The article discusses methods for identifying parameters of partial differential equations. Identification problems are often called poorly conditioned. However, the reason is the ambiguity of the solution, even at the point of minimality of the criterion. In particular, the article discusses: (1) An analysis of singular values for identify the unambiguous solution. The basis of these methods is a singular value decomposition of the matrix of experimental data, what makes it possible to abandon the inversion of matrices and, as a consequence, translate the problem of ill-conditioned problems into the problem of ambiguity of the solution. (2) The issues of anomalous measurements and combination of various experiments. (3) A universal optimization method for identifying parameters by their complete simple enumeration. The method is based on fast calculation of points on a multidimensional sphere. (4) The issues of identifiability of linear structure models and construction of experiments guaranteeing identification. (5) A method for identifying parameters via projecting linear structure model elements onto the plane of guarantors. (6) An approach to constructing histograms of unknown parameters of dynamic systems before calculating them using any algorithm. The approach is based on linear structure models with parameters on a sphere and a rather unexpected application of singular value decomposition. (7) The methods are accompanied by examples of heat equations. The Appendix containsalgorithmsintheMATLABlanguageforallexamples. (8)The presented optimization, projection and statistical methods based on the concept of linear structure models allow solving the same identification problem in fundamentally different ways, which significantly increases the reliability of the results obtained.
| Published in | International Journal of Systems Engineering (Volume 8, Issue 2) |
| DOI | 10.11648/j.ijse.20240802.12 |
| Page(s) | 40-65 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Identification, Identifiability, Guaranteeing Identification Experiment, Projection onto Planes of Guarantors, Histograms of Unknown Parameters
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| [3] | Golub G. H., Van Loan C. F. An analisis of the total least- squares problem. SIAM J. Numer. Anal. 1980, 17(6), pp. 883–893. |
| [4] | Van Huffel S., Vandewalle J. The total least squares problem. SIAM, Philadelphia. 1991. |
| [5] | Pearson K. On lines and planes of closed fit to system of points in space. Philosophical Magazine. 1901, 2, pp. 559–572. |
| [6] | Kopysov O. Yu. Identification of Order, Parameters and State Estimation via Projection onto the Plane of Guarantors. IFAC-PapersOnLine. 2023, 56(2), pp. 7771– 7777. 22th IFAC World Congress. |
| [7] | PearsonK.Onthemathematicaltheoryofevalution. Phil. Trans. R. Soc. Lond., 1895, A, pp. 343-424. |
APA Style
Kopysov, O. Y. (2024). Identification of Physical Dynamical Processes via Linear Structure Models (Part 1). International Journal of Systems Engineering, 8(2), 40-65. https://doi.org/10.11648/j.ijse.20240802.12
ACS Style
Kopysov, O. Y. Identification of Physical Dynamical Processes via Linear Structure Models (Part 1). Int. J. Syst. Eng. 2024, 8(2), 40-65. doi: 10.11648/j.ijse.20240802.12
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Download@article{10.11648/j.ijse.20240802.12,
author = {Oleg Yu. Kopysov},
title = {Identification of Physical Dynamical Processes via Linear Structure Models (Part 1)},
journal = {International Journal of Systems Engineering},
volume = {8},
number = {2},
pages = {40-65},
doi = {10.11648/j.ijse.20240802.12},
url = {https://doi.org/10.11648/j.ijse.20240802.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijse.20240802.12},
abstract = {The article discusses methods for identifying parameters of partial differential equations. Identification problems are often called poorly conditioned. However, the reason is the ambiguity of the solution, even at the point of minimality of the criterion. In particular, the article discusses: (1) An analysis of singular values for identify the unambiguous solution. The basis of these methods is a singular value decomposition of the matrix of experimental data, what makes it possible to abandon the inversion of matrices and, as a consequence, translate the problem of ill-conditioned problems into the problem of ambiguity of the solution. (2) The issues of anomalous measurements and combination of various experiments. (3) A universal optimization method for identifying parameters by their complete simple enumeration. The method is based on fast calculation of points on a multidimensional sphere. (4) The issues of identifiability of linear structure models and construction of experiments guaranteeing identification. (5) A method for identifying parameters via projecting linear structure model elements onto the plane of guarantors. (6) An approach to constructing histograms of unknown parameters of dynamic systems before calculating them using any algorithm. The approach is based on linear structure models with parameters on a sphere and a rather unexpected application of singular value decomposition. (7) The methods are accompanied by examples of heat equations. The Appendix containsalgorithmsintheMATLABlanguageforallexamples. (8)The presented optimization, projection and statistical methods based on the concept of linear structure models allow solving the same identification problem in fundamentally different ways, which significantly increases the reliability of the results obtained.},
year = {2024}
}
TY - JOUR T1 - Identification of Physical Dynamical Processes via Linear Structure Models (Part 1) AU - Oleg Yu. Kopysov Y1 - 2024/12/18 PY - 2024 N1 - https://doi.org/10.11648/j.ijse.20240802.12 DO - 10.11648/j.ijse.20240802.12 T2 - International Journal of Systems Engineering JF - International Journal of Systems Engineering JO - International Journal of Systems Engineering SP - 40 EP - 65 PB - Science Publishing Group SN - 2640-4230 UR - https://doi.org/10.11648/j.ijse.20240802.12 AB - The article discusses methods for identifying parameters of partial differential equations. Identification problems are often called poorly conditioned. However, the reason is the ambiguity of the solution, even at the point of minimality of the criterion. In particular, the article discusses: (1) An analysis of singular values for identify the unambiguous solution. The basis of these methods is a singular value decomposition of the matrix of experimental data, what makes it possible to abandon the inversion of matrices and, as a consequence, translate the problem of ill-conditioned problems into the problem of ambiguity of the solution. (2) The issues of anomalous measurements and combination of various experiments. (3) A universal optimization method for identifying parameters by their complete simple enumeration. The method is based on fast calculation of points on a multidimensional sphere. (4) The issues of identifiability of linear structure models and construction of experiments guaranteeing identification. (5) A method for identifying parameters via projecting linear structure model elements onto the plane of guarantors. (6) An approach to constructing histograms of unknown parameters of dynamic systems before calculating them using any algorithm. The approach is based on linear structure models with parameters on a sphere and a rather unexpected application of singular value decomposition. (7) The methods are accompanied by examples of heat equations. The Appendix containsalgorithmsintheMATLABlanguageforallexamples. (8)The presented optimization, projection and statistical methods based on the concept of linear structure models allow solving the same identification problem in fundamentally different ways, which significantly increases the reliability of the results obtained. VL - 8 IS - 2 ER -
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