Browsing by Author "Jerez-Hanckes, Carlos"

Now showing 1 - 9 of 9
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    Bi-parametric operator preconditioning
    (2021) Escapil-Inchauspe, Paul; Jerez-Hanckes, Carlos
    We extend the operator preconditioning framework Hiptmair (2006) [10] to Petrov-Galerkin methods while accounting for parameter-dependent 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 bi-parametric abstract setting leads to robust and controlled schemes. For Hilbert spaces, we derive exhaustive linear and super-linear convergence estimates for iterative solvers, such as h-independent convergence bounds, when preconditioning with low-accuracy or, equivalently, with highly compressed approximations.
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    Derivation of a bidomain model for bundles of myelinated axons
    (2023) Jerez-Hanckes, Carlos; avila, Isabel A. Martinez; Pettersson, Irina; Rybalko, Volodymyr
    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 FitzHugh- Nagumo membrane dynamics. The approach is based on the two-scale convergence machinery combined with the method of monotone operators. (c) 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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    HELMHOLTZ SCATTERING BY RANDOM DOMAINS: FIRST-ORDER SPARSE BOUNDARY ELEMENT APPROXIMATION
    (2020) Escapil-Inchauspe, Paul; Jerez-Hanckes, Carlos
    We consider the numerical solution of time-harmonic 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 first-order shape Taylor expansions, we derive tensor deterministic first-kind 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. 263-281; H. Harbrecht, M. Peters, and M. Siebenmorgen, J. Comput. Phys., 252 (2013), pp. 128-141]. 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.
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    High-order Galerkin method for Helmholtz and Laplace problems on multiple open arcs
    (2020) Jerez-Hanckes, Carlos; Pinto, Jose
    We present a spectral Galerkin numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on a finite collection of open arcs in two-dimensional space. A boundary integral method is employed, giving rise to a first kind Fredholm equation whose variational form is discretized using weighted Chebyshev polynomials. Well-posedness 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.
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    MESH-INDEPENDENT OPERATOR PRECONDITIONING FOR BOUNDARY ELEMENTS ON OPEN CURVES
    (2014) Hiptmair, Ralf; Jerez-Hanckes, Carlos; Urzua-Torres, Carolina
    Boundary value problems for the Poisson equation in the exterior of an open bounded Lipschitz curve C can be recast as first-kind boundary integral equations featuring weakly singular or hypersingular boundary integral operators (BIOs). Based on the recent discovery in [C. Jerez-Hanckes and J. Nedelec, SIAM J. Math. Anal., 44 (2012), pp. 2666-2694] of inverses of these BIOs for C = [-1, 1], we pursue operator preconditioning of the linear systems of equations arising from Galerkin-Petrov discretization by means of zeroth- and first-order boundary elements. The preconditioners rely on boundary element spaces defined on dual meshes and they can be shown to perform uniformly well independently of the number of degrees of freedom even for families of locally refined meshes.
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    Optimization methods for achieving high diffraction efficiency with perfect electric conducting gratings
    (2020) Aylwin, Ruben; Silva-Oelker, Gerardo; Jerez-Hanckes, Carlos; Fay, Patrick
    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
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    Sparse tensor edge elements
    (2013) Hiptmair, Ralf; Jerez-Hanckes, Carlos; Schwab, Christoph
    We consider the tensorized operator for the Maxwell cavity source problem in frequency domain. Such formulations occur when computing statistical moments of the fields under a stochastic volume excitation. We establish a discrete inf-sup condition for its Ritz-Galerkin discretization on sparse tensor product edge element spaces built on nested sequences of meshes. Our main tool is a generalization of the edge element Fortin projector to a tensor product setting. The techniques extend to the surface boundary edge element discretization of tensorized electric field integral equation operators.
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    The effect of quadrature rules on finite element solutions of Maxwell variational problems Consistency estimates on meshes with straight and curved elements
    (2021) Aylwin, Ruben; Jerez-Hanckes, Carlos
    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.
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    THE OUTGOING TIME-HARMONIC ELECTROMAGNETIC WAVE IN A HALF-SPACE WITH NON-ABSORBING IMPEDANCE BOUNDARY CONDITION
    (2019) Rojas, Sergio; Muga, Ignacio; Jerez-Hanckes, Carlos
    We show existence and uniqueness of the outgoing solution for the Maxwell problem with an impedance boundary condition of Leontovitch type in a half-space. Due to the presence of surface waves guided by an infinite surface, the established radiation condition differs from the classical one when approaching the boundary of the half-space. This specific radiation pattern is derived from an accurate asymptotic analysis of the Green's dyad associated to this problem.