# Investigating Bayesian Parameter Identification Using Non-Standard Laboratory Specimens
## 2 Modeling of Fracture Propagation
### General Polynomial Chaos Expansion (gPCE) Proxy Model  
Performing multiple fracture simulations often requires extensive computational efforts, which can be both time and resource-consuming. These can make solving the Bayesian inverse problem using the MCMC method excessively slow or even impossible. To address this issue, mathematical proxy models are necessary. These models, also known as surrogate, meta, or response surface models, provide approximate results of numerical simulations with a certain degree of precision, thereby significantly accelerating the computational process [51; 55; 56]:  
\[\mathcal{G}(\mathbf{z})\approx\mathcal{\hat{G}}(\mathbf{z}). \tag{18}\]  
Here, \(\mathcal{\hat{G}}(\mathbf{z})\) represents the proxy model's approximation of the forward model operator \(\mathcal{G}\). The choice of proxy model must be made carefully, taking into account the characteristics of the analyzed problem. The study [49] compares the accuracy of various proxy models used to approximate numerical simulation results in crack propagation problems. With respect to performance, for simpler fracture propagation cases, the gPCE model demonstrated accuracy comparable to other approaches, such as neural networks, and was effective in achieving precise parameter identification. From a practical standpoint, the gPCE model is well-regarded for its computational efficiency and ease of training. Additionally, a sufficiently accurate gPCE model provides an analytical representation of the numerical model's output, enabling efficient statistical analysis with minimal computational cost. Based on these considerations, the gPCE proxy model is selected. However, it is important to note that for more complex fracture problems, gPCE may not be the most suitable proxy model, as highlighted in the same study. The gPCE method approximates numerical model solutions through a linear combination of multidimensional polynomials and their corresponding coefficients [53; 57; 58], as expressed by the following:  
\[\mathbf{u}=\mathcal{G}(\mathbf{z})\approx\mathcal{\hat{G}}(\mathbf{z})=\bar{ \mathbf{u}}=\sum_{k=0}^{K-1}\mathbf{v}_{k}\Phi_{k}(\mathbf{z})=\mathbf{Y}\mathbf{ \Phi}(\mathbf{z}), \tag{19}\]  
where \(\bar{\mathbf{u}}\) represents the approximation of the numerical model's output, \(\mathbf{\Phi}\) is the selected set of polynomials collected in the vector \(\mathbf{\Phi}=[\Phi_{0}(\mathbf{z}),\Phi_{1}(\mathbf{z}),\ldots,\Phi_{K-1}( \mathbf{z})]^{T}\), and \(\mathbf{Y}=[\mathbf{v}_{0},\mathbf{v}_{1},\ldots,\mathbf{v}_{K-1}]\) denotes the coefficients of the polynomial expansion, where \(\mathbf{v}_{k}\in\mathbb{R}^{Q}\) corresponds to the polynomial \(\Phi_{k}\). To accurately approximate the forward model, a polynomial basis is first selected based on the prior distribution of the model parameters. The coefficients \(\mathbf{Y}\) are then determined by solving a linear regression problem that minimizes the mean squared error (MSE) [53]:  
\[\mathbb{E}[\hat{\mathbf{e}}^{2}]=\mathbb{E}\Big{[}(\mathbf{u}-\tilde{\mathbf{u}}) ^{2}\Big{]}=\int_{\mathbb{R}}(\mathbf{u}(\mathbf{z})-\tilde{\mathbf{u}}( \mathbf{z}))^{2}\pi(\mathbf{z})d\mathbf{z}\approx\frac{1}{N}\sum_{i=1}^{N} \Bigl{(}\mathbf{u}(\mathbf{z}_{i})-\tilde{\mathbf{u}}(\mathbf{z}_{i}))^{2} \Bigr{)}. \tag{20}\]  
One significant advantage of the gPCE method is its ability to compute sensitivities directly from its coefficients [59]. Sensitivity analysis plays an essential role in experimental design, helping to optimize sensor placement and identify the most informative measurable outputs [60]. In this study, we used Sobol Sensitivity Analysis, a method for evaluating the contribution of input variables (i.e., model parameters) to the variance in the model's output. This method decomposes the total variance in the output into partial variances, which can be calculated directly from the coefficients of the gPCE model.