The efficient solution of parametric problems (e.g., in optimisation and uncertainty quantification) requires reduced order models (ROM) for fast, many-queries evaluations. These technologies are particularly critical in the context of digital twins of complex systems such as industrial production pipelines and smart cities. The computational cost for constructing these ROMs depends upon the number of involved parameters, the size of the system under analysis and its complexity (i.e., the cost of each snapshot). While dimensionality reduction techniques have been proposed to deal with the former aspect [1], the latter can be tackled using multi-fidelity models [2], data augmentation techniques [3] and local surrogate models [4]. In this talk, a local surrogate model is devised for a parametric elliptic PDE, starting from a multi-domain formulation with local subproblems featuring arbitrary Dirichlet conditions [4]. The local ROM is obtained coupling proper generalised decomposition (PGD) and domain decomposition (DD) techniques. The proposed approach presents several advantages: low-dimensional subproblems with only few active boundary parameters are obtained from the linearity of the PDE; an overlapping Schwarz method is employed to glue the local surrogate models, with no need for Lagrange multipliers to enforce the continuity in the overlapping region; no additional problems need to be solved during the execution of the Schwarz algorithm in the online phase. Numerical results are presented to illustrate accuracy and robustness of the resulting DD-PGD approach and its suitability to significantly reduce the cost of applying standard high-fidelity DD methods in the context of parametric problems. [1] F. Anowar et al., Conceptual and empirical comparison of dimensionality reduction algorithms (PCA, KPCA, LDA, MDS, SVD, LLE, ISOMAP, LE, ICA, t-SNE). Comput. Sci. Rev., 40:100378 (2021) [2] J.D. Jakeman et al., Adaptive experimental design for multi‐fidelity surrogate modeling of multi‐disciplinary systems. Int. J. Numer. Methods Eng., 123, 2760-2790 (2022) [3] A. Muixí et al., Data augmentation for the POD formulation of the parametric laminar incompressible Navier-Stokes equations. arXiv:2312.14756 (2023) [4] M. Discacciati et al., An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalised decomposition. Comput. Methods Appl. Mech. Eng., 418:116484 (2024)