Figure 4.

A schematic illustration of the proposed data-driven surrogate modeling framework for UQ of computational head models: in the first stage (see Section 3.2), the available material properties of each of the four substructures, ${𝒳}_{\text{M}}^i\in {\mathbb{R}}^{4\times N_i}(N_i$ denotes number of voxels for substructure $i$) are used to generate 300 realizations of the input random vector of material properties for the 2D head model (i.e., ${𝒳}_{\text{M}}$). Simulations of these realizations yields input-output (${{𝒳}_{\text{M}}-𝒴}_{\text{M}}^{\text{MAS}}$) data sets for training the surrogate model in the second stage. The surrogate model is developed in three steps (see Section 3.3): 1. perform nonlinear dimensionality reduction on the output via Grassmannian diffusion maps, 2. create Gaussian process mappings between the input and the reduced solutions (i.e., ${\mathbf{\Theta }}_{\text{M}}^{\text{MAS}}$ and ${\mathbf{\Sigma }}_{\text{M}}^{\text{MAS}}$), and 3. for out-of-sample predictions, create geometric harmonics mappings between the diffusion coordinates ${\mathbf{\Theta }}_{\text{M}}^{\text{MAS}}$ and the matrices ${\mathbf{\Gamma }}_{\text{U},\text{M}}^{\text{MAS}}$ and ${\mathbf{\Gamma }}_{\text{V},\text{M}}^{\text{MAS}}$ of the tangent spaces of the Grassmann manifolds, followed by exponential mappings (${exp}_{\stackrel{-}{\mathit{U}}}$ and ${exp}_{\stackrel{-}{\mathit{V}}}$, about the Karcher means) to obtain ${\mathbf{U}}_{\text{M}}^{\text{MAS}}$ and ${\mathbf{V}}_{\text{M}}^{\text{MAS}}$, and reverse SVD to reconstruct the full strain field.