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Author (up) Ayala, A.; Claeys, X.; Escapil-Inchauspé, P.; Jerez-Hanckes, C.
Title Local Multiple Traces Formulation for electromagnetics: Stability and preconditioning for smooth geometries Type
Year 2022 Publication Journal of Computational and Applied Mathematics Abbreviated Journal J. Comput. Appl. Math.
Volume 413 Issue Pages 114356
Keywords Maxwell scattering; Multiple Traces Formulation; Vector spherical harmonics; Preconditioning; Boundary element method
Abstract We consider the time-harmonic 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 Jerez-Hanckes (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 second-order inverse approximation of the operator. Numerical experiments validate our claims and confirm the relevance of the preconditioning strategies.
Address
Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0377-0427 ISBN Medium
Area Expedition Conference
Notes Approved
Call Number UAI @ alexi.delcanto @ Serial 1554
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Author (up) Aylwin, R.; Jerez-Hanckes, C.
Title Finite-Element Domain Approximation for Maxwell Variational Problems on Curved Domains Type
Year 2023 Publication SIAM Journal on Numerical Analysis Abbreviated Journal SIAM J. Numer. Anal.
Volume 61 Issue 3 Pages 1139-1171
Keywords Ne'; de'; lec finite elements; curl-conforming elements; Maxwell equations; proximation; Strang lemma
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.
Address
Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0036-1429 ISBN Medium
Area Expedition Conference
Notes WOS:001044149800001 Approved
Call Number UAI @ alexi.delcanto @ Serial 1700
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Author (up) Aylwin, R.; Jerez-Hanckes, C.
Title The effect of quadrature rules on finite element solutions of Maxwell variational problems Consistency estimates on meshes with straight and curved elements Type
Year 2021 Publication Numerische Mathematik Abbreviated Journal Numer. Math.
Volume 147 Issue Pages 903-936
Keywords 35Q61; 65N30; 65N12
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.
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Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0029-599X ISBN Medium
Area Expedition Conference
Notes WOS:000622655300001 Approved
Call Number UAI @ alexi.delcanto @ Serial 1335
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Author (up) Aylwin, R.; Jerez-Hanckes, C.; Pinto, J.
Title On the Properties of Quasi-periodic Boundary Integral Operators for the Helmholtz Equation Type
Year 2020 Publication Integral Equations And Operator Theory Abbreviated Journal Integr. Equ. Oper. Theory
Volume 92 Issue 2 Pages 41 pp
Keywords Wave scattering; Gratings; Quasi-periodic functions; Boundary integral equations
Abstract We study the mapping properties of boundary integral operators arising when solving two-dimensional, time-harmonic waves scattered by periodic domains. For domains assumed to be at least Lipschitz regular, we propose a novel explicit representation of Sobolev spaces for quasi-periodic functions that allows for an analysis analogous to that of Helmholtz scattering by bounded objects. Except for Rayleigh-Wood frequencies, continuity and coercivity results are derived to prove wellposedness of the associated first kind boundary integral equations.
Address [Aylwin, Ruben; Pinto, Jose] Pontificia Univ Catolica Chile, Dept Elect Engn, Santiago, Chile, Email: rdaylwin@uc.cl;
Corporate Author Thesis
Publisher Springer Basel Ag Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0378-620x ISBN Medium
Area Expedition Conference
Notes WOS:000522040900001 Approved
Call Number UAI @ eduardo.moreno @ Serial 1127
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Author (up) Aylwin, R.; Jerez-Hanckes, C.; Schwab, C.; Zech, J.
Title Multilevel Domain Uncertainty Quantification in Computational Electromagnetics Type
Year 2023 Publication Mathematical Models and Methods in Applied Sciences Abbreviated Journal Math. Models Methods Appl. Sci.
Volume 33 Issue 04 Pages 877-921
Keywords
Abstract We continue our study [Domain Uncertainty Quantification in Computational Electromagnetics, JUQ (2020), 8:301--341] of the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affine-parametric 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 edge-element discretizations in the nominal domain combined with: (a) multilevel Monte Carlo sampling, and (b) multilevel, sparse-grid quadrature for computing the expectation of the solutions with respect to uncertain domain ensembles. In addition, we analyze sparse-grid interpolation to compute surrogates of the domain-to-solution 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 dimension-independent algebraic convergence is achieved. For the multilevel sparse-grid quadrature methods, we prove higher order convergence rates which are free from the so-called 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 sparse-grid methods.
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Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0218-2025 ISBN Medium
Area Expedition Conference
Notes WOS:000961040400001 Approved
Call Number UAI @ alexi.delcanto @ Serial 1699
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Author (up) Aylwin, R.; Jerez-Hanckes, C.; Schwab, C.; Zech, J.
Title Domain Uncertainty Quantification in Computational Electromagnetics Type
Year 2020 Publication Siam-Asa Journal On Uncertainty Quantification Abbreviated Journal SIAM-ASA J. Uncertain. Quantif.
Volume 8 Issue 1 Pages 301-341
Keywords computational electromagnetics; uncertainty quantification; finite elements; shape holomorphy; sparse grid quadrature; Bayesian inverse problems
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.
Address [Aylwin, Ruben] Pontificia Univ Catolica Chile, Sch Engn, Santiago 7820436, Chile, Email: rdaylwin@uc.cl;
Corporate Author Thesis
Publisher Siam Publications Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 2166-2525 ISBN Medium
Area Expedition Conference
Notes WOS:000551383300011 Approved
Call Number UAI @ eduardo.moreno @ Serial 1207
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Author (up) Aylwin, R.; Silva-Oelker, G.; Jerez-Hanckes, C.; Fay, P.
Title Optimization methods for achieving high diffraction efficiency with perfect electric conducting gratings Type
Year 2020 Publication Journal Of The Optical Society Of America A-Optics Image Science And Vision Abbreviated Journal J. Opt. Soc. Am. A-Opt. Image Sci. Vis.
Volume 37 Issue 8 Pages 1316-1326
Keywords
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
Address [Aylwin, Ruben] Pontificia Univ Catolica Chile, Dept Elect Engn, Santiago, Chile, Email: rdaylwin@uc.cl
Corporate Author Thesis
Publisher Optical Soc Amer Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 1084-7529 ISBN Medium
Area Expedition Conference
Notes WOS:000555713900009 Approved
Call Number UAI @ eduardo.moreno @ Serial 1216
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Author (up) Dölz, J.; Harbrecht, H.; Jerez-Hanckes, C.; Multerer M.
Title Isogeometric multilevel quadrature for forward and inverse random acoustic scattering Type
Year 2022 Publication Computer Methods in Applied Mechanics and Engineering Abbreviated Journal Comput. Methods in Appl. Mech. Eng.
Volume 388 Issue Pages 114242
Keywords Uncertainty quantification: Helmholtz scattering; Isogeometric Analysis; Boundary Integral Methods; Bayesian inversion; Multilevel quadrature
Abstract We study the numerical solution of forward and inverse time-harmonic acoustic scattering problems by randomly shaped obstacles in three-dimensional 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.
Address
Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0045-7825 ISBN Medium
Area Expedition Conference
Notes Approved
Call Number UAI @ alexi.delcanto @ Serial 1476
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Author (up) Escapil-Inchauspe, P.; Jerez-Hanckes, C.
Title Bi-parametric operator preconditioning Type
Year 2021 Publication Computers & Mathematics With Applications Abbreviated Journal Comput. Math. Appl.
Volume 102 Issue Pages 220-232
Keywords Operator preconditioning; Galerkin methods; Numerical approximation; Iterative linear solvers
Abstract 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.
Address
Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0898-1221 ISBN Medium
Area Expedition Conference
Notes Approved
Call Number UAI @ alexi.delcanto @ Serial 1471
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Author (up) Escapil-Inchauspe, P.; Jerez-Hanckes, C.
Title Helmholtz Scattering by Random Domains: First-Order Sparse Boundary Elements Approximation Type
Year 2020 Publication SIAM Journal of Scientific Computing Abbreviated Journal SIAM J. Sci. Comput.
Volume 42 Issue 5 Pages A2561-A2592
Keywords Helmholtz equation; shape calculus; uncertainty quantification; boundary element method; combination technique
Abstract 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.
Address
Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 1064-8275 ISBN Medium
Area Expedition Conference
Notes Approved
Call Number UAI @ eduardo.moreno @ Serial 1205
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Author (up) Fierro, I.; Jerez-Hanckes, C.
Title Fast Calderon preconditioning for Helmholtz boundary integral equations Type
Year 2020 Publication Journal Of Computational Physics Abbreviated Journal J. Comput. Phys.
Volume 409 Issue Pages 22 pp
Keywords Operator preconditioning; Calderon preconditioning; Helmholtz equations; Hierarchical matrices; Fast solvers; Boundary elements method
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 low-order 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 L-2-pairings. Based on coarser quadrature rules over dual cells and H-matrix 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 three-dimensional Helmholtz weakly and hyper-singular boundary integral operators. Several numerical experiments validate our claims and point towards further enhancements. (C) 2020 Elsevier Inc. All rights reserved.
Address [Fierro, Ignacia] UCL, Dept Math, Gower St, London, England, Email: carlos.jerez@uai.cl
Corporate Author Thesis
Publisher Academic Press Inc Elsevier Science Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0021-9991 ISBN Medium
Area Expedition Conference
Notes WOS:000522726000020 Approved
Call Number UAI @ eduardo.moreno @ Serial 1153
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Author (up) Fuenzalida, C.; Jerez-Hanckes, C.; McClarren, R.G.
Title Uncertainty Quantification For Multigroup Diffusion Equations Using Sparse Tensor Approximations Type
Year 2019 Publication Siam Journal On Scientific Computing Abbreviated Journal SIAM J. Sci. Comput.
Volume 41 Issue 3 Pages B545-B575
Keywords multigroup diffusion equation; uncertainty quantification; sparse tensor approximation; finite element method
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 energy-dependent 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.
Address [Fuenzalida, Consuelo] Pontificia Univ Catolica Chile, Sch Engn, Santiago, Chile, Email: mcfuenzalida@uc.cl;
Corporate Author Thesis
Publisher Siam Publications Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 1064-8275 ISBN Medium
Area Expedition Conference
Notes WOS:000473033300033 Approved
Call Number UAI @ eduardo.moreno @ Serial 1023
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Author (up) Hiptmair, R.; Jerez-Hanckes, C.; Urzúa-Torres, C.
Title Optimal Operator Preconditioning For Galerkin Boundary Element Methods On 3D Screens Type
Year 2020 Publication SIAM Journal on Numerical Analysis Abbreviated Journal SIAM J. Numer. Anal.
Volume 58 Issue 1 Pages 834-857
Keywords
Abstract We consider first-kind weakly singular and hypersingular boundary integral operators for the Laplacian on screens in $\mathbb{R}^{3}$ and their Galerkin discretization by means of low-order piecewise polynomial boundary elements. For the resulting linear systems of equations we propose novel Calderón-type 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 bi-Lipschitz 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.
Address
Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0036-1429 ISBN Medium
Area Expedition Conference
Notes Approved
Call Number UAI @ eduardo.moreno @ Serial 1011
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Author (up) Jerez-Hanckes, C.; Labarca, I.
Title Time-domain multiple traces boundary integral formulation for acoustic wave scattering in 2D Type
Year 2023 Publication Engineering Analysis with Boundary Elements Abbreviated Journal Eng. Anal. Bound. Elem.
Volume 157 Issue Pages 216-228
Keywords Acoustic wave scattering; Wave transmission problems; Convolution quadrature; Time-domain boundary integral operators; Multiple traces formulation; Domain decomposition
Abstract We present a novel computational scheme to solve acoustic wave transmission problems over two-dimensional composite scatterers, i.e. penetrable obstacles possessing junctions or triple points. The continuous problem is cast as a local multiple traces time-domain boundary integral formulation. For discretization in time and space, we resort to convolution quadrature schemes coupled to a non-conforming 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.
Address
Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0955-7997 ISBN Medium
Area Expedition Conference
Notes Approved
Call Number UAI @ alexi.delcanto @ Serial 1875
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Author (up) Jerez-Hanckes, C.; Martínez, I.A.; Pettersson, I.; Rybalko, V.
Title Derivation of a bidomain model for bundles of myelinated axons Type
Year 2023 Publication Nonlinear Analysis-Real World Applications Abbreviated Journal Nonlinear Anal.-Real World Appl.
Volume 70 Issue Pages 103789
Keywords
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 FitzHugh-Nagumo membrane dynamics. The approach is based on the two-scale convergence machinery combined with the method of monotone operators.
Address
Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 1468-1218 ISBN Medium
Area Expedition Conference
Notes WOS:000891375200009 Approved
Call Number UAI @ alexi.delcanto @ Serial 1653
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Author (up) Jerez-Hanckes, C.; Martínez, IA.; Pettersson, I.; Volodymyr, R.
Title Multiscale Analysis of Myelinated Axons Type
Year 2021 Publication SEMA SIMAI Springer Series Abbreviated Journal SEMA SIMAI Springer Ser.
Volume 10 Issue Pages 17-35
Keywords
Abstract We consider a three-dimensional 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 one-dimensional 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 one-dimensional 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.
Address
Corporate Author Thesis
Publisher Springer, Cham Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 2199-305X ISBN 978-3-030-62029-5 Medium
Area Expedition Conference
Notes Approved
Call Number UAI @ alexi.delcanto @ Serial 1321
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Author (up) Jerez-Hanckes, C.; Pettersson, I.; Rybalko, V.
Title Derivation Of Cable Equation By Multiscale Analysis For A Model Of Myelinated Axons Type
Year 2020 Publication Discrete And Continuous Dynamical Systems-Series B Abbreviated Journal Discrete Contin. Dyn. Syst.-Ser. B
Volume 25 Issue 3 Pages 815-839
Keywords Hodgkin-Huxley model; nonlinear cable equation; cellular electrophysiology; multiscale modeling; homogenization
Abstract We derive a one-dimensional 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 three-dimensional model.
Address [Jerez-Hanckes, Carlos] Univ Adolfo Ibanez, Fac Engn & Sci, Diagonal Torres 2700, Santiago, Chile, Email: carlos.jerez@uai.cl;
Corporate Author Thesis
Publisher Amer Inst Mathematical Sciences-Aims Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 1531-3492 ISBN Medium
Area Expedition Conference
Notes WOS:000501609800001 Approved
Call Number UAI @ eduardo.moreno @ Serial 1069
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Author (up) Jerez-Hanckes, C.; Pinto, J.
Title Spectral Galerkin Method for Solving Helmholtz Boundary Integral Equations on Smooth Screens Type
Year 2022 Publication IMA Journal of Numerical Analysis Abbreviated Journal IMA J. Numer. Anal.
Volume 42 Issue 4 Pages 3571-3608
Keywords boundary integral equations; spectral methods; wave scattering problems; screens problems; non-Lipschitz domains
Abstract We solve first-kind 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 Galerkin-Bubnov methods take as discretization elements pushed-forward weighted azimuthal projections of standard spherical harmonics onto the unit disk. By exactly depicting edge singular behaviors we show that these spectral or high-order bases yield super-algebraic 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.
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Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0272-4979 ISBN Medium
Area Expedition Conference
Notes WOS:000790079800001 Approved
Call Number UAI @ alexi.delcanto @ Serial 1446
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Author (up) Jerez-Hanckes, C.; Pinto, J.
Title Spectral Galerkin Method for Solving Helmholtz and Laplace Dirichlet Problems on Multiple Open Arcs Type
Year 2020 Publication Lecture Notes in Computational Science and Engineering Abbreviated Journal Lect. Notes Comput. Sci. Eng.
Volume 134 Issue Pages 383-393
Keywords
Abstract We present a spectral numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on an unbounded non-Lipschitz 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.
Address
Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 2197-7100 ISBN Medium
Area Expedition Conference
Notes Approved
Call Number UAI @ alexi.delcanto @ Serial 1449
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Author (up) Jerez-Hanckes, C.; Pinto, J.
Title High-order Galerkin method for Helmholtz and Laplace problems on multiple open arcs Type
Year 2020 Publication Mathematical Modelling and Numerical Analysis Abbreviated Journal ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer.
Volume 54 Issue 6 Pages 1975-2009
Keywords
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 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.
Address
Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0764-583X ISBN Medium
Area Expedition Conference
Notes Approved
Call Number UAI @ eduardo.moreno @ Serial 1126
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