## 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 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 language | English |
---|---|

Pages (from-to) | 567-590 |

Number of pages | 24 |

Journal | SIAM/ASA Journal on Uncertainty Quantification |

Volume | 11 |

Issue number | 2 |

Early online date | 18 May 2023 |

DOIs | |

Publication status | Published - 1 Jun 2023 |

### Bibliographical note

Funding: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.

## Keywords

- 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