Abstract
In this work we present address the combination of the Hierarchical Model (Hi-Mod) reduction approach with projection-based reduced order methods, exploiting either on Greedy Reduced Basis (RB) or Proper Orthogonal Decomposition (POD), in a parametrized setting. The Hi-Mod approach, introduced in, is suited to reduce problems in pipe-like domains featuring a dominant axial dynamics, such as those arising for instance in haemodynamics. The Hi-Mod approach aims at reducing the computational cost by properly combining a finite element discretization of the dominant dynamics with a modal expansion in the transverse direction. In a parametrized context, the Hi-Mod approach has been employed as the high-fidelity method during the offline stage of model order reduction techniques based on RB or POD. The resulting combined reduction methods, which have been named Hi-RB and Hi-POD, respectively, will be presented with applications in diffusion-advection problems, fluid dynamics and optimal control problems, focusing on the approximation stability of the proposed methods and their computational performance.
Highlights
Hierarchical Model Reduction (HiMod) is based on the decomposition of the domain into a dominant ow direction Ω1D and a transverse one, γx
These results are related to oine and online solutions for an Advection-DiusionReaction PDE, where the parameters are allowed to vary on very large ranges
The online solution manifold is composed by 20 HiMod basis functions
Summary
When dealing with Parametrized Partial Dierential Equations the computational cost required by a large number of solutions for each new value of the involved parameters may be unaordably large. Dierent methods have been studied in order to nd solutions in a more ecient way. We combine the Hierarchical Model Reduction (HiMod) technique [5, 4, 1], developed for non-parametric equations over domains with a dominant ow direction, with a Reduced Basis method, to eciently tackle parameter dependence [3]. Depending on the construction of the reduced basis space we end up with two reduced order techniques called HiRB (based on a Greedy algorithm [6]) and HiPOD (based on Proper Orthogonal Decomposition [2]). We present results related to saddle point problems, in particular to Stokes equations, and Optimal Control problems
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