Surrogates

For complex physics or characterization tests, the computational cost of models used for calibration can make the use of some calibration methods impractical. The ability to use surrogates that accurately reproduce model results can facilitate more rigorous calibrations. This section covers the surrogate generation tools available within MatCal and how to incorporate them in the calibration process.

Gaussian Process Surrogates Theory

These surrogates are based on those developed in [24].

In the cited work, surrogates were employed to approximate an expensive finite element (FE) model, providing predictions of load and displacement based on a specific set of constitutive model parameters. To enhance the surrogate representation of the full-field displacements, principal component analysis (PCA) was utilized. PCA is a dimension-reduction technique that transforms high-dimensional data into a reduced basis, preserving the maximum variance within the data. Through PCA, the high-dimensional data was effectively represented using only a few PCA modes and their corresponding singular values (a.k.a. amplitudes). Surrogates were subsequently constructed for each PCA singular value. Consequently, given a set of parameter values, the surrogate produced predictions of the PCA singular values, enabling the reconstruction of the high-dimensional displacement field with little information loss.

Response Dimensionality Reduction

Given a data matrix, $\mathcal{A} \in \mathbb{R}^{u \times v}$ , PCA is used to factorize the matrix into three new matrices, $\mathcal{A} = USV^{T}$ , where $U \in \mathbb{R}^{u \times r}$ and $V^{T} \in \mathbb{R}^{r \times v}$ are orthonormal bases with columns and rows, respectively, containing the left and right singular vectors of $\mathcal{A}$ . The matrix $S \in \mathbb{R}^{r \times r}$ is diagonal and contains the singular values, and $r$ is the rank of $\mathcal{A}$ . By retaining a number of principal components $p < r$ , matrix $\mathcal{A}$ can be expressed in a reduced-dimensional space. The retained components are written as $V^{*T} \subset V^{T}, V^{*T} in \mathbb{R}^{p times v}$ . The reduction of $\mathcal{A}$ is written as $\mathcal{A}^{pca} = \mathcal{A}'V^{*} \in \mathbb{R}^{u \times p}$ and its reconstruction by $\mathcal{A}^{recon} = \mathcal{A}^{pca}V^{*T}$ .

The data matrix $\mathcal{A}$ is representative of a high-dimensional data set, such a full-field data. The dimension $u$ is the number of repeat measurements, or model evaluations with different parameter values, and $v$ is the number of measurements in each data set. Generally speaking, for this application, $u < v$ .

The scikit-learn implementation of PCA is utilized in MatCal [21].

GP Surrogates of PCA Amplitudes

Results Reconstruction

Adaptive GP Surrogates

The fundamental concepts used for adaptive GP surrogate construction are based on those developed in [11],

This adaptive surrogate modeling approach iteratively selects new sample points to construct a surrogate more efficiently and with greater accuracy than traditional space-filling methods. It combines K-fold cross validation with Voronoi tessellations of the input space to assess local model error and identify regions where the surrogate performs poorly, guiding the refinement of the parameter space. A batch Voronoi adaptive sampling strategy is employed to select multiple informative samples simultaneously, reducing overall computational cost. By balancing exploration and exploitation through cross-validation error estimates and spatial partitioning, the method concentrates the sampling efforts to regions of high uncertainty or error, steadily improving surrogate accuracy while minimizing expensive model evaluations.