# Bayesian Uncertainty Quantification with Multi-Fidelity Data and Gaussian Processes for Impedance Cardiography of Aortic Dissection

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## Abstract

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## 1. Introduction

## 2. Bayesian Multi-Fidelity Scheme

#### 2.1. Statistical Model

#### 2.2. Prediction and Its Uncertainty

## 3. Algorithm and Mock Data Scrutiny

## 4. Application to Finite Element Simulations of Impedance Cardiography of Aortic Dissection

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Appendix A. Mathematical Proofs

#### Appendix A.1. Parameter Posterior and Parameter Estimates

#### Appendix A.2. Predictive Mean

#### Appendix A.3. Predictive Covariance

## References

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**Figure 1.**Mock data analysis: Posterior probability density functions of the nonlinear kernel parameters ${\alpha}_{1}$ and ${\alpha}_{2}$. Black dashed line: True value

**Figure 2.**Mock data analysis: Prediction. Note that the uncertainties have been multiplied by a factor of 10 for illustrative purposes.

**Figure 3.**Right: Mesh-converged HiFi model with 100,000–550,000 degrees of freedom. Left: LoFi model with 9000–15,000 with labels of the geometrical objects. Adapted from Reference [27]

**Figure 5.**Data, prediction, and prediction uncertainty of the absolute value of the admittance, i.e., the inverse impedance in units of inverse Ohm: ${z}_{1}({x}_{2})$ denotes LoFi data at the same pivot points as HiFi data.

Hyperparameter | Estimate (Multi-Fidelity) | Truth |
---|---|---|

${\beta}_{1}^{(1)}$ | $0.30\pm 0.07$ | $0.32$ |

${\beta}_{1}^{(2)}$ | $-0.30\pm 0.09$ | $-0.40$ |

${\beta}_{1}^{(3)}$ | $0.02\pm 0.08$ | $0.1$ |

${\beta}_{1}^{(4)}$ | $0.34\pm 0.06$ | $0.35$ |

${\beta}_{1}^{(5)}$ | $-0.50\pm 0.04$ | $-0.51$ |

${\beta}_{1}^{(6)}$ | $0.35\pm 0.02$ | $0.33$ |

${\beta}_{1}^{(7)}$ | $-0.033\pm 0.008$ | $-0.034$ |

${\beta}_{1}^{(8)}$ | $-0.146\pm 0.005$ | $-0.142$ |

${\beta}_{1}^{(9)}$ | $0.1745\pm 0.0008$ | $0.1750$ |

${\beta}_{1}^{(10)}$ | $-0.1031\pm 0.0004$ | $-0.1034$ |

${\beta}_{2}^{(1)}$ | $0.42\pm 0.59$ | $0.\dot{3}$ |

${\beta}_{2}^{(2)}$ | $-0.17\pm 0.55$ | $0.15$ |

${\beta}_{2}^{(3)}$ | $0.11\pm 0.17$ | $0.01\dot{6}$ |

${\beta}_{2}^{(4)}$ | $0.03\pm 0.10$ | 0 |

${\sigma}_{1}$ | $0.133\pm 0.096$ | $0.1$ |

${\sigma}_{2}$ | $0.47\pm 0.58$ | $0.01$ |

${\rho}_{1}$ | $2.97\pm 0.02$ | 3 |

${\alpha}_{1}$ | $8.6\pm 0.9$ | $10.1$ |

${\alpha}_{2}$ | $10\pm 3$ | $20.1$ |

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**MDPI and ACS Style**

Ranftl, S.; Melito, G.M.; Badeli, V.; Reinbacher-Köstinger, A.; Ellermann, K.; von der Linden, W.
Bayesian Uncertainty Quantification with Multi-Fidelity Data and Gaussian Processes for Impedance Cardiography of Aortic Dissection. *Entropy* **2020**, *22*, 58.
https://doi.org/10.3390/e22010058

**AMA Style**

Ranftl S, Melito GM, Badeli V, Reinbacher-Köstinger A, Ellermann K, von der Linden W.
Bayesian Uncertainty Quantification with Multi-Fidelity Data and Gaussian Processes for Impedance Cardiography of Aortic Dissection. *Entropy*. 2020; 22(1):58.
https://doi.org/10.3390/e22010058

**Chicago/Turabian Style**

Ranftl, Sascha, Gian Marco Melito, Vahid Badeli, Alice Reinbacher-Köstinger, Katrin Ellermann, and Wolfgang von der Linden.
2020. "Bayesian Uncertainty Quantification with Multi-Fidelity Data and Gaussian Processes for Impedance Cardiography of Aortic Dissection" *Entropy* 22, no. 1: 58.
https://doi.org/10.3390/e22010058