New Smoothing Approximations to Piecewise Smooth Functions and Applications

Yılmaz, Nurullah; Şahiner, Ahmet

doi:10.1080/01630563.2018.1561466

Cited by 6 publications

(7 citation statements)

References 27 publications

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“…The partition τ a is a special case of (5), and it is studied extensively; see for instance [26,28].…”

Section: Smoothing Approximations For One-dimensional Case Let Us Consider the Partitionmentioning

confidence: 99%

“…Another application of smoothing approximations for P-S functions is data modeling and compression. For example, the model smoothness is an important property for capturing patterns and removing noise from data sets [28]. For these purposes, smoothing is regarded as a fundamental process in data modeling.…”

mentioning

confidence: 99%

“…Definition 2.1. [21,28] A real-valued function f : Ω −→ R is called piecewise smooth function on Ω if it is continuous on Ω, there exists a finite collection of smooth functions f i : Ω −→ R, i = 1, 2, • • • , r, and there exists A i i=1,••• ,r a partition of Ω, such that…”

mentioning

confidence: 99%

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Smoothing approximations for piecewise smooth functions: A probabilistic approach

Amhraoui¹,

Masrour²

2022

NACO

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<p style='text-indent:20px;'>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.</p>

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“…The partition τ a is a special case of (5), and it is studied extensively; see for instance [26,28].…”

Section: Smoothing Approximations For One-dimensional Case Let Us Consider the Partitionmentioning

confidence: 99%

mentioning

confidence: 99%

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Smoothing approximations for piecewise smooth functions: A probabilistic approach

Amhraoui¹,

Masrour²

2022

NACO

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show abstract

“…The comprehensive overview on smoothing approaches can be found in [20,28,29]. Among the smoothing functions, the smoothing function studied in [30] distinguishes itself from the others due to different structure, formulations and useful properties.…”

Section: Introductionmentioning

confidence: 99%

“…In this study, we propose a new smoothing function inspiring from the smoothing methods given in [30] based on the problem (1.1). By the help of this smoothing approach, we design a family of smooth equations which is surrogate for the original problem (1.1).…”

Section: Introductionmentioning

confidence: 99%