We consider a family of boundary integral operators supported on a collection of parametrically defined bounded Lipschitz boundaries. Thus, the boundary integral operators themselves also depend on the parametric variables, leading to a parameter-to-operator map. One of the main results of this article is to establish the analytic or holomorphic dependence of said boundary integral operators upon the parametric variables, i.e., of the parameter-to-operator map. As a direct consequence we also establish holomorphic dependence of solutions to boundary integral equations, i.e., holomorphy of the parameter-to-solution map. To this end, we construct a holomorphic extension to complex-valued boundary deformations and investigate the complex Fr\'echet differentiability of boundary integral operators with respect to each parametric variable. The established parametric holomorphy results have been identified as a key property to derive best $N$-term approximation rates to overcome the so-called curse of dimensionality in the approximation of parametric maps with distributed, high-dimensional inputs. To demonstrate the applicability of the derived results, we consider as a concrete example the sound-soft Helmholtz acoustic scattering problem and its frequency-robust boundary integral formulations. For this particular application, we explore the consequences of our results in reduced order modelling, Bayesian shape inversion, and the construction of efficient surrogates using artificial neural networks.

翻译：本文考虑一族定义在参数化有界Lipschitz边界集合上的边界积分算子。由于边界积分算子本身也依赖于参数变量，从而产生参数到算子的映射。本文的主要结果之一是建立所述边界积分算子关于参数变量（即参数到算子映射）的解析或全纯依赖性。作为直接推论，我们还建立了边界积分方程解的全纯依赖性，即参数到解映射的全纯性。为此，我们构造了复值边界形变的全纯延拓，并研究了边界积分算子关于各参数变量的复Fréchet可微性。所建立的参数全纯性结果已被证实是导出最优$N$项逼近率的关键性质，可用于克服分布高维输入的参数化映射逼近中的所谓的维数灾难。为了展示所得结果的应用性，我们以声软Helmholtz声散射问题及其频率鲁棒边界积分公式为具体实例，探讨了该结果在降阶建模、贝叶斯形状反演以及利用人工神经网络构建高效代理模型等方面的应用。