Geometri Uydurma için En İyi Eşlenik Gradyan Yönteminin Tespit Edilmesi

Abstract

Bu çalışmada, ölçülen 2B profillere geometri uydurulması için en iyi eşlenik gradyan (EG) yönteminin tespit edilmesi hedeflenmektedir. Bu amaçla, iyi bilinen üç eşlenik gradyan yöntemleri, Fletcher-Reeves, Polak-Ribiere and Hestenes-Stiefel kullanıldı. Adı geçen yöntemlerin performansları test etmek için daire, kare, üçgen, elips ve dikdörtgen geometrilerini içeren test parçaları ilk olarak 3B yazıcı ile imal edildi ve daha sonra bu geometrilerin 2B profillerini elde etmek amacıyla adı geçen geometriler koordinat ölçme makinesi ile tarandı. Ölçülerek ve modellenerek elde edilen veriler arasındaki hatayı en aza indirmek için doğrusal olmayan en küçük kareler prosedürü uygulandı. Bu uygulama için bir iterativ doğru boyunca arama gerçekleştirildi. Arama yönü ise yukarıda adı geçen yöntemler kullanılarak hesaplandı. Geometri uydurma sürecinde her bir iterasyonda yapılan fonksiyon değerlendirme sayısı hesap edildi ve ilgili eşlenik gradyan yöntemi yakınsadığında ortaya çıkan toplam fonksiyon değerlendirme sayısı yöntemin performans ölçütü olarak belirlendi. Verimli bir şekilde en iyi eşlenik gradyan yöntemini tespit edebilmek için bu performans ölçütleri kullanılarak performans ve veri profilleri oluşturuldu. Performans profillerine dayanarak, Fletcher-Reeves ve Polak-Ribiere yöntemlerinin beş geometriden üçünde en hızlı olduğu ifade edilebilir. Buna ek olarak, tüm EG yöntemleri test geometrilerinin %80'inin geometri uydurmasını tamamlayabilmiştir. Öte yandan, veri profilleri incelenerek, Polak-Ribiere ve Hestenes-Stiefel yöntemlerinin Fletcher-Reeves yöntemine göre çok daha az sayıda fonksiyon değerlendirmesi ile maksimum geometri uydurma kabiliyetlerine (%80) ulaştıkları tespit edilmiştir. Ayrıca birçok geometride Polak-Ribiere yöntemi diğerlerinden daha iyi olduğundan bu yöntem geometri uydurma için en iyi yöntem olarak belirlendi. Sonuç olarak, çalışmada rapor edilen sonuçlar koordinat ölçme makinesi verilerinin işlenmesi ile ilgilenen son kullanıcılara verimli bir geometri uydurma gerçekleştirmelerinde yardımcı olabilir.

Keywords

• Eşlenik gradyan
• Geometri uydurma
• En iyileme

DETERMINATION OF OPTIMAL CONJUGATE GRADIENT METHOD FOR GEOMETRY FITTING

Abstract

In this study, it is aimed to determine the optimal conjugate gradient (CG) method for the geometry fitting of 2D measured profiles. To this end, the three well-known CG methods such as the Fletcher-Reeves, Polak-Ribiere and Hestenes-Stiefel were employed. For testing those methods performances, the five primitive geometries accommodating circle, square, triangle, ellipse and rectangle were first built with a 3D printer, and then they were scanned with a coordinate measuring machine (CMM) to achieve their 2D profiles. The nonlinear least squares procedure was implemented to minimize the error between those measured data and modeled ones. An iterative line search was utilized for this task. The search direction was calculated using the above-mentioned CG methods. During the geometry fitting process, the number of function evaluations at each iteration were computed and the total number of function evaluations were set to be a performance measure of the CG method in question when it converged. By using these performance measures, the performance and data profiles were created to efficiently determine the optimal CG method. Based on performance profiles, it can be stated that the Fletcher-Reeves and Polak-Ribiere methods are the fastest ones on three test geometries out of five. In addition to that, all the CG methods were able to complete the geometry fitting of 80% of test geometries. On the other hand, by examining the data profiles, it was determined that the Polak-Ribiere and Hestenes-Stiefel methods achieve their maximum capabilities of the completing geometry fitting (i.e., 80%) with much lower number of function evaluations than the Fletcher-Reeves method. Besides, in most geometries, the Polak-Ribiere method outperformed the others, thereby it was determined to be the optimal one for the geometry fitting. As a conclusion, the reported results in this work might help the end-users who study on the CMM data processing to conduct an efficient geometry fitting.

Keywords

• Conjugate gradient
• Geometry fitting
• Optimization

References

1. Cao, J., Wu, J., 2020, “A conjugate gradient algorithm and its applications in image restoration”, Applied Numerical Mathematics, 152, 243-252.
2. Chattopadhyay, S., Chattopadhyay, G., 2018, “Conjugate gradient descent learned ANN for Indian summer monsoon rainfall and efficiency assessment through Shannon-Fano coding”, Journal of Atmospheric and Solar-Terrestrial Physics, 179, 202-205.
3. Desmos, https://www.desmos.com, Access date:16.05.2021.
4. Dolan, E.D., More, J.J., 2002, “Benchmarking optimization software with performance profiles”, Mathematical programming, 91, 201-213.
5. Fatemi, M., 2016, “A new efficient conjugate gradient method for unconstrained optimization”, Journal of Computational and Applied Mathematics, 300, 207-216.
6. Fletcher, R., Reeves, C.M., 1964, “Function minimization by conjugate gradients”, The Computer Journal, 7(2), 149-154.
7. Helmig, T., Al-Sibai, F., Kneer, R., 2020, “Estimating sensor number and spacing for inverse calculation of thermal boundary conditions using the conjugate gradient method”, International Journal of Heat and Mass Transfer, 153, 119638.
8. Hestenes, M.R., Stiefel, E. 1952, “Methods of conjugate gradients for solving linear systems”, Journal of Research of the National Bureau of Standards, 49(6), 409-436.

9. Hu, W., Wu, J., Yuan, G., 2020, “Some modified Hestenes-Stiefel conjugate gradient algorithms with application in image restoration”, Applied Numerical Mathematics, 158, 360–376.
10. Jia, P., 2017, Fitting a parametric model to a cloud of points via optimization methods, Ph.D. thesis, Syracuse University, New York.
11. Jiang, X., Jian, J., 2019, “Improved Fletcher–Reeves and Dai–Yuan conjugate gradient methods with the strong Wolfe line search”, Journal of Computational and Applied Mathematics, 348, 525-534.
12. Joo, K. S., Bose, T., Xu, G. F., 1997, “Image restoration using a conjugate gradient-based adaptive filtering algorithm”, Circuits systems signal processing, 16(2), 197-206.
13. Li, Y., Chen, W., Zhou, H., Yang, L., 2020, “Conjugate gradient method with pseudospectral collocation scheme for optimal rocket landing guidance”, Aerospace Science and Technology, 104, 105999.
14. More, J. J., Wild, S. M., 2009, “Benchmarking derivative-free optimization algorithms”, SIAM Journal on Optimization, 20, 172-191.
15. Mtagulwa, P., Kaelo, P., 2019, “An efficient modified PRP-FR hybrid conjugate gradient method for solving unconstrained optimization problems”, Applied Numerical Mathematics, 145, 111–120.
16. Nocedal, J., Wright, S.J., 2006, Numerical optimization, 2nd ed., Springer Science & Business Media, New York. Polak, E., Ribiere, G., 1969, “Note sur la convergence de methodes de directions conjuguees. Revue francaise d’informatique et de recherche operationnelle”, Revue francaise d’informatique et de recherche operationnelle, 3(R1), 35-43.
17. Schwarz, H. R., 1979, “The Method of Conjugate Gradients in Finite Element Applications”, Journal of Applied Mathematics and Physics, 30, 342-354.
18. Wang, L., Han, X., Xie, Y., 2013, “A new conjugate gradient method for solving multi-source dynamic load identification problem”, Int J Mech Mater Des, 9, 191-197.
19. Wang, J., Zhang, B., Sun, Z., Hao, W. and Sun, Q., 2018, “A novel conjugate gradient method with generalized Armijo search for efficient training of feedforward neural networks”, Journal of Neurocomputing, 275, 308-316.
20. Xiong, P., Deng, J., Lu, T., Lu, Q., Liu, Y., Zhang, Y., 2020, “A sequential conjugate gradient method to estimate heat flux for nonlinear inverse heat conduction problem”, Annals of Nuclear Energy, 149, 107798.