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

Euan A. Spence, Jared Wunsch

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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 \simk - 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).

Original languageEnglish
Pages (from-to)567-590
Number of pages24
JournalSIAM/ASA Journal on Uncertainty Quantification
Issue number2
Early online date18 May 2023
Publication statusPublished - 1 Jun 2023

Bibliographical note

The work of the first author was supported by EPSRC grant EP/1025995/1. The work of the second author was partially supported by Simons Foundation grant 631302, NSF grant DMS-2054424, and a Simons Fellowship.


  • Helmholtz
  • high-frequency
  • parametric holomorphy

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Modelling and Simulation


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