# Comparison of the Probabilistic Ant Colony Optimization Algorithm and Some Iteration Method in Application for Solving the Inverse Problem on Model With the Caputo Type Fractional Derivative

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

- $x,t$—spatial and time variable,
- u—function representing the temperature distribution,
- c—specific heat,
- $\rho $—density,
- $\lambda $—thermal conductivity,
- $\alpha $—order of fractional derivative,
- g—additional heat source,
- f—function representing the initial condition,
- $\varphi ,\psi $—functions representing the boundary condition of the first type.

#### Description of the Considered Inverse Problem

## 3. Methods of Solution

#### 3.1. Solution of the Direct Problem

#### 3.2. Solution of the Inverse Problem

**Initialization of the algorithm**

- Setting the input parameters of the algorithm: $L,M,I,nT,q,\xi $.
- Generating L solutions playing the role of the pheromone spots. Assigning them to the set ${T}_{0}$ (initial archive).
- Computing the values of the objective function for all of the pheromone spots (parallel computing) and sorting the archive ${T}_{0}$ from the best solution to the worst.
**Iteration process** - Assigning the probabilities to the pheromone spots according to the pattern:$${p}_{l}=\frac{{\omega}_{l}}{{\displaystyle \sum _{l=1}^{L}}{\omega}_{l}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,L,$$$${\omega}_{l}=\frac{1}{qL\sqrt{2\pi}}\times {e}^{\frac{-{(l-1)}^{2}}{2{q}^{2}{L}^{2}}}.$$
- The ant randomly chooses the $l-$th solution with the probability ${p}_{l}$.
- The ant transforms the j-th coordinate ($j=1,2,\dots ,n$) of the $l-$th solution ${s}_{j}^{l}$ by sampling the neighborhood using the probability density function (in this case the Gauss function):$$g(x,\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi}}\xb7{e}^{\frac{-{(x-\mu )}^{2}}{2{\sigma}^{2}}},$$
- Steps 5–6 are being repeated by every ant. We get M new solutions (pheromone spots) by that.
- Partition of the new solutions into the $nT$ groups. Computing the value of the objective function for the new solutions (parallel computing).
- Adding the new solutions to the archive ${T}_{i}$, sorting them by the quality and rejecting the M worst solutions.
- The steps 3–9 are being repeated I times.

## 4. Numerical Examples

#### 4.1. Example 1

#### 4.2. Example 2

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Values of the coefficients ${\overline{a}}_{1}$ (

**a**) and ${\overline{a}}_{2}$ (

**b**) for the input data disturbed by the $5\%$ error.

**Figure 3.**Value of the coefficient ${\overline{a}}_{3}$ (

**a**) and value of the functional J (

**b**) for the input data disturbed by the $5\%$ error.

**Table 1.**Results of calculations for the RealACO algorithm (grid $100\times 900$) (${\overline{a}}_{i}$—reconstructed value of the coefficient ${a}_{i}$, ${\delta}_{{\overline{a}}_{i}}$—the relative error of reconstruction of the coefficient ${a}_{i}$, $\sigma $—standard deviation ($i=1,2,3$)).

Noise | ${\overline{\mathit{a}}}_{\mathit{i}}$ | ${\mathit{\delta}}_{{\overline{\mathit{a}}}_{\mathit{i}}}[\%]$ | $\mathit{\sigma}$ |
---|---|---|---|

$0\%$ | $0.9999$ | $0.01$ | $4.36\times {10}^{-4}$ |

$0.4999$ | $0.02$ | $7.81\times {10}^{-5}$ | |

$0.9999$ | $0.01$ | $4.36\times {10}^{-5}$ | |

$1\%$ | $1.0001$ | $0.01$ | $1.77\times {10}^{-4}$ |

$0.4998$ | $0.04$ | $1.74\times {10}^{-4}$ | |

$0.9999$ | $0.01$ | $7.21\times {10}^{-5}$ | |

$2\%$ | $0.9995$ | $0.05$ | $1.96\times {10}^{-4}$ |

$0.4996$ | $0.08$ | $3.67\times {10}^{-4}$ | |

$0.9997$ | $0.03$ | $1.96\times {10}^{-5}$ | |

$5\%$ | $1.0001$ | $0.01$ | $3.79\times {10}^{-4}$ |

$0.4996$ | $0.08$ | $5.82\times {10}^{-5}$ | |

$1.0003$ | $0.03$ | $4.84\times {10}^{-5}$ |

**Table 2.**Comparison of the algorithms according to the obtained errors of reconstructing the coefficient $\lambda $ for the input data from the example 1.

Noise | RealACO | Iteration Method | |
---|---|---|---|

Error | $\mathit{\epsilon}$ | Error | |

$0\%$ | $4.2781\times {10}^{-4}$ | $0.01$ | $4.8035\times {10}^{-3}$ |

$0.001$ | $4.0918\times {10}^{-4}$ | ||

$0.0001$ | $3.5545\times {10}^{-5}$ | ||

$1\%$ | $3.3106\times {10}^{-3}$ | $0.01$ | $9.0230\times {10}^{-2}$ |

$0.001$ | $8.9619\times {10}^{-2}$ | ||

$0.0001$ | $8.9578\times {10}^{-2}$ | ||

$2\%$ | $1.0889\times {10}^{-2}$ | $0.01$ | $1.8033\times {10}^{-1}$ |

$0.001$ | $1.8057\times {10}^{-1}$ | ||

$0.0001$ | $1.8059\times {10}^{-1}$ | ||

$5\%$ | $8.7965\times {10}^{-3}$ | $0.01$ | $4.4052\times {10}^{-1}$ |

$0.001$ | $4.4016\times {10}^{-1}$ | ||

$0.0001$ | $4.4012\times {10}^{-1}$ |

**Table 3.**Comparison of the algorithms according to the obtained errors of reconstructing the coefficient $\lambda $ for the input data from example 2.

Noise | RealACO | Iteration Method | |
---|---|---|---|

Error | $\mathit{\epsilon}$ | Error | |

$0\%$ | $0.6230$ | $0.01$ | $1.1821$ |

$0.001$ | $1.1825$ | ||

$0.0001$ | $1.1826$ | ||

$1\%$ | $0.5982$ | $0.01$ | $1.2050$ |

$0.001$ | $1.2054$ | ||

$0.0001$ | $1.2055$ | ||

$2\%$ | $0.6146$ | $0.01$ | $1.2369$ |

$0.001$ | $1.2374$ | ||

$0.0001$ | $1.2375$ | ||

$5\%$ | $0.6566$ | $0.01$ | $1.4903$ |

$0.001$ | $1.4907$ | ||

$0.0001$ | $1.4908$ |

**Table 4.**Results obtained by the RealACO algorithm (grid $100\times 1800$) (${\overline{a}}_{i}$ – reconstructed value of a coefficient ${a}_{i}$, ${\delta}_{{\overline{a}}_{i}}$ – relative error of reconstruction of the coefficient ${a}_{i}$, $\sigma $ – standard deviation ($i=1,2,3$)).

Noise | ${\overline{\mathit{a}}}_{\mathit{i}}$ | ${\mathit{\delta}}_{{\overline{\mathit{a}}}_{\mathit{i}}}[\%]$ | $\mathit{\sigma}$ |
---|---|---|---|

$0\%$ | $0.9889$ | $1.10$ | $0.1077$ |

$3.9999$ | $0.01$ | $0.0124$ | |

$1.0875$ | $1.14$ | $0.0507$ | |

$1\%$ | $0.9910$ | $0.90$ | $0.0128$ |

$3.9991$ | $0.03$ | $0.0040$ | |

$1.0874$ | $1.15$ | $0.0140$ | |

$2\%$ | $0.9900$ | $0.99$ | $0.0266$ |

$3.9998$ | $0.01$ | $0.0076$ | |

$1.0873$ | $1.15$ | $0.0156$ | |

$5\%$ | $0.9881$ | $1.18$ | $0.0277$ |

$4.0002$ | $0.01$ | $0.0182$ | |

$1.0869$ | $1.18$ | $0.0150$ |

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**MDPI and ACS Style**

Brociek, R.; Chmielowska, A.; Słota, D.
Comparison of the Probabilistic Ant Colony Optimization Algorithm and Some Iteration Method in Application for Solving the Inverse Problem on Model With the Caputo Type Fractional Derivative. *Entropy* **2020**, *22*, 555.
https://doi.org/10.3390/e22050555

**AMA Style**

Brociek R, Chmielowska A, Słota D.
Comparison of the Probabilistic Ant Colony Optimization Algorithm and Some Iteration Method in Application for Solving the Inverse Problem on Model With the Caputo Type Fractional Derivative. *Entropy*. 2020; 22(5):555.
https://doi.org/10.3390/e22050555

**Chicago/Turabian Style**

Brociek, Rafał, Agata Chmielowska, and Damian Słota.
2020. "Comparison of the Probabilistic Ant Colony Optimization Algorithm and Some Iteration Method in Application for Solving the Inverse Problem on Model With the Caputo Type Fractional Derivative" *Entropy* 22, no. 5: 555.
https://doi.org/10.3390/e22050555