
Ayala, A., Claeys, X., EscapilInchauspé, P., & JerezHanckes, C. (2022). Local Multiple Traces Formulation for electromagnetics: Stability and preconditioning for smooth geometries. J. Comput. Appl. Math., 413, 114356.
Abstract: We consider the timeharmonic electromagnetic transmission problem for the unit sphere. Appealing to a vector spherical harmonics analysis, we prove the first stability result of the local multiple traces formulation (MTF) for electromagnetics, originally introduced by Hiptmair and JerezHanckes (2012) for the acoustic case, paving the way towards an extension to general piecewise homogeneous scatterers. Moreover, we investigate preconditioning techniques for the local MTF scheme and study the accumulation points of induced operators. In particular, we propose a novel secondorder inverse approximation of the operator. Numerical experiments validate our claims and confirm the relevance of the preconditioning strategies.



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. (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., & 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., 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., 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



Dölz, J., Harbrecht, H., JerezHanckes, C., & Multerer M. (2022). Isogeometric multilevel quadrature for forward and inverse random acoustic scattering. Comput. Methods in Appl. Mech. Eng., 388, 114242.
Abstract: We study the numerical solution of forward and inverse timeharmonic acoustic scattering problems by randomly shaped obstacles in threedimensional space using a fast isogeometric boundary element method. Within the isogeometric framework, realizations of the random scatterer can efficiently be computed by simply updating the NURBS mappings which represent the scatterer. This way, we end up with a random deformation field. In particular, we show that it suffices to know the deformation field’s expectation and covariance at the scatterer’s boundary to model the surface’s Karhunen–Loève expansion. Leveraging on the isogeometric framework, we employ multilevel quadrature methods to approximate quantities of interest such as the scattered wave’s expectation and variance. By computing the wave’s Cauchy data at an artificial, fixed interface enclosing the random obstacle, we can also directly infer quantities of interest in free space. Adopting the Bayesian paradigm, we finally compute the expected shape and variance of the scatterer from noisy measurements of the scattered wave at the artificial interface. Numerical results for the forward and inverse problems validate the proposed approach.



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.



EscapilInchauspe, P., & JerezHanckes, C. (2021). Biparametric operator preconditioning. Comput. Math. Appl., 102, 220–232.
Abstract: We extend the operator preconditioning framework Hiptmair (2006) [10] to PetrovGalerkin methods while accounting for parameterdependent perturbations of both variational forms and their preconditioners, as occurs when performing numerical approximations. By considering different perturbation parameters for the original form and its preconditioner, our biparametric abstract setting leads to robust and controlled schemes. For Hilbert spaces, we derive exhaustive linear and superlinear convergence estimates for iterative solvers, such as hindependent convergence bounds, when preconditioning with lowaccuracy or, equivalently, with highly compressed approximations.



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., & Labarca, I. (2023). Timedomain multiple traces boundary integral formulation for acoustic wave scattering in 2D. Eng. Anal. Bound. Elem., 157, 216–228.
Abstract: We present a novel computational scheme to solve acoustic wave transmission problems over twodimensional composite scatterers, i.e. penetrable obstacles possessing junctions or triple points. The continuous problem is cast as a local multiple traces timedomain boundary integral formulation. For discretization in time and space, we resort to convolution quadrature schemes coupled to a nonconforming spatial spectral discretization based on second kind Chebyshev polynomials displaying fast convergence. Computational experiments confirm convergence of multistep and multistage convolution quadrature for a variety of complex domains.



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., & Pinto, J. (2020). Spectral Galerkin Method for Solving Helmholtz and Laplace Dirichlet Problems on Multiple Open Arcs. In Lecture Notes in Computational Science and Engineering (Vol. 134, pp. 383–393).
Abstract: We present a spectral numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on an unbounded nonLipschitz domain R2∖Γ¯¯¯ , where Γ is a finite collection of open arcs. Through an indirect method, a first kind formulation is derived whose variational form is discretized using weighted Chebyshev polynomials. This choice of basis allows for exponential convergence rates under smoothness assumptions. Moreover, by implementing a simple compression algorithm, we are able to efficiently account for large numbers of arcs as well as a wide wavenumber range.



JerezHanckes, C., & Pinto, J. (2022). Spectral Galerkin Method for Solving Helmholtz Boundary Integral Equations on Smooth Screens. IMA J. Numer. Anal., 42(4), 3571–3608.
Abstract: We solve firstkind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three dimensions with either Dirichlet or Neumann conditions. The proposed GalerkinBubnov methods take as discretization elements pushedforward weighted azimuthal projections of standard spherical harmonics onto the unit disk. By exactly depicting edge singular behaviors we show that these spectral or highorder bases yield superalgebraic error convergence in the corresponding energy norms whenever the screen is an analytic deformation of the unit disk. Moreover, we provide a fully discrete analysis of the method, including quadrature rules, based on analytic extensions of the spectral basis to complex neighborhoods. Finally, we include numerical experiments to support our claims as well as appendices with computational details for treating the associated singular integrals.



JerezHanckes, C., Martínez, I. A., Pettersson, I., & Rybalko, V. (2023). Derivation of a bidomain model for bundles of myelinated axons. Nonlinear Anal.Real World Appl., 70, 103789.
Abstract: The work concerns the multiscale modeling of a nerve fascicle of myelinated axons. We present a rigorous derivation of a macroscopic bidomain model describing the behavior of the electric potential in the fascicle based on the FitzHughNagumo membrane dynamics. The approach is based on the twoscale convergence machinery combined with the method of monotone operators.



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.

