We consider a family of boundary integral operators supported on a collection of parametrically defined bounded Lipschitz boundaries. Consequently, the boundary integral operators themselves also depend on the parametric variables, thus leading to a parameter-to-operator map. The main result 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 \emph{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 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：本文研究定义于一族参数化有界Lipschitz边界上的边界积分算子族。因此，边界积分算子本身也依赖于参数变量，从而形成了参数到算子的映射。本文的主要结果是证明所述边界积分算子（即参数到算子的映射）关于参数变量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，我们构造了针对复值边界形变的全纯延拓，并研究了边界积分算子关于每个参数变量的复Fréchet可微性。已建立的参数全纯性结果被证明是克服参数映射逼近中所谓维数灾难的关键性质，这类映射具有分布式高维输入。为展示所得结果的适用性，我们以声软Helmholtz声学散射问题及其频率鲁棒边界积分公式作为具体示例。针对该特定应用，我们探讨了研究结果在降阶建模、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中的意义。