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Author (up) Aylwin, R.; Henriquez, F.; Schwab, C. doi  openurl
  Title ReLU Neural Network Galerkin BEM Type
  Year 2023 Publication Journal of Scientific Computing Abbreviated Journal J. Sci. Comput.  
  Volume 95 Issue 2 Pages 41  
  Keywords BOUNDARY INTEGRAL METHOD; ELEMENT METHODS; P-VERSION; CONVERGENCE; FRAMEWORK; EQUATIONS; OPERATORS; SCREEN  
  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 formulation-based 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 so-called Rectified Linear Units (ReLU) as the underlying activation function. The ReLU-NNs 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 ReLU-NNs boils down to a fine tuning of these parameters, in network training. We argue that ReLU-NNs of fixed depth and with a variable width allow us to recover well-known approximation rate results for the standard Galerkin Boundary Element Method (BEM). This observation hinges on existing well-known 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 ReLU-NNs over different classes of functions. We prove that shallow ReLU-NNs, 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 ReLU-NNs for the numerical approximation of BIEs. We find that weighted residual estimators, which are reliable without further assumptions on the quasi-uniformity of the underlying mesh, can be employed for the construction of computationally efficient loss functions for ReLU-NN training. The proposed framework allows us to leverage on state-of-the-art computational deep learning technologies such as TENSORFLOW and TPUs for the numerical solution of BIEs using ReLU-NNs. Exploratory numerical experiments validate our theoretical findings and indicate the viability of the proposed ReLU-NN Galerkin BEM approach.  
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  Series Volume Series Issue Edition  
  ISSN 0885-7474 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000954568900001 Approved  
  Call Number UAI @ alexi.delcanto @ Serial 1769  
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Author (up) Aylwin, R.; Jerez-Hanckes, C. doi  openurl
  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.  
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  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. doi  openurl
  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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  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. doi  openurl
  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. doi  openurl
  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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  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. doi  openurl
  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. doi  openurl
  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) Pinto, J.; Aylwin, R.; Jerez-Hanckes, C. doi  openurl
  Title Fast solver for quasi-periodic 2D-Helmholtz scattering in layered media Type
  Year 2021 Publication Mathematical Modelling and Numerical Analysis Abbreviated Journal ESAIM-Math. Model. Numer. Anal.  
  Volume 55 Issue 5 Pages 2445 - 2472  
  Keywords Boundary integral equations; quasi-periodic scattering; spectral elements; gratings; multi-layered domain  
  Abstract We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding cut-off frequencies (also known as Rayleigh-Wood frequencies), we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic 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.  
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  Series Volume Series Issue Edition  
  ISSN 0764-583X ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000710646200001 Approved  
  Call Number UAI @ alexi.delcanto @ Serial 1445  
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Author (up) Pinto, J.; Aylwin, R.; Silva-Oelker, G.; Jerez-Hanckes, C. doi  openurl
  Title Diffraction efficiency optimization for multilayered parametric holographic gratings Type
  Year 2021 Publication Optics Letters Abbreviated Journal Opt. Lett.  
  Volume 46 Issue 16 Pages 3929-3932  
  Keywords MULTICORE FIBER; REFRACTIVE-INDEX; TEMPERATURE; SENSOR; STRAIN  
  Abstract Multilayered diffraction gratings are an essential component in many optical devices due to their ability to engineer light. We propose a first-order 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.  
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  Series Volume Series Issue Edition  
  ISSN 0146-9592 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000685553000028 Approved  
  Call Number UAI @ alexi.delcanto @ Serial 1444  
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