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

TL;DR

This paper investigates how the holomorphic dependence of Helmholtz solutions on stochastic parameters deteriorates as the wavenumber increases, revealing the influence of trapping properties and providing sharp bounds for high-frequency regimes.

Contribution

It generalizes previous results on parametric holomorphy for Helmholtz problems to a broader class with arbitrary stochastic dependence, explicitly characterizing the effect of wavenumber and trapping.

Findings

Holomorphic region decreases with increasing wavenumber $k$.

Trapping and nontrapping problems exhibit different rates of holomorphy decay.

Bounds on holomorphy are sharp, confirming the theoretical predictions.

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 [Ganesh, Kuo, Sloan 2021] 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 generalise the result in [Ganesh, Kuo, Sloan 2021] 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 [Galkowski, Marchand, Spence 2021]. 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 [Ganesh, Kuo, Sloan 2021] arise from a genuine feature of the Helmholtz equation with $k$ large (and not, for example, a suboptimal bound).