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Model reduction for a power grid model
Motion tomography via occupation kernels
1. | Oak Ridge National Laboratory, Computer Science and Mathematics Division |
2. | Oklahoma State University, School of Mechanical and Aerospace Engineering |
3. | Oregon State University, Collaborative Robotics and Intelligent Systems Institute |
4. | University of South Florida, Department of Mathematics and Statistics |
The goal of motion tomography is to recover a description of a vector flow field using measurements along the trajectory of a sensing unit. In this paper, we develop a predictor corrector algorithm designed to recover vector flow fields from trajectory data with the use of occupation kernels developed by Rosenfeld et al. [
References:
[1] |
N. Aronszajn,
Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.
doi: 10.1090/S0002-9947-1950-0051437-7. |
[2] |
A. S. Aweiss, B. D. Owens, J. Rios, J. R. Homola and C. P. Mohlenbrink, Unmanned aircraft systems (UAS) traffic management (UTM) national campaign Ⅱ, AIAA Inf. Syst. - AIAA Infotech @ Aerosp., Kissimmee, FL, 2018.
doi: 10.2514/6.2018-1727. |
[3] |
Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48, American Mathematical Society, Providence, RI, 2000.
doi: 10.1090/coll/048. |
[4] |
R. L. Burden and J. D. Faires, Numerical Analysis, $7^{th}$ edition, Brooks/Cole, 2001. |
[5] |
D. Chang, W. Wu, C. R. Edwards and F. Zhang,
Motion tomography: Mapping flow fields using autonomous underwater vehicles, Internat. J. Robotics Res., 36 (2017), 320-336.
doi: 10.1177/0278364917698747. |
[6] |
R. E. Moore, Computational Functional Analysis, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1985. |
[7] |
V. I. Paulsen and M. Raghupathi, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, Cambridge Studies in Advanced Mathematics, 152, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316219232.![]() ![]() ![]() |
[8] |
J. Petrich, C. A. Woolsey and D. J. Stilwell,
Planar flow model identification for improved navigation of small AUVs, Ocean Engrg., 36 (2009), 119-131.
doi: 10.1016/j.oceaneng.2008.10.002. |
[9] |
J. A. Rosenfeld, R. Kamalapurkar, L. F. Gruss and T. T. Johnson,
Dynamic mode decomposition for continuous time systems with the Liouville operator, preprint, J. Nonlinear Sci., 32 (2022).
doi: 10.1007/s00332-021-09746-w. |
[10] |
J. A. Rosenfeld, B. Russo, R. Kamalapurkar and T. T. Johnson, The occupation kernel method for nonlinear system identification, preprint, arXiv: 1909.11792. |
[11] |
V. Stepanyan and K. S. Krishnakumar, Estimation, navigation and control of multi-rotor drones in an urban wind field, AIAA Inf. Syst. - AIAA Infotech @ Aerosp., Grapevine, TX, 2017.
doi: 10.2514/6.2017-0670. |
[12] |
W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, 182, Readings in Mathematics, Springer-Verlag, New York, 1998.
doi: 10.1007/978-1-4612-0601-9. |
[13] |
H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511617539.![]() ![]() ![]() |
[14] |
W. Wu, D. Chang and F. Zhang,
Glider CT: Reconstructing flow fields from predicted motion of underwater gliders, Proceedings of the Eighth ACM International Conference on Underwater Networks and Systems, (2013), 1-8.
doi: 10.1145/2532378.2532403. |
show all references
References:
[1] |
N. Aronszajn,
Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.
doi: 10.1090/S0002-9947-1950-0051437-7. |
[2] |
A. S. Aweiss, B. D. Owens, J. Rios, J. R. Homola and C. P. Mohlenbrink, Unmanned aircraft systems (UAS) traffic management (UTM) national campaign Ⅱ, AIAA Inf. Syst. - AIAA Infotech @ Aerosp., Kissimmee, FL, 2018.
doi: 10.2514/6.2018-1727. |
[3] |
Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48, American Mathematical Society, Providence, RI, 2000.
doi: 10.1090/coll/048. |
[4] |
R. L. Burden and J. D. Faires, Numerical Analysis, $7^{th}$ edition, Brooks/Cole, 2001. |
[5] |
D. Chang, W. Wu, C. R. Edwards and F. Zhang,
Motion tomography: Mapping flow fields using autonomous underwater vehicles, Internat. J. Robotics Res., 36 (2017), 320-336.
doi: 10.1177/0278364917698747. |
[6] |
R. E. Moore, Computational Functional Analysis, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1985. |
[7] |
V. I. Paulsen and M. Raghupathi, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, Cambridge Studies in Advanced Mathematics, 152, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316219232.![]() ![]() ![]() |
[8] |
J. Petrich, C. A. Woolsey and D. J. Stilwell,
Planar flow model identification for improved navigation of small AUVs, Ocean Engrg., 36 (2009), 119-131.
doi: 10.1016/j.oceaneng.2008.10.002. |
[9] |
J. A. Rosenfeld, R. Kamalapurkar, L. F. Gruss and T. T. Johnson,
Dynamic mode decomposition for continuous time systems with the Liouville operator, preprint, J. Nonlinear Sci., 32 (2022).
doi: 10.1007/s00332-021-09746-w. |
[10] |
J. A. Rosenfeld, B. Russo, R. Kamalapurkar and T. T. Johnson, The occupation kernel method for nonlinear system identification, preprint, arXiv: 1909.11792. |
[11] |
V. Stepanyan and K. S. Krishnakumar, Estimation, navigation and control of multi-rotor drones in an urban wind field, AIAA Inf. Syst. - AIAA Infotech @ Aerosp., Grapevine, TX, 2017.
doi: 10.2514/6.2017-0670. |
[12] |
W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, 182, Readings in Mathematics, Springer-Verlag, New York, 1998.
doi: 10.1007/978-1-4612-0601-9. |
[13] |
H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511617539.![]() ![]() ![]() |
[14] |
W. Wu, D. Chang and F. Zhang,
Glider CT: Reconstructing flow fields from predicted motion of underwater gliders, Proceedings of the Eighth ACM International Conference on Underwater Networks and Systems, (2013), 1-8.
doi: 10.1145/2532378.2532403. |



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