# American Institute of Mathematical Sciences

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 & Optimization, doi: 10.3934/naco.2021033
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##### References:
(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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