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An a priori method for estimating the informativeness of the configuration of sensor placement when solving inverse problems of remote sensing

  • Dmitry V. Lukyanenko ORCID logo EMAIL logo , Bulat I. Valiakhmetov ORCID logo , Eugene E. Tyrtyshnikov ORCID logo and Anatoly G. Yagola ORCID logo
Published/Copyright: January 13, 2025

Abstract

The question of finding the optimal location and number of sensors is important when solving applied inverse problems of remote sensing. An incorrect answer to this question can significantly affect (1) the accuracy of reconstructing the unknown parameters of the object under study, and/or (2) the computational complexity of the inverse problem being solved. The work proposes an economical algorithm that allows among all equivalent (in the sense of the computational complexity of obtaining a solution to the inverse problem) sensor configurations, to select the one that will give a more accurate result when processing experimental data. This algorithm also allows to make a conclusion about whether it is really necessary to process all the experimental data obtained in the experiment, or whether it is possible to limit ourselves to only part of them without a significant loss of accuracy in the reconstructed solution. The algorithm is based on (1) the application of the mosaic-skeleton approximation method to the matrix of the system of equations to which the inverse problem being solved is reduced, and (2) the calculation of the compression rate of the approximating matrix relative to the original dense matrix for a given permissible relative approximation error. Thus, the algorithm is a priori, that is, it does not require a preliminary search for a solution to the inverse problem for the sensor configuration under study. Moreover, if the formulas that define the elements of the system matrix are known, then the algorithm requires the calculation of not all of these elements, but only part of them.

MSC 2020: 65R32; 68W25; 65F55

Award Identifier / Grant number: 23-41-00002

Funding statement: Supported by the Russian Science Foundation (project 23-41-00002).

References

[1] A. A. Alonso, I. G. Kevrekidis, J. R. Banga and C. E. Frouzakis, Optimal sensor location and reduced order observer design for distributed process systems, Comput. Chem. Eng. 28 (2004), no. 1, 27–35. 10.1016/S0098-1354(03)00175-3Search in Google Scholar

[2] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University, Cambridge, 2004. 10.1017/CBO9780511804441Search in Google Scholar

[3] L. CanXing, L. Xingshan, Z. Ping and D. Jing, A review on optimal sensor placement for health monitoring, 8th International Conference on Electronic Measurement and Instruments, IEEE Press, Piscataway (2007), 170–173. 10.1109/ICEMI.2007.4351109Search in Google Scholar

[4] S. P. Chepuri and G. Leus, Sparsity-promoting sensor selection for non-linear measurement models, IEEE Trans. Signal Process. 63 (2015), no. 3, 684–698. 10.1109/TSP.2014.2379662Search in Google Scholar

[5] D. J. Chmielewski, T. Palmer and V. Manousiouthakis, On the theory of optimal sensor placement, AIChE J. 48 (2002), no. 5, 1001–1012. 10.1002/aic.690480510Search in Google Scholar

[6] Y. M. Fu and L. Yu, Optimal sensor placement based on mac and spga algorithms, Advances in Industrial and Civil Engineering, Adv. Mater. Res. 594, Trans Tech, Stäfa (2012), 1118–1122. 10.4028/www.scientific.net/AMR.594-597.1118Search in Google Scholar

[7] K. Fukami, R. Maulik, N. Ramachandra, K. Fukagata and K. Taira, Global field reconstruction from sparse sensors with voronoi tessellation-assisted deep learning, Nature Mach. Intell. 3 (2021), no. 11, 945–951. 10.1038/s42256-021-00402-2Search in Google Scholar

[8] M. Galetto and B. Pralio, Optimal sensor positioning for large scale metrology applications, Precision Eng. 34 (2010), no. 3, 563–577. 10.1016/j.precisioneng.2010.02.001Search in Google Scholar

[9] S. A. Goreinov and E. E. Tyrtyshnikov, The maximal-volume concept in approximation by low-rank matrices, Structured Matrices in Mathematics, Computer Science, and Engineering, Contemp. Math. 280, American Mathematical Society, Providence (2001), 47–51. 10.1090/conm/280/4620Search in Google Scholar

[10] S. A. Goreinov, E. E. Tyrtyshnikov and N. L. Zamarashkin, A theory of pseudoskeleton approximations, Linear Algebra Appl. 261 (1997), 1–21. 10.1016/S0024-3795(97)80059-6Search in Google Scholar

[11] S. A. Goreinov, N. L. Zamarashkin and E. E. Tyrtyshnikov, Pseudo-skeleton approximations of matrices, Dokl. Akad. Nauk 343 (1995), no. 2, 151–152. Search in Google Scholar

[12] W. Hackbusch, Hierarchical Matrices: Algorithms and Analysis, Springer Ser. Comput. Math. 49, Springer, Heidelberg, 2015. 10.1007/978-3-662-47324-5Search in Google Scholar

[13] M. Hamdollahzadeh, S. Adelipour and F. Behnia, Optimal sensor configuration for two dimensional source localization based on tdoa/fdoa measurements, 17th International Radar Symposium, IEEE Press, Piscataway (2016), 1–6. 10.1109/IRS.2016.7497276Search in Google Scholar

[14] M. Hintermüller, C. N. Rautenberg, M. Mohammadi and M. Kanitsar, Optimal sensor placement: A robust approach, SIAM J. Control Optim. 55 (2017), no. 6, 3609–3639. 10.1137/16M1088867Search in Google Scholar

[15] H. Jamali-Rad, A. Simonetto, X. Ma and G. Leus, Distributed sparsity-aware sensor selection, IEEE Trans. Signal Process. 63 (2015), no. 22, 5951–5964. 10.1109/TSP.2015.2460224Search in Google Scholar

[16] C. Jiang, Z. Chen, R. Su and Y. C. Soh, Group greedy method for sensor placement, IEEE Trans. Signal Process. 67 (2019), no. 9, 2249–2262. 10.1109/TSP.2019.2903017Search in Google Scholar

[17] Y. Jiang, D. Li and G. Song, On the physical significance of the effective independence method for sensor placement, J. Phys. Conf. Ser. 842 (2017), Article ID 012030. 10.1088/1742-6596/842/1/012030Search in Google Scholar

[18] S. Joshi and S. Boyd, Sensor selection via convex optimization, IEEE Trans. Signal Process. 57 (2009), no. 2, 451–462. 10.1109/TSP.2008.2007095Search in Google Scholar

[19] M. Juhlin and A. Jakobsson, Optimal sensor placement for localizing structured signal sources, Signal Process. 202 (2023), Article ID 108679. 10.1016/j.sigpro.2022.108679Search in Google Scholar

[20] D. C. Kammer, Sensor placement for on-orbit modal identification and correlation of large space structures, J. Guidance Control Dynam. 14 (1991), no. 2, 251–259. 10.2514/3.20635Search in Google Scholar

[21] D. C. Kammer and L. Yao, Enhancement of on-orbit modal identification of large space structures through sensor placement, J. Sound Vibration 171 (1994), no. 1, 119–139. 10.1006/jsvi.1994.1107Search in Google Scholar

[22] N. Karnik, M. G. Abdo, C. E. Estrada Perez, J. S. Yoo, J. J. Cogliati, R. S. Skifton, P. Calderoni, S. L. Brunton and K. Manohar, Optimal sensor placement with adaptive constraints for nuclear digital twins, preprint (2023), https://arxiv.org/abs/2306.13637. Search in Google Scholar

[23] V. Kekatos, G. B. Giannakis and B. Wollenberg, Optimal placement of phasor measurement units via convex relaxation, IEEE Trans. Power Syst. 27 (2012), no. 3, 1521–1530. 10.1109/TPWRS.2012.2185959Search in Google Scholar

[24] A. Krause, A. Singh and C. Guestrin, Near-optimal sensor placements in gaussian processes: Theory, efficient algorithms and empirical studies, J. Mach. Learn. Res. 9 (2008), no. 2, 235–284. Search in Google Scholar

[25] E.-T. Lee and H.-C. Eun, Optimal sensor placement in reduced-order models using modal constraint conditions, Sensors 22 (2022), no. 2, Paper No. 589. 10.3390/s22020589Search in Google Scholar PubMed PubMed Central

[26] S. Liu, S. P. Chepuri, M. Fardad, E. Maşazade, G. Leus and P. K. Varshney, Sensor selection for estimation with correlated measurement noise, IEEE Trans. Signal Process. 64 (2016), no. 13, 3509–3522. 10.1109/TSP.2016.2550005Search in Google Scholar

[27] K. Manohar, B. W. Brunton, J. N. Kutz and S. L. Brunton, Data-driven sparse sensor placement for reconstruction: Demonstrating the benefits of exploiting known patterns, IEEE Control Syst. 38 (2018), no. 3, 63–86. 10.1109/MCS.2018.2810460Search in Google Scholar

[28] J. Neering, M. Bordier and N. Maizi, Optimal passive source localization, International Conference on Sensor Technologies and Applications, IEEE Press, Piscataway (2007), 295–300. 10.1109/SENSORCOMM.2007.4394937Search in Google Scholar

[29] N. H. Nguyen and K. Doğançay, Optimal geometry analysis for multistatic TOA localization, IEEE Trans. Signal Process. 64 (2016), no. 16, 4180–4193. 10.1109/TSP.2016.2566611Search in Google Scholar

[30] V. Nieminen and J. Sopanen, Optimal sensor placement of triaxial accelerometers for modal expansion, Mech. Syst. Signal Process. 184 (2023), Article ID 109581. 10.1016/j.ymssp.2022.109581Search in Google Scholar

[31] D. Pearson, S. U. Pillai and Y. Lee, An algorithm for near-optimal placement of sensor elements, IEEE Trans. Inform. Theory 36 (1990), no. 6, 1280–1284. 10.1109/18.59928Search in Google Scholar

[32] S. Ručevskis, T. Rogala and A. Katunin, Optimal sensor placement for modal-based health monitoring of a composite structure, Sensors 22 (2022), 10.3390/s22103867. 10.3390/s22103867Search in Google Scholar PubMed PubMed Central

[33] Y. Saito, T. Nonomura, K. Nankai, K. Yamada, K. Asai, Y. Sasaki and D. Tsubakino, Data-driven vector-measurement-sensor selection based on greedy algorithm, IEEE Sensors Lett. 4 (2020), no. 7, 1–4. 10.1109/LSENS.2020.2999186Search in Google Scholar

[34] Y. Saito, T. Nonomura, K. Yamada, K. Nakai, T. Nagata, K. Asai, Y. Sasaki and D. Tsubakino, Determinant-based fast greedy sensor selection algorithm, IEEE Access 9 (2021), 68535–68551. 10.1109/ACCESS.2021.3076186Search in Google Scholar

[35] A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Math. Appl. 328, Kluwer Academic, Dordrecht, 1995. 10.1007/978-94-015-8480-7Search in Google Scholar

[36] E. E. Tyrtyshnikov, A Brief Introduction to Numerical Analysis, Birkhäuser, Boston, 1997. 10.1007/978-0-8176-8136-4Search in Google Scholar

[37] E. E. Tyrtyshnikov, Mosaic-skeleton approximations, Calcolo 33 (1997), 47–57. 10.1007/BF02575706Search in Google Scholar

[38] E. E. Tyrtyshnikov, Incomplete cross approximation in the mosaic-skeleton method, Computing 64 (2000), 367–380. 10.1007/s006070070031Search in Google Scholar

[39] D. Uciński, D-optimal sensor selection in the presence of correlated measurement noise, Measurement 164 (2020), Article ID 107873. 10.1016/j.measurement.2020.107873Search in Google Scholar

[40] Y. Wang, D. Lukyanenko and A. Yagola, Magnetic parameters inversion method with full tensor gradient data, Inverse Probl. Imaging 13 (2019), no. 4, 745–754. 10.3934/ipi.2019034Search in Google Scholar

[41] K. Yamada, Y. Saito, T. Nonomura and K. Asai, Greedy sensor selection for weighted linear least squares estimation under correlated noise, IEEE Access 10 (2022), 79356–79364. 10.1109/ACCESS.2022.3194250Search in Google Scholar

[42] C. Yang and Z. X. Lu, An interval effective independence method for optimal sensor placement based on non-probabilistic approach, Sci. China Technol. Sci. 60 (2017), 186–198. 10.1007/s11431-016-0526-9Search in Google Scholar

[43] T. H. Yi, X. Wang and H. N. Li, Optimal placement of triaxial accelerometers using modal kinetic energy method, Progress in Structures, Appl. Mech. Mater. 166, Trans Tech, Stäfa (2012), 1583–1586. 10.4028/www.scientific.net/AMM.166-169.1583Search in Google Scholar

[44] J. Zhang, S. P. Chepuri, R. C. Hendriks and R. Heusdens, Microphone subset selection for mvdr beamformer based noise reduction, IEEE/ACM Trans. Audio Speech Language Process. 26 (2017), no. 3, 550–563. 10.1109/TASLP.2017.2786544Search in Google Scholar

[45] M. Zhdanov, Inverse Theory and Applications in Geophysics, Elsevier, Amsterdam, 2015. Search in Google Scholar

[46] D. A. Zheltkov and E. E. Tyrtyshnikov, A parallel implementation of the matrix cross approximation method, Numer. Methods Program. 16 (2015), 369–375. 10.26089/NumMet.v16r336Search in Google Scholar

Received: 2024-03-09
Revised: 2024-12-19
Accepted: 2025-01-01
Published Online: 2025-01-13
Published in Print: 2025-02-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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