doi: 10.3934/naco.2021033
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

D. P. Bertsekas, Nondifferentiable optimization via approximation, in Nondifferentiable Optimization, Springer, (1975), 1–25. doi: 10.1007/bfb0120696.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Lipman and D. Levin, Approximating piecewise-smooth functions, IMA Journal of Numerical Analysis, 30 (2010), 1159-1183.  doi: 10.1093/imanum/drn087.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. T. Rockafellar, A property of piecewise smooth functions, Computational Optimization and Applications, 25 (2003), 247-250.  doi: 10.1023/A:1022921624832.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. E. Xavier, The hyperbolic smoothing clustering method, Pattern Recognition, 43 (2010), 731-737.   Google Scholar

[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.  Google Scholar

[29]

I. Zang, A smoothing-out technique for minmax optimization, Mathematical Programming, 19 (1980), 61-77.  doi: 10.1007/BF01581628.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

D. P. Bertsekas, Nondifferentiable optimization via approximation, in Nondifferentiable Optimization, Springer, (1975), 1–25. doi: 10.1007/bfb0120696.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Lipman and D. Levin, Approximating piecewise-smooth functions, IMA Journal of Numerical Analysis, 30 (2010), 1159-1183.  doi: 10.1093/imanum/drn087.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. T. Rockafellar, A property of piecewise smooth functions, Computational Optimization and Applications, 25 (2003), 247-250.  doi: 10.1023/A:1022921624832.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. E. Xavier, The hyperbolic smoothing clustering method, Pattern Recognition, 43 (2010), 731-737.   Google Scholar

[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.  Google Scholar

[29]

I. Zang, A smoothing-out technique for minmax optimization, Mathematical Programming, 19 (1980), 61-77.  doi: 10.1007/BF01581628.  Google Scholar

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 $
[1]

Nurullah Yilmaz, Ahmet Sahiner. Generalization of hyperbolic smoothing approach for non-smooth and non-Lipschitz functions. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021170

[2]

Nurullah Yilmaz, Ahmet Sahiner. On a new smoothing technique for non-smooth, non-convex optimization. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 317-330. doi: 10.3934/naco.2020004

[3]

Paul Glendinning. Non-smooth pitchfork bifurcations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 457-464. doi: 10.3934/dcdsb.2004.4.457

[4]

Alessandro Colombo, Nicoletta Del Buono, Luciano Lopez, Alessandro Pugliese. Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2911-2934. doi: 10.3934/dcdsb.2018166

[5]

Giuseppe Tomassetti. Smooth and non-smooth regularizations of the nonlinear diffusion equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1519-1537. doi: 10.3934/dcdss.2017078

[6]

Luis Bayón, Jose Maria Grau, Maria del Mar Ruiz, Pedro Maria Suárez. A hydrothermal problem with non-smooth Lagrangian. Journal of Industrial & Management Optimization, 2014, 10 (3) : 761-776. doi: 10.3934/jimo.2014.10.761

[7]

Nicola Gigli, Sunra Mosconi. The Abresch-Gromoll inequality in a non-smooth setting. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1481-1509. doi: 10.3934/dcds.2014.34.1481

[8]

Hongwei Lou, Junjie Wen, Yashan Xu. Time optimal control problems for some non-smooth systems. Mathematical Control & Related Fields, 2014, 4 (3) : 289-314. doi: 10.3934/mcrf.2014.4.289

[9]

Yanni Xiao, Tingting Zhao, Sanyi Tang. Dynamics of an infectious diseases with media/psychology induced non-smooth incidence. Mathematical Biosciences & Engineering, 2013, 10 (2) : 445-461. doi: 10.3934/mbe.2013.10.445

[10]

Deepak Singh, Bilal Ahmad Dar, Do Sang Kim. Sufficiency and duality in non-smooth interval valued programming problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 647-665. doi: 10.3934/jimo.2018063

[11]

Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011

[12]

Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187

[13]

Jianhua Huang, Wenxian Shen. Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 855-882. doi: 10.3934/dcds.2009.24.855

[14]

Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109

[15]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2021, 11 (3) : 521-554. doi: 10.3934/mcrf.2020052

[16]

Jianfeng Lv, Yan Gao, Na Zhao. The viability of switched nonlinear systems with piecewise smooth Lyapunov functions. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1825-1843. doi: 10.3934/jimo.2020048

[17]

Jean-Marie Barbaroux, Dirk Hundertmark, Tobias Ried, Semjon Vugalter. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction. Kinetic & Related Models, 2017, 10 (4) : 901-924. doi: 10.3934/krm.2017036

[18]

Chao Zhang, Lihe Wang, Shulin Zhou, Yun-Ho Kim. Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2559-2587. doi: 10.3934/cpaa.2014.13.2559

[19]

Xiaoshan Chen, Xun Li, Fahuai Yi. Optimal stopping investment with non-smooth utility over an infinite time horizon. Journal of Industrial & Management Optimization, 2019, 15 (1) : 81-96. doi: 10.3934/jimo.2018033

[20]

Philippe Pécol, Pierre Argoul, Stefano Dal Pont, Silvano Erlicher. The non-smooth view for contact dynamics by Michel Frémond extended to the modeling of crowd movements. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 547-565. doi: 10.3934/dcdss.2013.6.547

 Impact Factor: 

Metrics

  • PDF downloads (95)
  • HTML views (149)
  • Cited by (0)

Other articles
by authors

[Back to Top]