Numerical methods for the design of large-scale nonlinear discrete ill-posed inverse problems. Inverse Problems\/ 26\/ (2), 025002

    2009

  8. [16]

    Magnant, C

    Haber, E., Z. Magnant, C. Lucero, and L. Tenorio (2012). Numerical methods for a-optimal designs with a sparsity constraint for ill-posed inverse problems. Computational Optimization and Applications\/ 52 , 293--314

    2012

  9. [17]

    Herring, J. L., J. G. Nagy, and L. Ruthotto (2018). Lap: a linearize and project method for solving inverse problems with coupled variables. Sampling Theory in Signal and Image Processing\/ 17 , 127--151

    2018

  10. [18]

    Kaminski, F

    Houweling, S., T. Kaminski, F. Dentener, J. Lelieveld, and M. Heimann (1999). Inverse modeling of methane sources and sinks using the adjoint of a global transport model. Journal of Geophysical Research: Atmospheres\/ 104\/ (D21), 26137--26160

    1999

  11. [19]

    Huan, X. and Y. Marzouk (2014). Gradient-based stochastic optimization methods in bayesian experimental design. International Journal for Uncertainty Quantification\/ 4\/ (6)

    2014

  12. [20]

    Huan, X. and Y. M. Marzouk (2013). Simulation-based optimal bayesian experimental design for nonlinear systems. Journal of Computational Physics\/ 232\/ (1), 288--317

    2013

  13. [21]

    Hwang, Y., H. J. Kim, W. Chang, K. Yeo, and Y. Kim (2019). Bayesian pollution source identification via an inverse physics model. Computational Statistics & Data Analysis\/ 134 , 76--92

    2019

  14. [22]

    Jakkala, K. and S. Akella (2023). Efficient sensor placement from regression with sparse gaussian processes in continuous and discrete spaces. arXiv preprint arXiv:2303.00028\/

    2023 arXiv

  15. [23]

    Joshi, S. and S. Boyd (2008). Sensor selection via convex optimization. IEEE Transactions on Signal Processing\/ 57\/ (2), 451--462

    2008

  16. [24]

    Tsaknakis, Y

    Khanduri, P., I. Tsaknakis, Y. Zhang, J. Liu, S. Liu, J. Zhang, and M. Hong (2023). Linearly constrained bilevel optimization: A smoothed implicit gradient approach

    2023

  17. [25]

    Klein, L. J., T. van Kessel, D. Nair, R. Muralidhar, H. Hamann, and N. Sosa (2017). Monitoring fugitive methane gas emission from natural gas pads. In International Electronic Packaging Technical Conference and Exhibition , Volume 58097, pp.\ V001T03A006. American Society of M...

    2017

  18. [26]

    Klise, K. A., B. L. Nicholson, and C. D. Laird (2017). Sensor placement optimization using chama. Technical report, Sandia National Lab.(SNL-NM), Albuquerque, NM (United States)

    2017

  19. [27]

    Singh, and C

    Krause, A., A. Singh, and C. Guestrin (2008). Near-optimal sensor placements in gaussian processes: Theory, efficient algorithms and empirical studies. Journal of Machine Learning Research\/ 9\/ (2)

    2008

  20. [28]

    Liu, S. and L. N. Vicente (2021). The stochastic multi-gradient algorithm for multi-objective optimization and its application to supervised machine learning. Annals of Operations Research\/ , 1--30

    2021

  21. [29]

    Liu, X. and K. Yeo (2023). Inverse models for estimating the initial condition of spatio-temporal advection-diffusion processes. Technometrics\/ , 1--14

    2023

  22. [30]

    Liu, X., K. Yeo, L. Klein, Y. Hwang, D. Phan, and X. Liu (2022). Optimal sensor placement for atmospheric inverse modelling. In 2022 IEEE International Conference on Big Data (Big Data) , pp.\ 4848--4853. IEEE

    2022

  23. [31]

    Mak, S. and V. R. Joseph (2018). Support points. The Annals of Statistics\/ 46\/ (6A), 2562--2592

    2018

  24. [32]

    Manohar, K., B. W. Brunton, J. N. Kutz, and S. L. Brunton (2018). Data-driven sparse sensor placement for reconstruction: Demonstrating the benefits of exploiting known patterns. IEEE Control Systems Magazine\/ 38\/ (3), 63--86

    2018

  25. [33]

    Manohar, K., J. N. Kutz, and S. L. Brunton (2021). Optimal sensor and actuator selection using balanced model reduction. IEEE Transactions on Automatic Control\/ 67\/ (4), 2108--2115

    2021

  26. [34]

    Meng, M. and X. Li (2020). Aug-pdg: Linear convergence of convex optimization with inequality constraints. arXiv preprint arXiv:2011.08569\/

    2020 arXiv

  27. [35]

    Narayanan, S. D., Z. B. Patel, A. Agnihotri, and N. Batra (2020). A toolkit for spatial interpolation and sensor placement. In Proceedings of the 18th Conference on Embedded Networked Sensor Systems , pp.\ 653--654

    2020

  28. [36]

    Juditsky, G

    Nemirovski, A., A. Juditsky, G. Lan, and A. Shapiro (2009). Robust stochastic approximation approach to stochastic programming. SIAM Journal on optimization\/ 19\/ (4), 1574--1609

    2009

  29. [37]

    Parise, F. and A. Ozdaglar (2017). Sensitivity analysis for network aggregative games. In 2017 IEEE 56th Annual Conference on Decision and Control (CDC) , pp.\ 3200--3205. IEEE

    2017

  30. [38]

    Chebira, and M

    Ranieri, J., A. Chebira, and M. Vetterli (2014). Near-optimal sensor placement for linear inverse problems. IEEE Transactions on signal processing\/ 62\/ (5), 1135--1146

    2014

  31. [39]

    Riddick, S. N., M. Mbua, J. C. Riddick, C. Houlihan, A. L. Hodshire, and D. J. Zimmerle (2023). Uncertainty quantification of methods used to measure methane emissions of 1 g ch4 h- 1. Sensors\/ 23\/ (22), 9246

    2023

  32. [40]

    Chung, and M

    Ruthotto, L., J. Chung, and M. Chung (2018). Optimal experimental design for inverse problems with state constraints. SIAM Journal on Scientific Computing\/ 40\/ (4), B1080--B1100

    2018

  33. [41]

    Shapiro, A. and A. Philpott (2007). A tutorial on stochastic programming. Manuscript. Available at www2. isye. gatech. edu/ashapiro/publications. html\/ 17

    2007

  34. [42]

    Sharrock, L. and N. Kantas (2022). Joint online parameter estimation and optimal sensor placement for the partially observed stochastic advection-diffusion equation. SIAM/ASA Journal on Uncertainty Quantification\/ 10\/ (1), 55--95

    2022

  35. [43]

    Shen, J. and T. F. Chan (2002). Mathematical models for local nontexture inpaintings. SIAM Journal on Applied Mathematics\/ 62\/ (3), 1019--1043

    2002

  36. [44]

    Sinsbeck, M. and W. Nowak (2017). Sequential design of computer experiments for the solution of bayesian inverse problems. SIAM/ASA Journal on Uncertainty Quantification\/ 5\/ (1), 640--664

    2017

  37. [45]

    Spantini, A., T. Cui, K. Willcox, L. Tenorio, and Y. Marzouk (2017). Goal-oriented optimal approximations of bayesian linear inverse problems. SIAM Journal on Scientific Computing\/ 39\/ (5), S167--S196

    2017

  38. [46]

    Stockie, J. M. (2011). The mathematics of atmospheric dispersion modeling. Siam Review\/ 53\/ (2), 349--372

    2011

  39. [47]

    Tarantola, A. (2005). Inverse problem theory and methods for model parameter estimation . SIAM

    2005

  40. [48]

    Khanduri, and M

    Tsaknakis, I., P. Khanduri, and M. Hong (2022). An implicit gradient-type method for linearly constrained bilevel problems. In ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) , pp.\ 5438--5442. IEEE

    2022

  41. [49]

    Wang, Z., J. M. Bardsley, A. Solonen, T. Cui, and Y. M. Marzouk (2017). Bayesian inverse problems with l\_1 priors: a randomize-then-optimize approach. SIAM Journal on Scientific Computing\/ 39\/ (5), S140--S166

    2017

  42. [50]

    Willoughby, R. A. (1979). Solutions of ill-posed problems (an tikhonov and vy arsenin). SIAM Review\/ 21\/ (2), 266

    1979

  43. [51]

    Chen, and O

    Wu, K., P. Chen, and O. Ghattas (2023). An offline-online decomposition method for efficient linear bayesian goal-oriented optimal experimental design: Application to optimal sensor placement. SIAM Journal on Scientific Computing\/ 45\/ (1), B57--B77

    2023

  44. [52]

    O’Leary-Roseberry, P

    Wu, K., T. O’Leary-Roseberry, P. Chen, and O. Ghattas (2023). Large-scale bayesian optimal experimental design with derivative-informed projected neural network. Journal of Scientific Computing\/ 95\/ (1), 30

    2023

  45. [53]

    Yang, C., M. Wen, C. Chen, C. Li, J. Huang, L. Song, and Y. Li (2025). Improving the accuracy of methane sensor with dual measurement modes based on off-axis integrated cavity output spectroscopy using white noise perturbation. Applied Sciences\/ 15\/ (10), 5562

    2025

  46. [54]

    Hwang, X

    Yeo, K., Y. Hwang, X. Liu, and J. Kalagnanam (2019). Development of hp-inverse model by using generalized polynomial chaos. Computer Methods in Applied Mechanics and Engineering\/ 347 , 1--20

    2019

  47. [55]

    Yu, J., V. M. Zavala, and M. Anitescu (2018). A scalable design of experiments framework for optimal sensor placement. Journal of Process Control\/ 67 , 44--55

    2018

  48. [56]

    Zou, H. and T. Hastie (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society Series B: Statistical Methodology\/ 67\/ (2), 301--320

    2005

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.