Helmholtz Scattering by Random Domains: First-Order Sparse Boundary Element Approximation

Escapil-Inchauspé, Paul; Jerez-Hanckes, Carlos

doi:10.1137/19m1279277

Cited by 7 publications

(8 citation statements)

References 50 publications

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“…Next, according to (19), λ = (γ 0 U) = γ U ∈ H ṡ(Γ). For β = 0, since U| Γ = 0 hence ∇ Γ U = 0, one can write σ = γ 1 U ∈ H s (Γ) by (20). Next, for…”

Section: Characterization Step (S1)mentioning

confidence: 99%

“…Besides, it paves the way towards a framework allowing to deal with general BVPs and their boundary reduction. Moreover, the formulas for the DDs of Cauchy data allow for numerous applications in first-order shape boundary methods [20]. Recent works [28,29] characterized the shape derivative for a variety of scattering problems appearing in acoustics and electromagnetism.…”

Section: Thenmentioning

confidence: 99%

“…Besides the obvious applications of this analysis in optimization, one sees it in inverse problems (cf. [9,Chapter 4 and 5] and [26,27,39,40]), first-order second statistical moment approximation for uncertainty quantification [18,20], and even shape holomorphy analysis [31]. Appealing to bi-Lipschitz diffeomorphisms between nominal and perturbed domains, one carries out shape calculus-see [1,25,42] for an exhaustive overview of these techniques.…”

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confidence: 99%

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Shape differentiability of Helmholtz scattering problems via explicit shape calculus

Escapil-Inchauspé,

Jerez-Hanckes

2020

Preprint

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We consider the time-harmonic scalar wave scattering problems with Dirichlet, Neumann, impedance and transmission boundary conditions. Under this setting, we analyze how sensitive diffracted fields and Cauchy data are to small perturbations of a given nominal shape. To this end, we follow [K. Eppler, Int. J. Appl. Math. Comput. Sci. 10(3) (2000), pp. 487-516], referred to as explicit shape calculus, and which places great emphasis on the characterization of the domain derivatives as boundary value problems. It consists in combining elliptic regularity theorems, shape calculus and functional analysis, allowing to deduce sharp differentiability results for the domain-tosolution and domain-to-Cauchy data mappings. The technique is applied to both classic and limit Sobolev regularity cases, leading to new and comprehensive differentiability results.

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“…Next, according to (19), λ = (γ 0 U) = γ U ∈ H ṡ(Γ). For β = 0, since U| Γ = 0 hence ∇ Γ U = 0, one can write σ = γ 1 U ∈ H s (Γ) by (20). Next, for…”

Section: Characterization Step (S1)mentioning

confidence: 99%

Section: Thenmentioning

confidence: 99%

mentioning

confidence: 99%

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Shape differentiability of Helmholtz scattering problems via explicit shape calculus

Escapil-Inchauspé,

Jerez-Hanckes

2020

Preprint

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“…UQ for the Helmholtz equation and k-explicit parametric regularity. While a large amount of initial UQ theory concerned Poisson's equation \nabla \cdot (A(\bfx , \bfy )\nabla u(\bfx , \bfy )) = - f (\bfx ), there has been increasing interest in UQ of the Helmholtz equation with (large) wavenumber k (see, e.g., [26,44,23,64,39,6,35]) and the time-harmonic Maxwell equations [46,47,27,1]. The Helmholtz equation with wavenumber k and random coefficients is…”

Section: Introductionmentioning

confidence: 99%

Wavenumber-Explicit Parametric Holomorphy of Helmholtz Solutions in the Context of Uncertainty Quantification

Spence

Wunsch

2023

SIAM/ASA J. Uncertainty Quantification

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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 [35] 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 \sim k - 1 into the complex plane. In this paper, we generalize the result in [35] 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 [31]. An immediate implication of these results is that the k-dependent restrictions imposed on the randomness in the analysis of quasi-Monte Carlo methods in [35] arise from a genuine feature of the Helmholtz equation with k large (and not, for example, a suboptimal bound).

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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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Helmholtz Scattering by Random Domains: First-Order Sparse Boundary Element Approximation

Cited by 7 publications

References 50 publications

Shape differentiability of Helmholtz scattering problems via explicit shape calculus

Shape differentiability of Helmholtz scattering problems via explicit shape calculus

Wavenumber-Explicit Parametric Holomorphy of Helmholtz Solutions in the Context of Uncertainty Quantification

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

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