We propose a parametric sampling strategy for the reduction of large-scale PDE systems with multidimensional input parametric spaces by leveraging models of different fidelity. The design of this methodology allows a user to adaptively sample points ad hoc from a discrete training set with no prior requirement of error estimators.It is achieved by exploiting low-fidelity models throughout the parametric space to sample points using an efficient sampling strategy, and at the sampled parametric points, high-fidelity models are evaluated to recover the reduced basis functions. The low-fidelity models are then adapted with the reduced order models ( ROMs) built by projection onto the subspace spanned by the recovered basis functions. The process continues until the low-fidelity model can represent the high-fidelity model adequately for all the parameters in the parametric space. Since the proposed methodology leverages the use of low-fidelity models to assimilate the solution database, it significantly reduces the computational cost in the offline stage. The highlight of this article is to present the construction of the initial low-fidelity model, and a sampling strategy based on the discrete empirical interpolation method (DEIM). We test this approach on a 2D steady-state heat conduction problem for two different input parameters and make a qualitative comparison with the classical greedy reduced basis method (RBM), and further test on a 9-dimensional parametric non-coercive elliptic problem and analyze the computational performance based on different tuning of greedy selection of points.

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The study of viscoelastic fluid flows are very complexdue to the nature of the governing equations making numerical simulations a complex task.A key challenge in such flows is the stabilisation of numerical techniques which is quite often difficult to meetspecially when the Weissenberg number is high.The momentum and continuity equations form an elliptic saddle point problem for velocity and pressure and the constitutive equation is hyperbolic in nature [1]. In such flows, due to the dependency of the viscoelastic stress tensor on velocity gradients, there arises some compatibility issues in the discretisation space between velocity field and viscoelastic stress tensor. There are many improvements in the numerical techniques to treat and maintain the elliptic nature of the momentum equation. Here, we implement discrete elastic viscous split stress-Galerkin (DEVSS-G) [1,2] for smooth interpolation of velocity gradient in the constitutive equation by considering velocity gradient as an additional dependent variable. Also, due to the presence of convection term in the constitutive equation, there can arise spurious numerical oscillations in convection dominated case. So stabilisation techniques like SUPG [2,3] is applied where the test function is modified and more weightage is being put in the upwind direction.

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We propose a parametric sampling strategy for reduction of large scale PDE systems with multidimensional input parametric spaces by leveraging models of different fidelity. The design of this methodology allows a user to adaptively sample points ad hoc from a discrete training set with no prior requirement of error estimators. It is achieved by exploiting low‐fidelity models throughout the parametric space to sample points using an efficient sampling strategy, and at the sampled parametric points, high‐fidelity models are evaluated to recover the reduced basis functions. The low‐fidelity models are then adapted with the reduced order models built by projection onto the subspace spanned by the recovered basis functions. The process continues until the low‐fidelity model can represent the high‐fidelity model adequately for all the parameters in the parametric space. Since the proposed methodology leverages the use of low‐fidelity models to assimilate the solution database, it significantly reduces the computational cost in the offline stage. The highlight of this article is to present the construction of the initial low‐fidelity model, and a sampling strategy based on the discrete empirical interpolation method. We test this approach on a 2D steady‐state heat conduction problem for two different input parameters and make a qualitative comparison with the classical greedy reduced basis method and with random selection of points. Further, we test the efficacy of the proposed method on a 9‐dimensional parametric non‐coercive elliptic problem and analyze the computational performance based on different tuning of greedy selection of points.