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., Early Access.
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. (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

EscapilInchauspe, P., & JerezHanckes, C. (2020). Helmholtz Scattering by Random Domains: FirstOrder Sparse Boundary Elements Approximation. SIAM J. Sci. Comput., 42(5), A2561–A2592.
Abstract: We consider the numerical solution of timeharmonic acoustic scattering by obstacles with uncertain geometries for Dirichlet, Neumann, impedance, and transmission boundary conditions. In particular, we aim to quantify diffracted fields originated by small stochastic perturbations of a given relatively smooth nominal shape. Using firstorder shape Taylor expansions, we derive tensor deterministic firstkind boundary integral equations for the statistical moments of the scattering problems considered. These are then approximated by sparse tensor Galerkin discretizations via the combination technique [M. Griebel, M. Schneider, and C. Zenger, A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, P. de Groen and P. Beauwens, eds., Elsevier, Amsterdam, 1992, pp. 263281; H. Harbrecht, M. Peters, and M. Siebenmorgen, J. Comput. Phys., 252 (2013), pp. 128141]. We supply extensive numerical experiments confirming the predicted error convergence rates with polylogarithmic growth in the number of degrees of freedom and accuracy in approximation of the moments. Moreover, we discuss implementation details such as preconditioning to finally point out further research avenues.

Fierro, I., & JerezHanckes, C. (2020). Fast Calderon preconditioning for Helmholtz boundary integral equations. J. Comput. Phys., 409, 22 pp.
Abstract: Calderon multiplicative preconditioners are an effective way to improve the condition number of first kind boundary integral equations yielding provable mesh independent bounds. However, when discretizing by local loworder basis functions as in standard Galerkin boundary element methods, their computational performance worsens as meshes are refined. This stems from the barycentric mesh refinement used to construct dual basis functions that guarantee the discrete stability of L2pairings. Based on coarser quadrature rules over dual cells and Hmatrix compression, we propose a family of fast preconditioners that significantly reduce assembly and computation times when compared to standard versions of Calderon preconditioning for the threedimensional Helmholtz weakly and hypersingular boundary integral operators. Several numerical experiments validate our claims and point towards further enhancements. (C) 2020 Elsevier Inc. All rights reserved.

Fuenzalida, C., JerezHanckes, C., & McClarren, R. G. (2019). Uncertainty Quantification For Multigroup Diffusion Equations Using Sparse Tensor Approximations. SIAM J. Sci. Comput., 41(3), B545–B575.
Abstract: We develop a novel method to compute first and second order statistical moments of the neutron kinetic density inside a nuclear system by solving the energydependent neutron diffusion equation. Randomness comes from the lack of precise knowledge of external sources as well as of the interaction parameters, known as cross sections. Thus, the density is itself a random variable. As Monte Carlo simulations entail intense computational work, we are interested in deterministic approaches to quantify uncertainties. By assuming as given the first and second statistical moments of the excitation terms, a sparse tensor finite element approximation of the first two statistical moments of the dependent variables for each energy group can be efficiently computed in one run. Numerical experiments provided validate our derived convergence rates and point to further research avenues.

Hiptmair, R., JerezHanckes, C., & UrzúaTorres, C. (2020). Optimal Operator Preconditioning For Galerkin Boundary Element Methods On 3D Screens. SIAM J. Numer. Anal., 58(1), 834–857.
Abstract: We consider firstkind weakly singular and hypersingular boundary integral operators for the Laplacian on screens in $\mathbb{R}^{3}$ and their Galerkin discretization by means of loworder piecewise polynomial boundary elements. For the resulting linear systems of equations we propose novel Calderóntype preconditioners based on (i) new boundary integral operators, which provide the exact inverses of the weakly singular and hypersingular operators on flat disks, and (ii) stable duality pairings relying on dual meshes. On screens obtained as images of the unit disk under biLipschitz transformations, this approach achieves condition numbers uniformly bounded in the meshwidth even on locally refined meshes. Comprehensive numerical tests also confirm its excellent preasymptotic performance.

JerezHanckes, C., & Pinto, J. (2020). Highorder Galerkin method for Helmholtz and Laplace problems on multiple open arcs. ESAIMMath. Model. Numer. Anal.Model. Math. Anal. Numer., 54(6), 1975–2009.
Abstract: We present a spectral Galerkin numerical scheme for solving Helmholtz and Laplace prob lems with Dirichlet boundary conditions on a finite collection of open arcs in twodimensional space. A boundary integral method is employed, giving rise to a first kind Fredholm equation whose variational form is discretized using weighted Chebyshev polynomials. Wellposedness of the discrete problems is established as well as algebraic or even exponential convergence rates depending on the regularities of both arcs and excitations. Our numerical experiments show the robustness of the method with respect to number of arcs and large wavenumber range. Moreover, we present a suitable compression algorithm that further accelerates computational times.

JerezHanckes, C., Martínez, I. A., Pettersson, I., & Volodymyr, R. (2021). Multiscale Analysis of Myelinated Axons. In SEMA SIMAI Springer Series (Vol. 10, pp. 17–35). Springer, Cham.
Abstract: We consider a threedimensional model for a myelinated neuron, which includes Hodgkin–Huxley ordinary differential equations to represent membrane dynamics at Ranvier nodes (unmyelinated areas). Assuming a periodic microstructure with alternating myelinated and unmyelinated parts, we use homogenization methods to derive a onedimensional nonlinear cable equation describing the potential propagation along the neuron. Since the resistivity of intracellular and extracellular domains is much smaller than the myelin resistivity, we assume this last one to be a perfect insulator and impose homogeneous Neumann boundary conditions on the myelin boundary. In contrast to the case when the conductivity of the myelin is nonzero, no additional terms appear in the onedimensional limit equation, and the model geometry affects the limit solution implicitly through an auxiliary cell problem used to compute the effective coefficient. We present numerical examples revealing the forecasted dependence of the effective coefficient on the size of the Ranvier node.

JerezHanckes, C., Pettersson, I., & Rybalko, V. (2020). Derivation Of Cable Equation By Multiscale Analysis For A Model Of Myelinated Axons. Discrete Contin. Dyn. Syst.Ser. B, 25(3), 815–839.
Abstract: We derive a onedimensional cable model for the electric potential propagation along an axon. Since the typical thickness of an axon is much smaller than its length, and the myelin sheath is distributed periodically along the neuron, we simplify the problem geometry to a thin cylinder with alternating myelinated and unmyelinated parts. Both the microstructure period and the cylinder thickness are assumed to be of order epsilon, a small positive parameter. Assuming a nonzero conductivity of the myelin sheath, we find a critical scaling with respect to epsilon which leads to the appearance of an additional potential in the homogenized nonlinear cable equation. This potential contains information about the geometry of the myelin sheath in the original threedimensional model.

SilvaOelker, G., JerezHanckes, C., & Fay, R. (2019). Hightemperature tungstenhafnia optimized selective thermal emitters for thermophotovoltaic applications. J. Quant. Spectrosc. Radiat. Transf., 231, 61–68.
Abstract: Tungstenhafnia (WHfO2) selective thermal emitters with high hemispherical emittance for thermophotovoltaic (TPV) applications are explored through numerical simulations. Two structures were analyzed: a planar multilayer stack and a grating. In both cases, through suitable design choices high thermal emittance with low directional sensitivity can be obtained. The designs are obtained by optimization of the structures using a genetic algorithm and a suitable cost function, along with simulations of the structures' emittance by using rigorous coupled wave analysis. Calculations show that these optimized structures possess high hemispherical thermal emittance for the wavelength range that matches the optical response of GaSb photovoltaic cells. For each structure, both the output power from the TPV cell and the conversion efficiency are studied as a function of emitter temperature and physical understanding of the optimized structures is developed. (C) 2019 Elsevier Ltd. All rights reserved.
