Advances in Hierarchical Model Reduction and combination with other computational reduction methods

Giulia Meglioli,Matteo Zancanaro,Simona Perotto,Gianluigi Rozza,Francesco Ballarin

doi:10.14293/p2199-8442.1.sop-math.jownzf.v1

Giulia Meglioli, Matteo Zancanaro + Show 3 more

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https://doi.org/10.14293/p2199-8442.1.sop-math.jownzf.v1

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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.

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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

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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

Γlat Γout γx

Number of basis functions