Least-Squares Padé approximation of parametric and stochastic Helmholtz maps

Bonizzoni, Francesca; Nobile, Fabio; Perugia, Ilaria; Pradovera, Davide

doi:10.1007/s10444-020-09749-3

Cited by 15 publications

(16 citation statements)

References 34 publications

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“…Moreover, each pole has order 1, regardless of its multiplicity as Laplace eigenvalue. This result (except for the pole order) was extended to scattering problems (2.5) in [BNPP20b], where, in contrast, the solution map u(z) is a function of the wavenumber, i.e., z = k. In the following, when referring to the frequency response map u, we will have in mind the solution to either problem (2.1) or (2.5).…”

Section: Meromorphicity Of the Solution Mapmentioning

confidence: 93%

“…Concerning scattering problems like (2.5) (with first-order absorbing boundary condition), in [BNPP20b] it was proven that the frequency response map is meromorphic, and that all its poles have negative imaginary part. However, a characterization as precise as in formula (2.7) is not available: the poles might have order larger than 1, and the corresponding residues will not be X -orthogonal.…”

Section: Meromorphicity Of the Solution Mapmentioning

confidence: 99%

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

Bonizzoni¹,

Pradovera²,

Ruggeri³

2021

Preprint

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We introduce several spatially adaptive model order reduction approaches tailored to noncoercive 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 on a different discrete space that resolves the local singularities of the 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 the standard rational interpolation method (SRI), a vector-or function-valued version of it (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 V-SRI and MRI work comparably well. Instead, when dealing with exterior scattering problems, whose frequency response is mostly smooth, the V-SRI method seems to be the best-performing one.

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Section: Meromorphicity Of the Solution Mapmentioning

confidence: 93%

Section: Meromorphicity Of the Solution Mapmentioning

confidence: 99%

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

Bonizzoni¹,

Pradovera²,

Ruggeri³

2021

Preprint

Self Cite

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“…In particular, we choose to approximate problem (46) using P 1 Finite Elements (FE) on a sufficiently fine tetrahedral discretization of Ω. The FE discretization of (46) defines a parametric problem of the form (6), whose solution map u can be proven meromorphic using compactness arguments [7,37]. Thus, we decide to approximate it for ν ∈ K = [1,4] (black), for η ∈ {0, 500} Hz.…”

Section: Time-harmonic Vibrations Of a Tuning Forkmentioning

confidence: 99%

“…More recently, in the wake of these methods for dynamical systems, and somehow trying to profit from their main advantages, univariate LS Padé approximants have been introduced and studied, both in a standard [5,7] and a fast [6] version, in the context of a single parameter. Such techniques are based on multiple solves of the FOM at a single parameter value, in the same spirit as Krylov subspace methods, but yield an explicit rational approximant like VF.…”

Section: Introductionmentioning

confidence: 99%

Interpolatory rational model order reduction of parametric problems lacking uniform inf-sup stability

Pradovera

2019

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We present a technique for the approximation of a class of Hilbert space-valued maps which arise within the framework of Model Order Reduction for parametric partial differential equations, whose solution map has a meromorphic structure. Our MOR stategy consists in constructing an explicit rational approximation based on few snapshots of the solution, in an interpolatory fashion. Under some restrictions on the structure of the original problem, we describe a priori convergence results for our technique, hereafter called minimal rational interpolation, which show its ability to identify the main features (e.g. resonance locations) of the target solution map. We also investigate some procedures to obtain a posteriori error indicators, which may be employed to adapt the degree and the sampling points of the minimal rational interpolant. Finally, some numerical experiments are carried out to confirm the theoretical results and the effectiveness of our technique. * D. Pradovera has been funded by the Swiss National Science Foundation (SNSF) through project 182236.

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“…UQ for the Helmholtz equation and k-explicit parametric regularity. Whilst a large amount of initial UQ theory concerned Poisson's equation ∇ • (A(x, y)∇u(x, y)) = −f (x), there has been increasing interest in UQ of Helmholtz equation with (large) wavenumber k [89,84,8,38,28,24,30,59,49,3,75,25,69,43,46,7,39,87] and the time-harmonic Maxwell equations [51,52,29,1]. The Helmholtz equation with wavenumber k and random coefficients is k −2 ∇ • (A(x, y)∇u(x, y)) + n(x, y)u(x, y) = −f (x) (1.2) where A and n depend on both the spatial variable x and the stochastic variable y.…”

Section: Introduction 1motivation: Wavenumber-explicit Uncertainty Qu...mentioning

confidence: 99%

Wavenumber-explicit parametric holomorphy of Helmholtz solutions in the context of uncertainty quantification

Spence¹,

Wunsch²

2022

Preprint

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A crucial role in the theory of uncertainty quantification (UQ) of PDEs is played by the regularity of the solution with respect to the stochastic parameters; indeed, a key property one seeks to establish is that the solution is holomorphic with respect to (the complex extensions of) the parameters. In the context of UQ for the high-frequency Helmholtz equation, a natural question is therefore: how does this parametric holomorphy depend on the wavenumber k?The recent paper [39] showed for a particular nontrapping variable-coefficient Helmholtz problem with affine dependence of the coefficients on the stochastic parameters that the solution operator can be analytically continued a distance ∼ k −1 into the complex plane.In this paper, we generalise the result in [39] about k-explicit parametric holomorphy to a much wider class of Helmholtz problems with arbitrary (holomorphic) dependence on the stochastic parameters; we show that in all cases the region of parametric holomorphy decreases with k, and show how the rate of decrease with k is dictated by whether the unperturbed Helmholtz problem is trapping or nontrapping. We then give examples of both trapping and nontrapping problems where these bounds on the rate of decrease with k of the region of parametric holomorphy are sharp, with the trapping examples coming from the recent results of [34].An immediate implication of these results is that the k-dependent restrictions imposed on the randomness in the analysis of quasi-Monte Carlo (QMC) methods in [39] arise from a genuine feature of the Helmholtz equation with k large (and not, for example, a suboptimal bound).

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Least-Squares Padé approximation of parametric and stochastic Helmholtz maps

Abstract: The present work deals with rational model order reduction methods based on the single-point Least-Square (LS) Padé approximation techniques introduced in Bonizzoni et al.

Cited by 15 publications

References 34 publications

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

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

Interpolatory rational model order reduction of parametric problems lacking uniform inf-sup stability

Wavenumber-explicit parametric holomorphy of Helmholtz solutions in the context of uncertainty quantification

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