Shape Holomorphy of the Calderón Projector for the Laplacian in $${\mathbb {R}}^2$$

Higher-order Quasi-Monte Carlo Training of Deep Neural Networks

“…[8]), boundary integral equations on parametric boundaries (e.g. [21]), and parametric, dynamical systems described by large systems of initial-value ODEs (e.g. [39] and, for a proof of parametric holomorphy of solution manifolds, [20]), and [22] for DtO maps for Bayesian Inverse Problems for PDEs.…”

“…, L − 1}, we define z = W ( ) z + b ( ) . Then, since W ( ) , b ( ) are real valued, (21) gives max i=1,...,d…”

“…Moreover, in [6] the authors introduced two sufficient criteria that ensure holomorphy in a variety of cases, including UQ for non-linear PDEs and shape holomorphy (e.g. [6,8,21,2] and the references here).…”

On efficient algorithms for computing near-best polynomial approximations to high-dimensional, Hilbert-valued functions from limited samples

“…A key characteristic of parametric model problems is that the target function f is often smooth. Indeed, there is a large body of literature [17,25,29,37,[42][43][44]49,61,[73][74][75]77,79,80,83,87,134,139] that has established that solution maps of a wide range of different parametric DEs are holomorphic (i.e., analytic) functions of their parameters.…”

Shape analyticity and singular perturbations for layer potential operators

“…For instance, Ammari et al consider a one-parametric regular perturbation and compute in [3] a series expansion of the single layer potential as a function of the perturbation parameter. Another example is given by the recent work on the "shape holomorphy" by Henríquez and Schwab [30], where the authors consider the layer potential operators supported on a 𝐶 2 Jordan curve in R 2 . In Henríquez and Schwab's paper a suitable parametrization of the Jordan curve plays the role of the (regular) perturbation parameter, which they think as an element in a complex Banach space, and, among other results, they show that the Calderón projector of the two-dimensional Laplacian is an holomorphic map of such parametrization.…”