Continuity of approximation by least-squares multivariate Padé approximants

Huard, Alain; Robin, Vincent

doi:10.1016/s0377-0427(99)00177-6

Cited by 4 publications

(5 citation statements)

References 7 publications

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“…Taking the limit as h → 0 in (19) and using the continuity of S (see Proposition 3.1), we derive problem (17), which is well-posed provided that z / ∈ Λ. Hence, for any z ∈ C \ Λ, the complex derivative dS dz (z) exists and is the unique solution of (17). ✷ Proposition 3.2 states that the solution map S is holomorphic in C except in the set of isolated points Λ = {λ j }, i.e., S ∈ H (C \ Λ; V ), where H (U ; V ) is the space of holomorphic mappings from U ⊂ C with values in V .…”

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Convergence analysis of Padé approximations for Helmholtz frequency response problems

2018

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The present work concerns the approximation of the solution map S associated to the parametric Helmholtz boundary value problem, i.e., the map which associates to each (real) wavenumber belonging to a given interval of interest the corresponding solution of the Helmholtz equation. We introduce a least squares rational Padé-type approximation technique applicable to any meromorphic Hilbert space-valued univariate map, and we prove the uniform convergence of the Padé approximation error on any compact subset of the interval of interest that excludes any pole. This general result is then applied to the Helmholtz solution map S, which is proven to be meromorphic in ℂ, with a pole of order one in every (single or multiple) eigenvalue of the Laplace operator with the considered boundary conditions. Numerical tests are provided that confirm the theoretical upper bound on the Padé approximation error for the Helmholtz solution map.

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Convergence analysis of Padé approximations for Helmholtz frequency response problems

2018

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“…Further we would like to point out to the reader who is interested in numerical applications that two variables Padé approximants were used in [12] to compute the resonance frequencies of an elastic structure with respect to a damping coefficient.…”

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confidence: 99%

Multivariate Padé-Bergman approximants

Boichu

Robin

2002

Numerische Mathematik

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“…Model order reduction methods aim at significantly reducing the computational cost by approximating the quantity of interest starting from evaluations at only few wavenumbers. They rely on a two-step strategy: the offline stage consists in the computation of a finite dimensional basis-e.g., the basis of snapshots (see, e.g., [6,11,17,18,23,27,31,32,34,35]), or evaluations of the frequency response map and its derivatives at fixed centers (Padé method, see, e.g., [3,10,12,13,16]); the output of this phase, whose computational cost may be very high, is stored, to be used during the online phase, in which the approximation of the frequency response map corresponding to a given new value of the parameter is constructed. This stage does not involve the numerical solution of any large-scale PDE, and is expected to provide the output in real time.…”

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confidence: 99%