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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. AmatS. BusquierA. 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. AslamS. 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. BagirovA. 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. JiangR. ShenX. 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. MazrouiD. 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. MengC. 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. NguyenB. SaheyaY.-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. SaheyaC.-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. WuP. 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. AmatS. BusquierA. 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. AslamS. 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. BagirovA. 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. JiangR. ShenX. 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. MazrouiD. 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. MengC. 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. NguyenB. SaheyaY.-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. SaheyaC.-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. WuP. 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.

Figure 1.  (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 $
Figure 2.  (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 $
Figure 3.  (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 $
Figure 4.  (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 $
Figure 5.  (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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