2020
DOI: 10.1007/s10444-020-09749-3
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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.

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Cited by 15 publications
(16 citation statements)
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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%
See 1 more Smart Citation
“…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%
“…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%
“…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%