论文标题
在不确定性定量的背景下
Wavenumber-explicit parametric holomorphy of Helmholtz solutions in the context of uncertainty quantification
论文作者
论文摘要
PDES的不确定性定量理论（UQ）的关键作用是由解决随机参数的规律性扮演的。实际上，人们寻求确定的关键属性是该解决方案是相对于（复杂的）参数的全体形态。在高频Helmholtz方程的UQ的背景下，一个自然的问题是：此参数Holomorphy如何取决于WaveNumber $ k $？ 最近的论文[Ganesh，Kuo，Sloan 2021]显示了特定的非捕获可变的Helmholtz问题，其系数对随机参数的仿射依赖性，可以在分析上可以在分析上继续进行距离$ \ sim k^{-1} $，以延长解决方案操作员。 在本文中，我们将结果概括为[Ganesh，Kuo，Sloan 2021]，大约$ k $ - 宣传参数holomorphy to helmholtz对随机参数的任意（Holomorphic）依赖性的较宽类别的问题；我们表明，在所有情况下，参数性霍明氏菌的区域都会随$ k $而降低，并显示出$ k $的降低率是如何取决于未受扰动的Helmholtz问题是陷阱还是不陷入困境。然后，我们给出了诱捕问题和非捕获问题的例子，在这些限制下，参数性霍明氏菌区域的$ k $降低率很清晰，捕获示例来自[Galkowski，Marchand，Spence，Spence 2021]的最新结果。 这些结果的直接含义是，在[Ganesh，Kuo，Sloan 2021]中分析Quasi-Monte Carlo（QMC）方法时，对$ k $依赖的限制施加的限制是由helmholtz方程的真正特征引起的。
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).
