Rational-approximation-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots

Francesca Bonizzoni; Davide Pradovera; Michele Ruggeri; Francesca Bonizzoni; Davide Pradovera; Michele Ruggeri

doi:10.3934/mine.2023074

Mathematics in Engineering

2023, Volume 5, Issue 4: 1-38. doi: 10.3934/mine.2023074

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Research article

Rational-approximation-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots

1.
MOX - Department of Mathematics, Politecnico di Milano, 20133 Milano, Italy
2.
Department of Mathematics, University of Vienna, 1090 Vienna, Austria
3.
Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, United Kingdom

Received: 24 October 2022 Revised: 13 January 2023 Accepted: 16 January 2023 Published: 31 January 2023

We introduce several spatially adaptive model order reduction approaches tailored to non-coercive elliptic boundary value problems, specifically, parametric-in-frequency Helmholtz problems. The offline information is computed by means of adaptive finite elements, so that each snapshot lives in a different discrete space that resolves the local singularities of the analytical solution and is adjusted to the considered frequency value. A rational surrogate is then assembled adopting either a least-squares or an interpolatory approach, yielding a function-valued version of the the standard rational interpolation method ($ \mathcal{V} $-SRI) and the minimal rational interpolation method (MRI). In the context of building an approximation for linear or quadratic functionals of the Helmholtz solution, we perform several numerical experiments to compare the proposed methodologies. Our simulations show that, for interior resonant problems (whose singularities are encoded by poles on the real axis), the spatially adaptive $ \mathcal{V} $-SRI and MRI work comparably well. Instead, when dealing with exterior scattering problems, whose frequency response is mostly smooth, the $ \mathcal{V} $-SRI method seems to be the best-performing one.
- model order reduction,
- rational approximation,
- parametric Helmholtz equation,
- frequency response,
- adaptive mesh refinement
Citation: Francesca Bonizzoni, Davide Pradovera, Michele Ruggeri. Rational-approximation-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots[J]. Mathematics in Engineering, 2023, 5(4): 1-38. doi: 10.3934/mine.2023074

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Abstract

We introduce several spatially adaptive model order reduction approaches tailored to non-coercive elliptic boundary value problems, specifically, parametric-in-frequency Helmholtz problems. The offline information is computed by means of adaptive finite elements, so that each snapshot lives in a different discrete space that resolves the local singularities of the analytical solution and is adjusted to the considered frequency value. A rational surrogate is then assembled adopting either a least-squares or an interpolatory approach, yielding a function-valued version of the the standard rational interpolation method ($ \mathcal{V} $-SRI) and the minimal rational interpolation method (MRI). In the context of building an approximation for linear or quadratic functionals of the Helmholtz solution, we perform several numerical experiments to compare the proposed methodologies. Our simulations show that, for interior resonant problems (whose singularities are encoded by poles on the real axis), the spatially adaptive $ \mathcal{V} $-SRI and MRI work comparably well. Instead, when dealing with exterior scattering problems, whose frequency response is mostly smooth, the $ \mathcal{V} $-SRI method seems to be the best-performing one.

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