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V

= V
rg local,i
rg global
Ξ_i (3.1)

with V
rg local,i
^{n N}m ^rg global
∈ R^n×(Ns×Nm) and Ξ ∈ R^(Ns×Nm)×Nm , where N_s is the total number of generated

∈ ^×R , V
snapshots and N_m is the number of independent components contained in each local basis, which are further assumed to

×

× ×
be independent to the N_l components of the obtained local bases. In this manner, only an interpolation of the coefficient matrix per validation point Ξ is required in order to obtain a local projection basis. This interpolation scheme is graphi- cally presented in Figure 1 by the light blue surface, which represents the domain spanned by the coefficient matrices, while the interpolation of the coefficient matrices is represented by the corresponding spline. Therefore, the proposed scheme requires interpolation of the coefficient matrix, which comprises only (N_s N_m) N_m entries, removing thus the dependency on the large dimension n of the full problem. To the contrary, a straightforward interpolation scheme would need to estimate n N_m entries, which would be probably accomplished by N_m interpolation schemes, one for each n-length vector.
As briefly discussed in section 2, the interpolation is carried out on the tangent space of the manifold and aims to exploit the fact that the solution in the vicinity of each parameter sample can be spanned by a local subspace, which is herein referred to as the local basis. Within this context, the local basis at an unseen point of the parameter space can be expressed as a combination of independent groups of basis vectors, with each one of those groups extracted from the

Table 1: Reduction Framework Algorithmic Process. N_s denotes the number of training snapshots, N_dof the degrees of freedom of the model,N_t the number of simulated timesteps and N_modes the independent components of the reduction basis. Notation is kept similar to the Equations of sections 2 and 3