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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.

Abstract. Many industrial and environmental processes 702Chettapong Janya-anurak, Thomas Bernard and Jürgen Beyerer INTRODUCTIONNowadays the computer simulation based on mathematical model is commonly applied in every branch of natural science and engineering disciplines. Simulations are essential tools for engineers to analysis, design and control technical processes. Many industrial and environmental processes are characterized as complex spatio-temporal systems. By using physical laws, the systems can be described mathematically with Partial Differential Equations (PDEs).In practice the modeling of the real process often leads to nonlinear coupled PDEs. Such models are often highly complex and their relationships between model inputs, model output and parameters may be poorly understood. Moreover, due to incomplete knowledge on underlying physics, simplifying assumptions or inevitable intrinsic variability, the solutions of physics-based models commonly differ from the real measurements.These problems are widely recognized in the scientific community and have also led the uncertainty quantification (UQ) framework to constitute an active research area recently. Uncertainty Quantification covers a wide range of topics. The relevant issues are, for example, propagation of uncertainty, sensitivity analysis and inverse problem, which are mainly employed through this paper.Under the uncertainty quantification framework, uncertainties in models are quantified using different mathematical tools. Expressing the uncertainties with a probabilistic description seems to be the one mostly chosen in practice. The stochastic approach of uncertainty modeling is achieved by representing uncertainties in the models as random variables, stochastic processes or random fields. However, solving coupled nonlinear PDEs with random variables requires generally extensive computational effort.The generalized Polynomial Chaos has been proposed on the last few years as an efficient methodology to computing in uncertainty quantification framework. The gPC is the extension of the original Polynomial Chaos expansion (PCE), proposed by Wiener in 1938 [1]. The original Wiener's Polynomial Chaos employs Hermite polynomial to represent the Gaussian random processes. The gPC extend the PCE toward some parametric statistical non-Gaussian distribution, based on the Askey scheme of orthogonal polynomials [2].Besides many academic examples have proven the potential of gPC, for example, the uncertainties propagation in PDE [3], calculation of Sobols' Indices for the sensitivity analysis [4] [9]and Bayesian inference in inverse problem [5], the application of gPC are found in a variety of areas, such as fluid dynamic, stricture-flow interactions, material deformations, internal combustion engine and biological problems.In this paper, we propose a concept for sensitivity analysis and parameter calibration of coupled nonlinear PDEs based on gPC-approximation. From user defined parameter uncertainties, the system responses are expanded with Polynomial Cha...