Estimate of the Convergence Rate of Monte Carlo Method for Normal Distribution on SO(3) when Modeling the Orientations of Polycrystal Grains
https://doi.org/10.1134/S2304487X21030032
Abstract
The electron microscopy method is actively applied in texture analysis to study the properties and characteristics of polycrystalline materials. To reconstruct the orientation distribution function from the experimentally obtained pole figures, methods of approximation of the normal distributions on SO(3) are used. One of these methods is the specialized Monte Carlo method. The modeling algorithm is specified by the following values: parameters of a given distribution (coordinates of the center and 6 parameters of an analogue of the covariance matrix), the number of n convolutions of infinitesimal rotations satisfying the central limit theorem (CLT) on the rotation group of the three-dimensional Euclidean space SO(3) and the sample size N from the obtained approximations. The result of the calculations is obtained in the form of rotation matrices, from which the Euler angles can be uniquely determined. The accuracy of approximations of the class of sequences for which the central limit theorem is valid (CLT sequences) has been studied on the three-dimensional rotation group SO(3) depending on the parameters. The dependence of the central normal distribution on the sharpness parameter ε (analogue of the standard deviation) has been studied. Orientations for the central normal distribution have been simulated by the specialized Monte Carlo method. An analytical estimate has been obtained of the accuracy of the approximations of the method of CLT sequences depending on the sharpness parameter in the sense of weak convergence.
About the Authors
D. V. Belyavskiy
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
T. I. Savyolova
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
References
1. Ivankina T. I., Matthies S. O razvitii kolichestvennogo teksturnogo analiza i primenenii ego v reshenii zadach nauk o Zemle [On the development of the quantitative texture analysis and its application in solving problems of the Earth sciences]. Physics of Particles and Nuclei, 2015. vol. 46. no. 3. pp. 665–768. (in Russian)
2. Borovkov M. V., Savyolova T. I. Vychislenie normalnyh raspredeleniy na gruppe vrashcheniy metodom Monte-Karlo [Calculation of normal distributions on a group of rotations by the Monte Carlo method]. Comput. Math. Math. Phys., 2002, vol. 42, no. 1, pp. 112–128. (in Russian)
3. Borovkov M., Savyolova T. The Computational Approaches to Calculate Normal Distributions on the Rolation Group. J. Appl. Crystallogr. 2007. V. 40. P. 449–455.
4. Borovkov M. V., Savyolova T. I. Normalnye raspredeleniya na SO(3) [Normal distributions on SO(3)]. Мoscow, National Research Nuclear University MEPhI Publ., 2002. 94 p.
5. Savyolova T. I., Korenkova E. F. Otsenka tochnosti nekotoryh statisticheskih harakteristik v teksturnom analize [Estimation of the accuracy of some statistical characteristics in texture analysis]. Zav. lab., 2006. vol. 72. no. 12. pp. 29–34. (in Russian)
6. Shaeben H. A Unified View of Methods to Resolve the Inverse Problem of Texture Geometry. Textures and Microstructures, 1996. V. 25. P. 171–181.
7. Qiu Y. Isotropic Distributions for 3-Dimention Rotations and One-Sample Bayes Inference. Graduate Thesis and Dissertation, 2013. Paper 13007.
8. Frolov A. N. Predelnye teoremy teorii veroyatnostey [Limit theorems of probability theory]. Saint-Petersburg University Publ., 2014. 152 p.
9. Ermakov S. M. Metod Monte-Karlo i smezhnye voprosy [Monte Carlo method and related issues]. Moscow, Nauka Publ., 1975. 471 p.
10. Randle V., Engler O. Introduction to texture analysis: macrotexture, microtexture and orientation mapping. 2nd ed. Boca Raton, CRC Press, 2010. 488 c.
11. Ram F., Zaefferer S., Jäpel T., Raabe D. Error analysis of the crystal orientation and disorientation obtained by the classical electron backscatter diffraction technique. J. Appl. Crystallogr. 2015. V. 48. P. 1–17.
12. Korolev V., Dorofeeva A. O neravnomernykh otsenkakh tochnosti normalnoy approksimatsii dlya raspredeleniy nekotorykh sluchaynykh summ pri oslablennykh momentnykh usloviyakh [On nonuniform estimates of accuracy of normal approximation for distributions of some random sums under relaxed moment conditions]. Inform. Primen., 2018. vol. 12. no. 4. pp. 86–91. (in Russian)
13. Korolev V., Dorofeeva A. Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions. Lithuanian Mathematical Journal. 2017. V. 57. № 1. P. 38–58.
14. Fan X. Exact rates of convergence in some martingale central limit theorems. Journal of Mathematical Analysis and Applications, 2018.
15. Sunklodas J. K. On the rate of convergence in the global central limit theorem for random sums of independent random variables. Lithuanian Mathematical Journal. 2017. V. 57. № 2. P. 244–258.
16. Belyavskiy D. V., Savyolova T. I. Otsenka tochnosti priblizheniy TsPT-posledovatelnoctey dlya normalnykh raspredeleniy na SO(3) [Estimation of Accuracy of CLT-Sequence Approximations for Normal Distributions on SO(3)]. Vestnik NIYaU MIFI, 2019, vol. 8, no. 2, pp. 188–194. (in Russian)
17. Savyolova T. I., Ivanova T. M., Sypchenko M. V. Metody resheniya nekorrektnyh zadach teksturnogo analiza i ih prilozheniya [Methods of solving incorrect problems of texture analysis and their applications]. Moscow, National Research Nuclear University MEPhI Publ., 2012. 268 p.
18. Savyolova T. I., Sypchenko M. V., Korenkova E. F. Otsenki tochnosti statisticheskikh kharakteristik normalnykh raspredeleniy na gruppe vrashcheniy SO(3) [Estimation of the accuracy of statistical characteristics for normal distributions on the roration group SO(3)]. Preprint 002-2006. Moscow, National Research Nuclear University MEPhI Publ., 2006. 32 p.
19. Savyolova T. I., Buharova T. I. Predstavleniya gruppy SU(2) i ikh primeneniye [Representations of the SU(2) group and their application]. Moscow, National Research Nuclear University MEPhI Publ., 1996. 114 p.
20. Antonova A. O., Savyolova T. I. Issledovanie metodami elektronnoy mikroskopii vliyaniya parametrov eksperimenta na vychislenie polyusnyh figur polikristallicheskih materialov [Investigation by electron microscopy of the influence of experimental parameters on the calculation of pole figures of polycrystalline materials]. Kristallografiya, 2016, vol. 61, no. 3, pp. 495–503. (in Russian)
21. Antonova A. O., Dzhumaev P. S., Savyolova T. I. Study of EBSD Experiment Parameters Influence on Computation of Polycrystalline Pole Figures and Orientation Distribution Function. MATEC Web of Conferences (ICMIE), 2016. V. 75. 09007.
Review
For citations:
Belyavskiy D.V., Savyolova T.I. Estimate of the Convergence Rate of Monte Carlo Method for Normal Distribution on SO(3) when Modeling the Orientations of Polycrystal Grains. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2021;10(3):230-238. (In Russ.) https://doi.org/10.1134/S2304487X21030032
Views:
180
Email this article 