
Aylwin, R., Henriquez, F., & Schwab, C. (2023). ReLU Neural Network Galerkin BEM. J. Sci. Comput., 95(2), 41.
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 formulationbased 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 socalled Rectified Linear Units (ReLU) as the underlying activation function. The ReLUNNs 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 ReLUNNs boils down to a fine tuning of these parameters, in network training. We argue that ReLUNNs of fixed depth and with a variable width allow us to recover wellknown approximation rate results for the standard Galerkin Boundary Element Method (BEM). This observation hinges on existing wellknown 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 ReLUNNs over different classes of functions. We prove that shallow ReLUNNs, 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 ReLUNNs for the numerical approximation of BIEs. We find that weighted residual estimators, which are reliable without further assumptions on the quasiuniformity of the underlying mesh, can be employed for the construction of computationally efficient loss functions for ReLUNN training. The proposed framework allows us to leverage on stateoftheart computational deep learning technologies such as TENSORFLOW and TPUs for the numerical solution of BIEs using ReLUNNs. Exploratory numerical experiments validate our theoretical findings and indicate the viability of the proposed ReLUNN Galerkin BEM approach.



Aylwin, R., & JerezHanckes, C. (2023). FiniteElement Domain Approximation for Maxwell Variational Problems on Curved Domains. SIAM J. Numer. Anal., 61(3), 1139–1171.
Abstract: We consider the problem of domain approximation in finite element methods for Maxwell equations on curved domains, i.e., when affine or polynomial meshes fail to exactly cover the domain of interest. In such cases, one is forced to approximate the domain by a sequence of polyhedral domains arising from inexact meshes. We deduce conditions on the quality of these approximations that ensure rates of error convergence between discrete solutions  in the approximate domains  to the continuous one in the original domain.



Aylwin, R., & JerezHanckes, C. (2021). The effect of quadrature rules on finite element solutions of Maxwell variational problems Consistency estimates on meshes with straight and curved elements. Numer. Math., 147, 903–936.
Abstract: We study the effects of numerical quadrature rules on error convergence rates when solving Maxwelltype variational problems via the curlconforming or edge finite element method. A complete a priori error analysis for the case of bounded polygonal and curved domains with nonhomogeneous 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.



Aylwin, R., JerezHanckes, C., & Pinto, J. (2020). On the Properties of Quasiperiodic Boundary Integral Operators for the Helmholtz Equation. Integr. Equ. Oper. Theory, 92(2), 41 pp.
Abstract: We study the mapping properties of boundary integral operators arising when solving twodimensional, timeharmonic waves scattered by periodic domains. For domains assumed to be at least Lipschitz regular, we propose a novel explicit representation of Sobolev spaces for quasiperiodic functions that allows for an analysis analogous to that of Helmholtz scattering by bounded objects. Except for RayleighWood frequencies, continuity and coercivity results are derived to prove wellposedness of the associated first kind boundary integral equations.



Aylwin, R., JerezHanckes, C., Schwab, C., & Zech, J. (2023). Multilevel Domain Uncertainty Quantification in Computational Electromagnetics. Math. Models Methods Appl. Sci., 33(04), 877–921.
Abstract: We continue our study [Domain Uncertainty Quantification in Computational Electromagnetics, JUQ (2020), 8:301341] of the numerical approximation of timeharmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affineparametric shape parametrization framework, mapping the physical domains to a nominal polygonal domain with piecewise smooth maps. The regularity of the pullback solutions on the nominal domain is characterized in piecewise Sobolev spaces. We prove error convergence rates and optimize the algorithmic steering of parameters for edgeelement discretizations in the nominal domain combined with: (a) multilevel Monte Carlo sampling, and (b) multilevel, sparsegrid quadrature for computing the expectation of the solutions with respect to uncertain domain ensembles. In addition, we analyze sparsegrid interpolation to compute surrogates of the domaintosolution mappings. All calculations are performed on the polyhedral nominal domain, which enables the use of standard simplicial finite element meshes. We provide a rigorous fully discrete error analysis and show, in all cases, that dimensionindependent algebraic convergence is achieved. For the multilevel sparsegrid quadrature methods, we prove higher order convergence rates which are free from the socalled curse of dimensionality, i.e. independent of the number of parameters used to parametrize the admissible shapes. Numerical experiments confirm our theoretical results and verify the superiority of the sparsegrid methods.



Aylwin, R., JerezHanckes, C., Schwab, C., & Zech, J. (2020). Domain Uncertainty Quantification in Computational Electromagnetics. SIAMASA J. Uncertain. Quantif., 8(1), 301–341.
Abstract: We study the numerical approximation of timeharmonic, electromagnetic fields inside a lossy cavity of uncertain geometry. Key assumptions are a possibly highdimensional parametrization of the uncertain geometry along with a suitable transformation to a fixed, nominal domain. This uncertainty parametrization results in families of countably parametric, Maxwelllike 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 highdimensional parameter domain and an Hcurl conforming edge element discretization of the parametric problem in the nominal domain. As a steppingstone in the analysis, we derive a novel Strangtype lemma for Maxwelllike 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 edgeelement consistency error analysis for the nominal problem are shown to imply convergence rates for multilevel Monte Carlo and for quasiMonte 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 shapetosolution maps in this setting. Finally, our computational experiments confirm the presented theoretical results.



Aylwin, R., SilvaOelker, G., JerezHanckes, C., & Fay, P. (2020). Optimization methods for achieving high diffraction efficiency with perfect electric conducting gratings. J. Opt. Soc. Am. AOpt. Image Sci. Vis., 37(8), 1316–1326.
Abstract: This work presents the implementation, numerical examples, and experimental convergence study of first and secondorder optimization methods applied to onedimensional 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 firstorder method (gradient descent); a secondorder 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 nonconvexity; and the BroydenFletcherGoldfarbShanno (BFGS) algorithm, a quasiNewton 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



Pinto, J., Aylwin, R., & JerezHanckes, C. (2021). Fast solver for quasiperiodic 2DHelmholtz scattering in layered media. ESAIMMath. Model. Numer. Anal., 55(5), 2445–2472.
Abstract: We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from twodimensional Helmholtz transmission problems in multilayered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding cutoff frequencies (also known as RayleighWood frequencies), we rigorously establish the wellposedness of both continuous and discrete problems, and prove superalgebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attained via Nyström methods.



Pinto, J., Aylwin, R., SilvaOelker, G., & JerezHanckes, C. (2021). Diffraction efficiency optimization for multilayered parametric holographic gratings. Opt. Lett., 46(16), 3929–3932.
Abstract: Multilayered diffraction gratings are an essential component in many optical devices due to their ability to engineer light. We propose a firstorder 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.

