Shakibi, K., Amirfakhrian, M., Kansa, E. (2019). Scattered data approximation of fully fuzzy data by quasi-interpolation. Iranian Journal of Fuzzy Systems, 16(3), 63-72. doi: 10.22111/ijfs.2019.4645

K. Shakibi; M. Amirfakhrian; E. J. Kansa. "Scattered data approximation of fully fuzzy data by quasi-interpolation". Iranian Journal of Fuzzy Systems, 16, 3, 2019, 63-72. doi: 10.22111/ijfs.2019.4645

Shakibi, K., Amirfakhrian, M., Kansa, E. (2019). 'Scattered data approximation of fully fuzzy data by quasi-interpolation', Iranian Journal of Fuzzy Systems, 16(3), pp. 63-72. doi: 10.22111/ijfs.2019.4645

Shakibi, K., Amirfakhrian, M., Kansa, E. Scattered data approximation of fully fuzzy data by quasi-interpolation. Iranian Journal of Fuzzy Systems, 2019; 16(3): 63-72. doi: 10.22111/ijfs.2019.4645

Scattered data approximation of fully fuzzy data by quasi-interpolation

^{1}Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran

^{2}Convergent Solutions, LLC, Livermore, CA, USA.

Abstract

Fuzzy quasi-interpolations help to reduce the complexity of solving a linear system of equations compared with fuzzy interpolations. Almost all fuzzy quasi-interpolations are focused on the form of $\widetilde{f}^{*}:\mathbb{R}\rightarrow F(\mathbb{R})$ or $\widetilde{f}^{*}:F(\mathbb{R})\rightarrow \mathbb{R}$. In this paper, we intend to offer a novel fuzzy radial basis function by the concept of source distance. Then, we will construct a fuzzy linear combination of such basis functions in order to introduce a fully fuzzy quasi-interpolation in the form of $\widetilde{f}^{*}:F(\mathbb{R})\rightarrow F(\mathbb{R})$. Also the error estimation of the proposed method is proved in terms of the fully fuzzy modulus of continuity which will be introduced in this paper. Finally some examples have been given to emphasize the acceptable accuracy of our method.

[1] S. Abbasbandy, Interpolation of fuzzy data by complete splines, The Korean Journal of Computational & Applied Mathematics, 8(3) (2001), 587{594. [2] S. Abbasbandy, M. Amirfakhrian, A new approach to universal approximation of fuzzy functions on a discrete set of points, Applied Mathematical Modelling, 30 (2006), 1525{1534. [3] S. Abbasbandy, M. Amirfakhrian, Numerical approximation of fuzzy functions by fuzzy polynomials, Applied Mathematics and Computation, 174 (2006), 1001{1006. [4] S. Abbasbandy, E. Babolian, Interpolation of fuzzy data by natural splines, Journal of Applied Mathematics and Computing, 5 (1998), 457-463. [5] M. Adabitabar Firozja, Fully fuzzy interpolation problem with signed distance, Communications in Numerical Analysis, 2012 (2012), 1{7, doi: 10.5899/2012/cna-00102. [6] M. Amirfakhrian, Some Approximation Methods in fuzzy Logic, LAP Lambert Academic publishing 2012. [7] M. Amirfakhrian, M. Arghand, E. J. Kansa, A new approximate method for an inverse time-dependent heat source problem using fundamental solutions and RBFs, Engineering Analysis with Boundary Elements, 64 (2016), 278{289. [8] M. Amirfakhrian, K. Shakibi, R. Rodrguez Lopez, Fuzzy quasi-interpolation solution for Fredholm fuzzy integral equations of second kind, Soft Computing, 21 (2017), 4323-4333. [9] P. Baranyi, T. D. Gedeon, Rule interpolation by spetial representation, IPMU 96: Proceedings Sixth International Conference, Granada 1996.

[10] R. K. Beatson, M. J. D. powell, Univariate MQ approximation quasi-interpolation to scattered data, Construc. approx, 8 (1992), 275- 288. [11] B. Bouchon-Meunier, J. Deellouli, M. Rifqi, L. Zerrouki, Analogy and fuzzy interpolation in case of sparse rules, Proc of the Eurofuse- SIC joint conf, 1999. [12] S. Dragievei, Multi-Dimentional Interpolation with Fuzzy Sets, Modeling with Spatial Information for Geographic Problems, (2005), 143{158. [13] D. Dubois, R. Martin-Clouaire, H. Prade, Practical computing in fuzzy logic, (M. Gupta, M., T. Yamakawa, Eds.), North-Holland, Amsterdam, Holland (1988), 11{34. [14] R. L. Hardy, Multiquadric equation of topography and other irregular surfaces, Journal of Geophysical Research, 76 (1971), 1905{1915. [15] R. L. Hardy, Theory and application of the multiquadric-biharmonic method, 20 years of discovery 1968-1988, Computers & Mathematics with Applications , 19 (1990), 163{208. [16] O. Kaleva, Interpolation of fuzzy data, Fuzzy Sets and Systems, 61 (1994), 63{70. [17] E. J. Kansa, Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics 1, Computers & Mathematics with Applications , 19 (1990), 127{145. [18] P. Liu, Analysis of approximation of continuous fuzzy functions by multivariate fuzzy polynomials, Fuzzy Sets and Systems, 127 (2005), 299{313. [19] R. Lowen, A fuzzy Lagrange interpolation theorem, Fuzzy Sets and Systems, 34 (1990), 33{38. [20] I. Perfilieva, Fuzzy function as an approximate solution to a system of fuzzy relation equation to a system of fuzzy relation equations, Fuzzy Sets and Systems, 147 (2004), 263{383. [21] M. J. D. Powell, Univariate multiquadric approximation: reproduction of linear polynomials, Univariate Multiquadric Approximation: Reproduction of Linear Polynomials, 94 (1990), 227{240. [22] A. N. Tikhonov, V. Y. Arsenin, Solutions of ill-posed problems. W. H. Winston and Sons: Washington, D. C.: John Wiley and Sons: New York; Translated from the Russian, Preface by translation editor Fritz John, Scripta series in Mathematics, 1977. [23] W. Voxman, Some remarks on distance between fuzzy numbers, Fuzzy Sets and Systems, 100 (1998), 353{365. [24] K. Wong, C. Che.Fung, H. Eren, T. Gedeon, Fuzzy Rule Interpo-lation for Multidimensional Input Spaces in Determining d50c of Hydro-cyclones, IEEE Transactions on Instrumentation and Measurement, 52(6) (2003), 1865{ 1869. [25] B. Wright, Radial basis function interpolation: numerical and analytical developments. A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy: Department of Applied Mathematics (2003), 1{139. [26] Z.M. Wu, R. Schaback, Shape preserving properties and convergence of univariate multiquadric quasi- interpolation,Acta Mathematica Sinica, 10 (1994), 441{446. [27] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.