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Efficient Uncertainty Assessment in EM Problems via Dimensionality Reduction of Polynomial-Chaos Expansions^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Brief Literature Review of Related Works

## 3. Proposed Methodology

#### 3.1. Polynomial Chaos Expansions

#### 3.2. The Morris Method

- For all $j,k=1,\dots ,N$.
- Let
**g**be the cells in the grid that satisfy ${m}_{j}^{*}\ge {m}_{k}^{*}$. - Calculate the mean ${m}_{j}^{*}$, for the cells in
**g**. Let this be $M\left\{{m}_{j}^{*}\left[\mathbf{g}\right]\right\}$. - Compute the product $len\left(\mathbf{g}\right)M\left\{{m}_{j}^{*}\left[\mathbf{g}\right]\right\}$, where $len\left(\mathbf{g}\right)$ is the number of values in g.

#### 3.3. The Finite-Difference Time-Domain Technique

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Mean value and (

**b**) standard deviation of the electric field for the first case of the 1D transmission-line problem. PC, Polynomial Chaos.

**Figure 3.**Mean elementary effects for each random variable in the first case of the 1D transmission-line problem.

**Figure 4.**(

**a**) Mean value and (

**b**) standard deviation of the electric field for the second case of the 1D transmission-line problem.

**Figure 6.**(

**a**) Mean value and (

**b**) standard deviation of the magnetic field for the first case of the second problem.

**Figure 7.**(

**a**) Mean value and (

**b**) standard deviation of the magnetic field for the second case of the second problem.

**Figure 10.**(

**a**) Mean value and (

**b**) standard deviation of the reflection coefficient for the first case of the path-antenna problem.

**Figure 12.**(

**a**) Mean value and (

**b**) standard deviation of the reflection coefficient for the second case of the path-antenna problem.

Dielectric Materials | Mean Dielectric Permittivities |
---|---|

${\u03f5}_{1}$ | $8.0{\u03f5}_{0}$ |

${\u03f5}_{2}$ | $2.4{\u03f5}_{0}$ |

${\u03f5}_{3}$ | $5.6{\u03f5}_{0}$ |

${\u03f5}_{4}$ | $5.6{\u03f5}_{0}$ |

${\u03f5}_{5}$ | $6.0{\u03f5}_{0}$ |

${\u03f5}_{6}$ | $7.2{\u03f5}_{0}$ |

${\u03f5}_{7}$ | $9.4{\u03f5}_{0}$ |

${\u03f5}_{8}$ | $8.7{\u03f5}_{0}$ |

Dielectric Materials | Mean Dielectric Permittivities |
---|---|

${\u03f5}_{1}$ | $8.0{\u03f5}_{0}$ |

${\u03f5}_{2}$ | $2.4{\u03f5}_{0}$ |

${\u03f5}_{3}$ | $5.6{\u03f5}_{0}$ |

${\u03f5}_{4}$ | $4.2{\u03f5}_{0}$ |

${\u03f5}_{5}$ | $6.0{\u03f5}_{0}$ |

${\u03f5}_{6}$ | $7.2{\u03f5}_{0}$ |

Parameters | Mean Values | Standard Deviations |
---|---|---|

${F}_{i}$ | $12.5$ mm | $0.360$ mm |

${G}_{pf}$ | $1.0$ mm | $0.028$ mm |

${W}_{f}$ | $8.5$ mm | $0.245$ mm |

W | $51.0$ mm | $1.472$ mm |

L | $38.0$ mm | $0.438$ mm |

$\u03f5$ | $4.3{\u03f5}_{0}$ | $0.049{\u03f5}_{0}$ |

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**MDPI and ACS Style**

Salis, C.; Kantartzis, N.; Zygiridis, T.
Efficient Uncertainty Assessment in EM Problems via Dimensionality Reduction of Polynomial-Chaos Expansions. *Technologies* **2019**, *7*, 37.
https://doi.org/10.3390/technologies7020037

**AMA Style**

Salis C, Kantartzis N, Zygiridis T.
Efficient Uncertainty Assessment in EM Problems via Dimensionality Reduction of Polynomial-Chaos Expansions. *Technologies*. 2019; 7(2):37.
https://doi.org/10.3390/technologies7020037

**Chicago/Turabian Style**

Salis, Christos, Nikolaos Kantartzis, and Theodoros Zygiridis.
2019. "Efficient Uncertainty Assessment in EM Problems via Dimensionality Reduction of Polynomial-Chaos Expansions" *Technologies* 7, no. 2: 37.
https://doi.org/10.3390/technologies7020037