#
Bayesian Surrogate Analysis and Uncertainty Propagation^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Bayesian Uncertainty Quantification

#### 2.1. General Structure of the Problem

#### 2.2. Bayesian Analysis and Selection of the Surrogate Model

#### 2.3. Bayesian Uncertainty Propagation with Surrogate Models

## 3. Numerical Example

## 4. Discussion

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mathematical Proofs

## Appendix B. The Transformation Invariant Prior for the Surrogate Coefficients

## References

- Xiu, D.; Karniadakis, G.E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput.
**2005**, 27, 1118–1139. [Google Scholar] [CrossRef] - O’Hagan, A. Polynomial Chaos: A Tutorial and Critique from a Statistician’s Perspective. 2013. Available online: http://tonyohagan.co.uk/academic/pdf/Polynomial-chaos.pdf (accessed on 25 June 2019).
- Crestaux, T.; Le Maître, O.P.; Martinez, J.-M. Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf.
**2009**, 94, 1161–1172. [Google Scholar] [CrossRef] - O’Hagan, A. Curve Fitting and Optimal Design for Prediction. J. R. Stat. Soc. Ser. B
**1978**, 40, 1–42. [Google Scholar] [CrossRef] - Rasmussen, C.E.; Williams, C.K. Gaussian Processes for Machine Learning; The MIT Press: Cambridge, MA, USA, 2006. [Google Scholar]
- Sraj, I.; Le Maître, O.P.; Knio, O.M.; Hoteit, I. Coordinate transformation and Polynomial Chaos for the Bayesian inference of a Gaussian process with parametrized prior covariance function. Comput. Methods Appl. Mech. Eng.
**2016**, 298, 205–228. [Google Scholar] [CrossRef] [Green Version] - O’Hagan, A.; Kennedy, M.C.; Oakley, J.E. Uncertainty analysis and other inference tools for complex computer codes. Bayesian Stat.
**1999**, 6, 503–524. [Google Scholar] - Kennedy, M.C.; O’Hagan, A. Predicting the output from a complex computer code when fast approximations are available. Biometrika
**2000**, 87, 1–13. [Google Scholar] [CrossRef] [Green Version] - Arnst, M.; Ghanem, R.G.; Soize, C. Identification of Bayesian posteriors for coefficients of chaos expansions. J. Comput. Phys.
**2010**, 229, 3134–3154. [Google Scholar] [CrossRef] [Green Version] - Madankan, R.; Singla, P.; Singh, T.; Scott, P.D. Polynomial-chaos-based Bayesian approach for state and parameter estimations. J. Guid. Control Dyn.
**2013**, 36, 1058–1074. [Google Scholar] [CrossRef] [Green Version] - Karagiannis, G.; Lin, G. Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs. J. Comput. Phys.
**2014**, 259, 114–134. [Google Scholar] [CrossRef] [Green Version] - Lu, F.; Morzfeld, M.; Tu, X.; Chorin, A.J. Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems. J. Comput. Phys.
**2015**, 282, 138–147. [Google Scholar] [CrossRef] [Green Version] - Hwai, M.; Tan, Y. Sequential Bayesian Polynomial Chaos Model Selection for Estimation of Sensitivity Indices. SIAM/ASA J. Uncertain. Quantif.
**2015**, 3, 146–168. [Google Scholar] [CrossRef] - Ghanem, R.G.; Owhadi, H.; Higdon, D. Handbook of Uncertainty Quantification; Springer: New York, NY, USA, 2017. [Google Scholar] [CrossRef]
- O’Hagan, A. Bayesian analysis of computer code outputs: A tutorial. Reliab. Eng. Syst. Saf.
**2006**, 91, 1290–1300. [Google Scholar] [CrossRef] - von der Linden, W.; Dose, V.; von Toussaint, U. Bayesian Probability Theory: Applications in the Physical Sciences, 1st ed.; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar] [CrossRef]
- Oladyshkin, S.; Nowak, W. Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliab. Eng. Syst. Saf.
**2012**, 106, 179–190. [Google Scholar] [CrossRef] - Jakeman, J.D.; Franzelin, F.; Narayan, A.; Eldred, M.; Plfüger, D. Polynomial chaos expansions for dependent random variables. Comput. Methods Appl. Mech. Eng.
**2019**, 351, 643–666. [Google Scholar] [CrossRef] [Green Version] - von der Linden, W.; Preuss, R.; Hanke, W. Consistent Application of Maximum Entropy to Quantum-Monte-Carlo Data. J. Physics: Condens. Matter
**1996**, 8, 1–13. [Google Scholar] [CrossRef] - Ranftl, S.; Müller, T.; Windberger, U.; von der Linden, W.; Brenn, G. Data and Codes for ’A Bayesian Approach to Blood Rheological Uncertainties in Aortic Hemodynamcis’; Zenodo Digital Repository: Genève, Switzerland, 2021. [Google Scholar] [CrossRef]
- Torre, E.; Marelli, S.; Embrechts, P.; Sudret, B. A general framework for data-driven uncertainty quantification under complex input dependencies using vine copulas. Probabilistic Eng. Mech.
**2019**, 55, 1–16. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Simulation data (black dots) and simulation uncertainty ($1\sigma $) according to our Bayesian approach (red, including surrogate uncertainty) as well as the naive simulation uncertainty (blue, neglecting surrogate uncertainty). The black line is the surrogate mean.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ranftl, S.; von der Linden, W.
Bayesian Surrogate Analysis and Uncertainty Propagation. *Phys. Sci. Forum* **2021**, *3*, 6.
https://doi.org/10.3390/psf2021003006

**AMA Style**

Ranftl S, von der Linden W.
Bayesian Surrogate Analysis and Uncertainty Propagation. *Physical Sciences Forum*. 2021; 3(1):6.
https://doi.org/10.3390/psf2021003006

**Chicago/Turabian Style**

Ranftl, Sascha, and Wolfgang von der Linden.
2021. "Bayesian Surrogate Analysis and Uncertainty Propagation" *Physical Sciences Forum* 3, no. 1: 6.
https://doi.org/10.3390/psf2021003006