Articlegreen2023

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

7 citations0/0 files storedSIAM/ASA Journal on Uncertainty Quantification

No readable copy stored yet — this record is metadata only.

No open-access location is known for this record yet.

Literature note Cite in a project Citation map

Probabilistic and Robust Engineering Design Spectral Theory in Mathematical Physics Numerical methods in inverse problems

Abstract

Abstract. 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 [Formula: see text]? 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 [Formula: see text] into the complex plane. In this paper, we generalize the result in [ 35 ] about [Formula: see text]-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 [Formula: see text] and show how the rate of decrease with [Formula: see text] 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 [Formula: see text] 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 [Formula: see text]-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 [Formula: see text] large (and not, for example, a suboptimal bound).

Paper brief

Loading…

Cite this work

Citation unavailable for this work.

BibTeX RIS CSL-JSON

No public copy is stored for this record, so only the metadata is cited.

Citations in this library

Citation map →

Stored only when both papers are in the corpus — 7 citations exist upstream.

References

2010Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs 2016Analytic regularity and collocation approximation for elliptic PDEs with random domain deformations 2016Analysis of the domain mapping method for elliptic diffusion problems on random domains 2013Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs 2018Large deformation shape uncertainty quantification in acoustic scattering 2020Domain Uncertainty Quantification in Computational Electromagnetics

Explore the map →

Papers that cite the same references — the relational fallback while embeddings are still backfilling.

Imported from OpenAlex (CC0) — nothing here is generated.