# Generalized Beta Prime Distribution Applied to Finite Element Error Approximation

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## Abstract

**:**

## 1. Introduction

## 2. Abstract Problem and Finite Element Error Estimates

**Lemma**

**1.**

- For a fixed value of the mesh size h, a random trial corresponds to the grid generation and the associated approximation ${u}_{h}^{\left(k\right)}$.
- The probability space $\mathsf{\Omega}$, therefore, contains all the possible results for a given random trial, namely, all of the possible grids that the mesh generator may construct, or equivalently, all of the corresponding associated approximations ${u}_{h}^{\left(k\right)}$.

## 3. Numerical Statistics and Probability-Law Comparison

## 4. A New Probability Law Based on GBP

- 1.
- In the previous works that enabled us to derive the “Sigmoid” probability law, we had made some assumptions (i.e., uniform distribution and independency). Since we now aim at obtaining a better fit between the probabilistic law and the statistical data, we will now relax the hypothesis of uniformity we had made on the densities of the random variables ${X}^{\left({k}_{i}\right)}\left(h\right),(i=1,2)$, (see [1,2,3]).
- 2.
- In order to obtain more degrees of freedom for the shape of these densities, let us first consider the random variable $Z={X}^{\left({k}_{2}\right)}-{X}^{\left({k}_{1}\right)}$. Our goal is to obtain, for the cumulative distribution function ${F}_{Z}$ defined by (12) at point $z=0$, a curve whose shape looks like a “Sigmoid”. We thus have to enrich our modeling process by adding more degrees of freedom, the “Sigmoid” probability law (13) and (14) containing only one parameter ${h}_{{k}_{1},{k}_{2}}^{*}$. For this reason, we now consider a density ${f}_{Z}$ for the random variable Z, whose associated cumulative distribution function ${F}_{Z}$ at point $z=0$ will include two exogenous parameters; these will be statistically estimated.

- 1.
- The support of the Beta density ${f}_{X}$ of the random variable X, denoted $SuppX$, is included in $[0,1]$. Nevertheless, that of the random variable Z is $[-{\beta}_{{k}_{1}},{\beta}_{{k}_{2}}]$, since $Z={X}^{\left({k}_{2}\right)}-{X}^{\left({k}_{1}\right)}$ and $Supp\phantom{\rule{0.166667em}{0ex}}{X}^{\left({k}_{i}\right)}\subset [0,{\beta}_{{k}_{i}}],(i=1,2)$. This will make us use a suitable transformation of the density ${f}_{X}$ to guarantee the correct support of the density ${f}_{Z}$ of Z.
- 2.
- We are looking for a probability law for the event $\left(\right)open="\{"\; close="\}">{X}^{\left({k}_{2}\right)}\le {X}^{\left({k}_{1}\right)}$ as a function of h, that belongs to $[0,+\infty [$. Consequently, we have to apply another transformation to the density ${f}_{Z}$ to guarantee this property for the support of h.

**Lemma**

**2.**

**Proof.**

**Remark**

**1.**

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

- 1.
- From (26) and (27), or equivalently from (30), we can obtain the asymptotic behavior of the probability of the event $\left(\right)open="\{"\; close="\}">{X}^{\left({k}_{1}\right)}\le {X}^{\left({k}_{2}\right)}$ as the mesh size h goes to 0.Clearly, $Prob\left(\right)open="\{"\; close="\}">{X}^{\left({k}_{1}\right)}\le {X}^{\left({k}_{2}\right)}$ goes to 0 with h. In other words, we have found with the new probabilistic law the usual result, which claims that the finite element ${P}_{{k}_{2}}$ is more accurate than ${P}_{{k}_{1}}$, since from (4) and (5), ${h}^{{k}_{2}}$ goes faster to zero than ${h}^{{k}_{1}}$ when ${k}_{1}<{k}_{2}$.Here, by (30), the same property is expressed in terms of probability, namely: the event $\left(\right)open="\{"\; close="\}">{X}^{\left({k}_{1}\right)}\le {X}^{\left({k}_{2}\right)}$ occurs almost never, or equivalently, the event $\left(\right)open="\{"\; close="\}">{X}^{\left({k}_{2}\right)}\le {X}^{\left({k}_{1}\right)}$ is an almost surely one, since its probability is equal to 1.
- 2.
- From (26), using the positivity of the density ${f}_{H}$, we conclude that ${\mathcal{P}}_{{k}_{1},{k}_{2}}\left(h\right)$ is a decreasing function of h. This property was already observed with the “Sigmoid” model in [1,3] where uniformity of the random variables ${X}^{\left({k}_{i}\right)},(i=1,2),$ was assumed.In other words, this property claims that the event $\left(\right)open="\{"\; close="\}">{P}_{{k}_{2}}ismoreaccuratethan{P}_{{k}_{1}}$ is true only for small h, and the more h increases the less likely this property is. Moreover, when h becomes large, asymptotically the event $\left(\right)open="\{"\; close="\}">{P}_{{k}_{2}}ismoreaccuratethan{P}_{{k}_{1}}$ occurs almost never.

## 5. Numerical Comparisons

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**General shape of probabilistic law (13) and (14) together with the limit Heaviside case (16).

**Figure 3.**${P}_{1}$ versus ${P}_{3}$ for the Runge function with $\alpha =3000$. Comparison between the statistical frequencies (blue) and the “Sigmoid law” (orange).

**Figure 4.**${P}_{2}$ versus ${P}_{3}$ for the smooth case. Comparison between the statistical frequencies (blue) and the two probabilistic “Sigmoid” laws (orange).

**Figure 5.**${P}_{1}$ versus ${P}_{3}$ for the Runge function with $\alpha =3000$. Comparison between the statistical frequencies (blue) and the GBP law (red).

**Figure 6.**${P}_{1}$ versus ${P}_{4}$.

**Up**: comparison between the statistical frequencies (blue) and the Sigmoid (orange).

**Down**: comparison between the statistical frequencies (blue) and the GBP law (red).

**Figure 7.**${P}_{2}$ versus ${P}_{3}$.

**Up**: comparison between the statistical frequencies (blue) and the Sigmoid (orange).

**Down**: comparison between the statistical frequencies (blue) and the GBP law (red).

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Chaskalovic, J.; Assous, F.
Generalized Beta Prime Distribution Applied to Finite Element Error Approximation. *Axioms* **2022**, *11*, 84.
https://doi.org/10.3390/axioms11030084

**AMA Style**

Chaskalovic J, Assous F.
Generalized Beta Prime Distribution Applied to Finite Element Error Approximation. *Axioms*. 2022; 11(3):84.
https://doi.org/10.3390/axioms11030084

**Chicago/Turabian Style**

Chaskalovic, Joël, and Franck Assous.
2022. "Generalized Beta Prime Distribution Applied to Finite Element Error Approximation" *Axioms* 11, no. 3: 84.
https://doi.org/10.3390/axioms11030084