2 G.N. Absi and S. Mahadevan [9, 10]. The generated matrices are faithful to modal analyses, but do not always maintain structural connectivity, and may not always retain physical significance. Iterative methods study the changes in model parameterization to evaluate the type and the location of the erroneous parameters. and try to minimize the difference between the experimental data and the FE model predictions by varying these parameters [11]. In these cases, the mathematical model used in the model updating can sometimes be ill conditioned. Liang and Mahadevan [12] replaced the expensive computational model with a surrogate model using Polynomial Chaos Expansion (PCE), and developed a systematic error quantification methodology. This approach facilitates running inexpensive simulations, while taking into account the resulting errors and uncertainties. This paper considers Bayesian calibration of model parameters with experimental data, using a corrected low-fidelity model. It uses the information available in high-fidelity simulation to adjust the low-fidelity model, for better agreement with experimental results. The aim is to reduce the uncertainty in the parameters ahead of the final calibration (especially when a small number of experimental data is available), thus providing a stronger prior that takes into account additional high-fidelity information that may be missing in the low-fidelity model (such as non-linearity, additional variables, etc.). In the same way, information available in the low-fidelity model and missing from the high-fidelity one is retained, such as modeling the full domain (i.e. the full time history vs. a small segment). The corrected low-fidelity model is inexpensive, and becomes more accurate. This is particularly useful when limited experimental data are available, and the need of a reliable, yet fast model is essential. 1.2 Multi-fidelity Calibration Method In this section, the concept of calibration is extended from a simple calibration using experimental data with a single model, to a sequential one, combining models of different fidelities. Assume that we have two models G1(X) and G2(X) of different fidelities. In order to achieve computational efficiency in Bayesian calibration, each model is replaced by fast running surrogate models S1(X) andS2(X). In order to build these surrogate models, the original models need to be evaluated multiple times. Assuming higher fidelity models run much slower than lower fidelity ones, time constraints will only allow fewer higher fidelity simulations. 1.2.1 Surrogate Model: Polynomial Chaos Expansion Many surrogate modeling techniques have been developed in the literature, such as linear/quadratic polynomial-based response-surface [13], artificial neural networks [14], support vector machine (SVM) [15], polynomial chaos expansion (PCE) [16], and Gaussian process (GP) interpolation (or Kriging) [17]. In this paper, a PCE is used to replace the original models for inexpensive sampling in the calibration process. PCE is a regression-based surrogate model that represents the output of a model with a series expansion in terms of standard random variables (SRVs). Consider a model yDf (x) where xDfx1, x2, : : : , xkg T is a vector of input random variables. We construct a PCE to replace f (x) using n multi-dimensional Hermite polynomials as basis functions: y D n X jD0 j j . / D T'. / C"surr (1.1) where is a vector of independent standard normal random variables which correspond to the original input x [18]. '(.)Df 0(.), 1(.), : : : , n(.)g T are the Hermite polynomial basis functions, and ™Df 0, 1, : : : , n,g T are the corresponding coefficients that can be estimated by the least squares method. A collocation point method can be used to efficiently select training points where the original model is evaluated [19]. Suppose that mtraining points ( i, yi), i D1, 2, : : : , mare available. Under the Gauss-Markov assumption [20], the surrogate model error "surr asymptotically follows a normal distribution with zero mean and variance given by Var Œ"surr s 2 Cs2'. /T ˆTˆ 1 '. / (1.2) where ˆDf'. 1/ ;'. 2/ ; : : : ;'. m/g T and s2 D 1 m n m X iD1 yi T'. i / 2.
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