scholarly journals Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function

2018 ◽  
Vol 42 (17) ◽  
pp. 5649-5667 ◽  
Author(s):  
Julia Calatayud ◽  
Juan Carlos Cortés ◽  
Marc Jornet
Keyword(s):  
Probability Density Function ◽  
Boundary Conditions ◽  
Uncertainty Quantification ◽  
Probability Density ◽  
Density Function ◽  
Bounded Domain ◽  
Parabolic Equations ◽  
Nonhomogeneous Boundary ◽  
Nonhomogeneous Boundary Conditions

Uncertainty quantification of nonlinear distributed parameter systems using generalized polynomial chaos

2019 ◽  
Vol 67 (4) ◽  
pp. 283-303
Author(s):  
Chettapong Janya-anurak ◽  
Thomas Bernard ◽  
Jürgen Beyerer
Keyword(s):  
Probability Density Function ◽  
Uncertainty Quantification ◽  
Probability Density ◽  
Density Function ◽  
Polynomial Chaos ◽  
Distributed Parameter Systems ◽  
Computational Effort ◽  
Distributed Parameter ◽  
Model Parameters ◽  
Generalized Polynomial

Abstract Many industrial and environmental processes are characterized as complex spatio-temporal systems. Such systems known as distributed parameter systems (DPSs) are usually highly complex and it is difficult to establish the relation between model inputs, model outputs and model parameters. Moreover, the solutions of physics-based models commonly differ somehow from the measurements. In this work, appropriate Uncertainty Quantification (UQ) approaches are selected and combined systematically to analyze and identify systems. However, there are two main challenges when applying the UQ approaches to nonlinear distributed parameter systems. These are: (1) how uncertainties are modeled and (2) the computational effort, as the conventional methods require numerous evaluations of the model to compute the probability density function of the response. This paper presents a framework to solve these two issues. Within the Bayesian framework, incomplete knowledge about the system is considered as uncertainty of the system. The uncertainties are represented by random variables, whose probability density function can be achieved by converting the knowledge of the parameters using the Principle of Maximum Entropy. The generalized Polynomial Chaos (gPC) expansion is employed to reduce the computational effort. The framework using gPC based on Bayesian UQ proposed in this work is capable of analyzing systems systematically and reducing the disagreement between model predictions and measurements of the real processes to fulfill user defined performance criteria. The efficiency of the framework is assessed by applying it to a benchmark model (neutron diffusion equation) and to a model of a complex rheological forming process. These applications illustrate that the framework is capable of systematically analyzing the system and optimally calibrating the model parameters.

scholarly journals Bayesian Inference in Credibility Theory

1975 ◽  
Vol 8 (2) ◽  
pp. 164-174
Author(s):  
L. D'Hooge ◽  
M. J. Goovaerts
Keyword(s):  
Bayesian Inference ◽  
Probability Density Function ◽  
Boundary Conditions ◽  
Probability Density ◽  
Density Function ◽  
Prior Probability ◽  
Correction Term ◽  
Credibility Theory ◽  
Bayesian Forecast ◽  
Inference Techniques

AbstractIn 1967, Bühlmann has shown that the credibility formula was the best linearized approximation to the exact Bayesian forecast.His result for the credibility factor can be found back by means of some Bayesian inference techniques. Introducing a uniform prior probability density function for the credibility factor provides us with a method for estimating z, a correction term to the Bühlmann's result is obtained. It is shown how prior boundary conditions can be introduced.

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