Parametric Shape Holomorphy of Boundary Integral Operators with Applications
Jürgen Dölz, Fernando Henríquez
TL;DR
The paper addresses the analytic dependence of boundary integral operators and their solutions on shape deformations of Lipschitz boundaries via a parametric holomorphy framework. It develops a complex-extension approach to affine shape perturbations and proves $(oldsymbol{b},p,oldsymbol{ u})$-holomorphy for parameter-to-operator and parameter-to-solution maps in 3D, for both bounded and singular kernels. The results are specialized to sound-soft Helmholtz scattering, establishing holomorphy for single-, double-, adjoint-double layer, and combined-field operators, and for the far-field pattern, with implications for best-$n$-term approximations, reduced-order modeling, Bayesian shape inversion, and ANN surrogates. This provides dimension-robust convergence rates for high-dimensional parameter spaces and offers a rigorous foundation for efficient surrogates and uncertainty quantification in acoustic scattering and related BIO applications. The work paves the way for extending parametric holomorphy to broader boundary classes and for integrating these insights into multilevel and neural-network-based surrogate frameworks.
Abstract
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} Fréchet 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.