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doi: 10.3934/naco.2021033
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Smoothing approximations for piecewise smooth functions: A probabilistic approach

 Laboratory of Mathematical Modeling, Simulation and Smart Systems (L2M3S), ENSAM-Meknes, Moulay ISMAIL University, Meknes, Morocco

* Corresponding author: Elmehdi Amhraoui

Received  May 2021 Revised  July 2021 Early access August 2021

Fund Project: The first author's work is supported by the national center for scientific and technical research, Morocco

In this article, we present a new approach to construct smoothing approximations for piecewise smooth functions. This approach proposes to formulate any piecewise smooth function as the expectation of a random variable. Based on this formulation, we show that smoothing all elements of a defined space of piecewise smooth functions is equivalent to smooth a single probability distribution. Furthermore, we propose to use the Boltzmann distribution as a smoothing approximation for this probability distribution. Moreover, we present the theoretical results, error estimates, and some numerical examples for this new smoothing method in both one-dimensional and multiple-dimensional cases.

Citation: Elmehdi Amhraoui, Tawfik Masrour. Smoothing approximations for piecewise smooth functions: A probabilistic approach. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2021033
References:
 [1] S. Amat, S. Busquier, A. Escudero and J. C. Trillo, Lagrange interpolation for continuous piecewise smooth functions, Journal of computational and applied mathematics, 221 (2008), 47-51.  doi: 10.1016/j.cam.2007.10.011. [2] M. Aslam, S. Riemenschneider and L. Shen, Smoothing transforms for wavelet approximation of piecewise smooth functions, IET Image Processing, 2 (2008), 239-248.  doi: 10.1049/iet-ipr:20080063. [3] A. Bagirov, A. Al Nuaimat and N. Sultanova, Hyperbolic smoothing function method for minimax problems, Optimization, 62 (2013), 759-782.  doi: 10.1080/02331934.2012.675335. [4] G. Z. Bai, S. Y. Liu, H. Wang, W. Zhi and Z. J. Yin, A novel method for smooth blending cylindrical surfaces whose axes are non-coplanar with intersecting line of two tapered surfaces, in Applied Mechanics and Materials, Trans. Tech. Publ., 687 (2014), 1470–1473. [5] B. Belkhatir and A. Zidna, Construction of flexible blending parametric surfaces via curves, Mathematics and Computers in Simulation, 79 (2009), 3599-3608.  doi: 10.1016/j.matcom.2009.04.015. [6] D. P. Bertsekas, Nondifferentiable optimization via approximation, in Nondifferentiable Optimization, Springer, (1975), 1–25. doi: 10.1007/bfb0120696. [7] M. Fukushima and L. Qi, Reformulation: Nonsmooth, Piecewise Smooth, Semismooth And Smoothing Methods, Vol. 22, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4757-6388-1. [8] X. Han, A degree by degree recursive construction of hermite spline interpolants, Journal of computational and Applied Mathematics, 225 (2009), 113-123.  doi: 10.1016/j.cam.2008.07.005. [9] E. Hartmann, Blending an implicit with a parametric surface, Computer Aided Geometric Design, 12 (1995), 825-835.  doi: 10.1016/0167-8396(95)00002-1. [10] F. Hashemi and S. Ketabchi, Numerical comparisons of smoothing functions for optimal correction of an infeasible system of absolute value equations, Numerical Algebra, Control & Optimization, 10 (2020), 13. doi: 10.3934/naco.2019029. [11] M. Jiang, R. Shen, X. Xu and Z. Meng, Second-order smoothing objective penalty function for constrained optimization problems, Numerical Functional Analysis and Optimization, 35 (2014), 294-309.  doi: 10.1080/01630563.2013.811421. [12] X. Jiang and Y. Zhang, A smoothing-type algorithm for absolute value equations, Journal of Industrial & Management Optimization, 9 (2013), 789. doi: 10.3934/jimo.2013.9.789. [13] L. Kuntz and S. Scholtes, Qualitative aspects of the local approximation of a piecewise differentiable function, Nonlinear Analysis: Theory, Methods & Applications, 25 (1995), 197-215.  doi: 10.1016/0362-546X(94)00202-S. [14] S. Lian and Y. Duan, Smoothing of the lower-order exact penalty function for inequality constrained optimization, Journal of Inequalities and Applications, 2016 (2016), 1-12.  doi: 10.1186/s13660-016-1126-9. [15] Y. Lipman and D. Levin, Approximating piecewise-smooth functions, IMA Journal of Numerical Analysis, 30 (2010), 1159-1183.  doi: 10.1093/imanum/drn087. [16] A. Mazroui, D. Sbibih and A. Tijini, A simple method for smoothing functions and compressing hermite data, Advances in Computational Mathematics, 23 (2005), 279-297.  doi: 10.1007/s10444-004-1783-y. [17] Z. Meng, C. Dang and X. Yang, On the smoothing of the square-root exact penalty function for inequality constrained optimization, Computational Optimization and Applications, 35 (2006), 375-398.  doi: 10.1007/s10589-006-8720-6. [18] J. Min, Z. Meng, G. Zhou and R. Shen, On the smoothing of the norm objective penalty function for two-cardinality sparse constrained optimization problems, Neurocomputing. [19] C. T. Nguyen, B. Saheya, Y.-L. Chang and J.-S. Chen, Unified smoothing functions for absolute value equation associated with second-order cone, Applied Numerical Mathematics, 135 (2019), 206-227.  doi: 10.1016/j.apnum.2018.08.019. [20] P. Prandoni and M. Vetterli, Approximation and compression of piecewise smooth functions, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 357 (1999), 2573-2591.  doi: 10.1098/rsta.1999.0449. [21] L. Qi and P. Tseng, On almost smooth functions and piecewise smooth functions, Nonlinear Analysis: Theory, Methods & Applications, 67 (2007), 773-794.  doi: 10.1016/j.na.2006.06.029. [22] D. Ralph and S. Scholtes, Sensitivity analysis of composite piecewise smooth equations, Mathematical Programming, 76 (1997), 593-612.  doi: 10.1016/S0025-5610(96)00063-9. [23] R. T. Rockafellar, A property of piecewise smooth functions, Computational Optimization and Applications, 25 (2003), 247-250.  doi: 10.1023/A:1022921624832. [24] B. Saheya, C.-H. Yu and J.-S. Chen, Numerical comparisons based on four smoothing functions for absolute value equation, Journal of Applied Mathematics and Computing, 56 (2018), 131-149.  doi: 10.1007/s12190-016-1065-0. [25] A. Sahiner, G. Kapusuz and N. Yilmaz, A new smoothing approach to exact penalty functions for inequality constrained optimization problems, Numerical Algebra, Control & Optimization, 6 (2016), 161. doi: 10.3934/naco.2016006. [26] H. Wu, P. Zhang and G.-H. Lin, Smoothing approximations for some piecewise smooth functions, Journal of the Operations Research Society of China, 3 (2015), 317-329.  doi: 10.1007/s40305-015-0091-1. [27] A. E. Xavier, The hyperbolic smoothing clustering method, Pattern Recognition, 43 (2010), 731-737. [28] N. Yilmaz and A. Sahiner, New smoothing approximations to piecewise smooth functions and applications, Numerical Functional Analysis and Optimization, 40 (2019), 513-534.  doi: 10.1080/01630563.2018.1561466. [29] I. Zang, A smoothing-out technique for minmax optimization, Mathematical Programming, 19 (1980), 61-77.  doi: 10.1007/BF01581628.

show all references

References:
 [1] S. Amat, S. Busquier, A. Escudero and J. C. Trillo, Lagrange interpolation for continuous piecewise smooth functions, Journal of computational and applied mathematics, 221 (2008), 47-51.  doi: 10.1016/j.cam.2007.10.011. [2] M. Aslam, S. Riemenschneider and L. Shen, Smoothing transforms for wavelet approximation of piecewise smooth functions, IET Image Processing, 2 (2008), 239-248.  doi: 10.1049/iet-ipr:20080063. [3] A. Bagirov, A. Al Nuaimat and N. Sultanova, Hyperbolic smoothing function method for minimax problems, Optimization, 62 (2013), 759-782.  doi: 10.1080/02331934.2012.675335. [4] G. Z. Bai, S. Y. Liu, H. Wang, W. Zhi and Z. J. Yin, A novel method for smooth blending cylindrical surfaces whose axes are non-coplanar with intersecting line of two tapered surfaces, in Applied Mechanics and Materials, Trans. Tech. Publ., 687 (2014), 1470–1473. [5] B. Belkhatir and A. Zidna, Construction of flexible blending parametric surfaces via curves, Mathematics and Computers in Simulation, 79 (2009), 3599-3608.  doi: 10.1016/j.matcom.2009.04.015. [6] D. P. Bertsekas, Nondifferentiable optimization via approximation, in Nondifferentiable Optimization, Springer, (1975), 1–25. doi: 10.1007/bfb0120696. [7] M. Fukushima and L. Qi, Reformulation: Nonsmooth, Piecewise Smooth, Semismooth And Smoothing Methods, Vol. 22, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4757-6388-1. [8] X. Han, A degree by degree recursive construction of hermite spline interpolants, Journal of computational and Applied Mathematics, 225 (2009), 113-123.  doi: 10.1016/j.cam.2008.07.005. [9] E. Hartmann, Blending an implicit with a parametric surface, Computer Aided Geometric Design, 12 (1995), 825-835.  doi: 10.1016/0167-8396(95)00002-1. [10] F. Hashemi and S. Ketabchi, Numerical comparisons of smoothing functions for optimal correction of an infeasible system of absolute value equations, Numerical Algebra, Control & Optimization, 10 (2020), 13. doi: 10.3934/naco.2019029. [11] M. Jiang, R. Shen, X. Xu and Z. Meng, Second-order smoothing objective penalty function for constrained optimization problems, Numerical Functional Analysis and Optimization, 35 (2014), 294-309.  doi: 10.1080/01630563.2013.811421. [12] X. Jiang and Y. Zhang, A smoothing-type algorithm for absolute value equations, Journal of Industrial & Management Optimization, 9 (2013), 789. doi: 10.3934/jimo.2013.9.789. [13] L. Kuntz and S. Scholtes, Qualitative aspects of the local approximation of a piecewise differentiable function, Nonlinear Analysis: Theory, Methods & Applications, 25 (1995), 197-215.  doi: 10.1016/0362-546X(94)00202-S. [14] S. Lian and Y. Duan, Smoothing of the lower-order exact penalty function for inequality constrained optimization, Journal of Inequalities and Applications, 2016 (2016), 1-12.  doi: 10.1186/s13660-016-1126-9. [15] Y. Lipman and D. Levin, Approximating piecewise-smooth functions, IMA Journal of Numerical Analysis, 30 (2010), 1159-1183.  doi: 10.1093/imanum/drn087. [16] A. Mazroui, D. Sbibih and A. Tijini, A simple method for smoothing functions and compressing hermite data, Advances in Computational Mathematics, 23 (2005), 279-297.  doi: 10.1007/s10444-004-1783-y. [17] Z. Meng, C. Dang and X. Yang, On the smoothing of the square-root exact penalty function for inequality constrained optimization, Computational Optimization and Applications, 35 (2006), 375-398.  doi: 10.1007/s10589-006-8720-6. [18] J. Min, Z. Meng, G. Zhou and R. Shen, On the smoothing of the norm objective penalty function for two-cardinality sparse constrained optimization problems, Neurocomputing. [19] C. T. Nguyen, B. Saheya, Y.-L. Chang and J.-S. Chen, Unified smoothing functions for absolute value equation associated with second-order cone, Applied Numerical Mathematics, 135 (2019), 206-227.  doi: 10.1016/j.apnum.2018.08.019. [20] P. Prandoni and M. Vetterli, Approximation and compression of piecewise smooth functions, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 357 (1999), 2573-2591.  doi: 10.1098/rsta.1999.0449. [21] L. Qi and P. Tseng, On almost smooth functions and piecewise smooth functions, Nonlinear Analysis: Theory, Methods & Applications, 67 (2007), 773-794.  doi: 10.1016/j.na.2006.06.029. [22] D. Ralph and S. Scholtes, Sensitivity analysis of composite piecewise smooth equations, Mathematical Programming, 76 (1997), 593-612.  doi: 10.1016/S0025-5610(96)00063-9. [23] R. T. Rockafellar, A property of piecewise smooth functions, Computational Optimization and Applications, 25 (2003), 247-250.  doi: 10.1023/A:1022921624832. [24] B. Saheya, C.-H. Yu and J.-S. Chen, Numerical comparisons based on four smoothing functions for absolute value equation, Journal of Applied Mathematics and Computing, 56 (2018), 131-149.  doi: 10.1007/s12190-016-1065-0. [25] A. Sahiner, G. Kapusuz and N. Yilmaz, A new smoothing approach to exact penalty functions for inequality constrained optimization problems, Numerical Algebra, Control & Optimization, 6 (2016), 161. doi: 10.3934/naco.2016006. [26] H. Wu, P. Zhang and G.-H. Lin, Smoothing approximations for some piecewise smooth functions, Journal of the Operations Research Society of China, 3 (2015), 317-329.  doi: 10.1007/s40305-015-0091-1. [27] A. E. Xavier, The hyperbolic smoothing clustering method, Pattern Recognition, 43 (2010), 731-737. [28] N. Yilmaz and A. Sahiner, New smoothing approximations to piecewise smooth functions and applications, Numerical Functional Analysis and Optimization, 40 (2019), 513-534.  doi: 10.1080/01630563.2018.1561466. [29] I. Zang, A smoothing-out technique for minmax optimization, Mathematical Programming, 19 (1980), 61-77.  doi: 10.1007/BF01581628.
(a), (b) and (c) contain the graphs of the function $f(x) = |x|$ and its smoothing approximation $\hat{f}_{\alpha}(x)$ for $\alpha = 1$, $\alpha = 2$, and $\alpha = 3$, respectively. (d) The error of the smoothing approximation for different values of $\alpha$
(a), (b) and (c) contain the graphs of the function $g(x) = \max(0,x)$ and its smoothing approximation $\hat{g}_{\alpha}(x)$ for $\alpha = 1,\alpha = 2\text{ and } \alpha = 3$, respectively. (d) The error of the smoothing approximation for different values of $\alpha$
(a), (b) and (c) contain the graphs of the function $h(x) = |x|+\max(0,x)$ and its smoothing approximation $\hat{h}_{\alpha}(x)$ for $\alpha = 1,\alpha = 2\text{ and } \alpha = 3$, respectively. (d) The error of the smoothing approximation for different values of $\alpha$
(a) The graph of $f(x,y)$. (b) The graph of the smoothing approximation $\hat{f}_\alpha(x,y)$ for $\alpha = 4$. (c) The error of the smoothing approximation for $\alpha = 3$. (d) The error of the smoothing approximation for $\alpha = 4$
(a) The graph of the function $f(x,y) = |x|+|y|$. (b) The graph of the smoothing approximation $\hat{f}_{\alpha}(x,y)$ for $\alpha = 1$. (c) The graph of the smoothing approximation $\hat{f}_{\alpha}(x,y)$ for $\alpha = 5$. (d) The error of the smoothing approximation for $\alpha = 5$
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