Abstract: We study the effects of numerical quadrature rules on error convergence rates when solving Maxwell-type variational problems via the curl-conforming or edge finite element method. A complete a priori error analysis for the case of bounded polygonal and curved domains with non-homogeneous coefficients is provided. We detail sufficient conditions with respect to mesh refinement and precision for the quadrature rules so as to guarantee convergence rates following that of exact numerical integration. On curved domains, we isolate the error contribution of numerical quadrature rules.

Abstract: We introduce Neural Network (NN for short) approximation architectures for the numerical solution of Boundary Integral Equations (BIEs for short). We exemplify the proposed NN approach for the boundary reduction of the potential problem in two spatial dimensions. We adopt a Galerkin formulation-based method, in polygonal domains with a finite number of straight sides. Trial spaces used in the Galerkin discretization of the BIEs are built by using NNs that, in turn, employ the so-called Rectified Linear Units (ReLU) as the underlying activation function. The ReLU-NNs used to approximate the solutions to the BIEs depend nonlinearly on the parameters characterizing the NNs themselves. Consequently, the computation of a numerical solution to a BIE by means of ReLU-NNs boils down to a fine tuning of these parameters, in network training. We argue that ReLU-NNs of fixed depth and with a variable width allow us to recover well-known approximation rate results for the standard Galerkin Boundary Element Method (BEM). This observation hinges on existing well-known properties concerning the regularity of the solution of the BIEs on Lipschitz, polygonal boundaries, i.e. accounting for the effect of corner singularities, and the expressive power of ReLU-NNs over different classes of functions. We prove that shallow ReLU-NNs, i.e. networks having a fixed, moderate depth but with increasing width, can achieve optimal order algebraic convergence rates. We propose novel loss functions for NN training which are obtained using computable, local residual a posteriori error estimators with ReLU-NNs for the numerical approximation of BIEs. We find that weighted residual estimators, which are reliable without further assumptions on the quasi-uniformity of the underlying mesh, can be employed for the construction of computationally efficient loss functions for ReLU-NN training. The proposed framework allows us to leverage on state-of-the-art computational deep learning technologies such as TENSORFLOW and TPUs for the numerical solution of BIEs using ReLU-NNs. Exploratory numerical experiments validate our theoretical findings and indicate the viability of the proposed ReLU-NN Galerkin BEM approach.

Abstract: We study the numerical approximation of time-harmonic, electromagnetic fields inside a lossy cavity of uncertain geometry. Key assumptions are a possibly high-dimensional parametrization of the uncertain geometry along with a suitable transformation to a fixed, nominal domain. This uncertainty parametrization results in families of countably parametric, Maxwell-like cavity problems that are posed in a single domain, with inhomogeneous coefficients that possess finite, possibly low spatial regularity, but exhibit holomorphic parametric dependence in the differential operator. Our computational scheme is composed of a sparse grid interpolation in the high-dimensional parameter domain and an Hcurl -conforming edge element discretization of the parametric problem in the nominal domain. As a stepping-stone in the analysis, we derive a novel Strang-type lemma for Maxwell-like problems in the nominal domain, which is of independent interest. Moreover, we accommodate arbitrary small Sobolev regularity of the electric field and also cover uncertain isotropic constitutive or material laws. The shape holomorphy and edge-element consistency error analysis for the nominal problem are shown to imply convergence rates for multilevel Monte Carlo and for quasi-Monte Carlo integration, as well as sparse grid approximations, in uncertainty quantification for computational electromagnetics. They also imply expression rate estimates for deep ReLU networks of shape-to-solution maps in this setting. Finally, our computational experiments confirm the presented theoretical results.

Abstract: This work presents the implementation, numerical examples, and experimental convergence study of first- and second-order optimization methods applied to one-dimensional periodic gratings. Through boundary integral equations and shape derivatives, the profile of a grating is optimized such that it maximizes the diffraction efficiency for given diffraction modes for transverse electric polarization. We provide a thorough comparison of three different optimization methods: a first-order method (gradient descent); a second-order approach based on a Newton iteration, where the usual Newton step is replaced by taking the absolute value of the eigenvalues given by the spectral decomposition of the Hessian matrix to deal with non-convexity; and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, a quasi-Newton method. Numerical examples are provided to validate our claims. Moreover, two grating profiles are designed for high efficiency in the Littrow configuration and then compared to a high efficiency commercial grating. Conclusions and recommendations, derived from the numerical experiments, are provided as well as future research avenues. (C) 2020 Optical Society of America

Abstract: Multilayered diffraction gratings are an essential component in many optical devices due to their ability to engineer light. We propose a first-order optimization strategy to maximize diffraction efficiencies of such structures by a fast approximation of the underlying boundary integral equations for polarized electromagnetic fields. A parametric representation of the structure interfaces via trigonometric functions enables the problem to be set as a parametric optimization one while efficiently representing complex structures. Derivatives of the efficiencies with respect to geometrical parameters are computed using shape calculus, allowing a straightforward implementation of gradient descent methods. Examples of the proposed strategy in chirped pulse amplification show its efficacy in designing multilayered gratings to maximize their diffraction efficiency.