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

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