arXiv subjects

Rubén Aylwin

Publications and source records attributed to Rubén Aylwin.

6 recordsLinked to original sources

arXiv2026

A Multilevel Interacting Particle System Method for the estimation of Failure Probabilities

We propose a novel method that combines a multilevel decomposition with the framework of Interacting Particle Systems to compute the probability that given quantities of interest take values above or below a certain pre-specified value. Through the application of a sequential sampling scheme, the algorithm is able to achieve convergence rates faster than those achieved by the classical Multilevel Monte Carlo algorithm, and competes with the current state of the art techniques, while also being dimension independent. Furthermore, the sample points generated by the algorithm form a sequence which concentrates near the critical value of the quantity of interest. Our results are rigorously established and later verified through numerical experiments.

math.NA↗

arXiv2022

Multilevel Domain Uncertainty Quantification in Computational Electromagnetics

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.

math.NA↗

arXiv2022

Finite-Element Domain Approximation for Maxwell Variational Problems on Curved Domains

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.

math.NA↗

arXiv2020

The effect of quadrature rules on finite element solutions of Maxwell variational problems. Consistency estimates on meshes with straight and curved elements

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 {\em 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 to numerical quadrature rules.

math.NA↗

arXiv2020

Fast Solver for Quasi-Periodic 2D-Helmholtz Scattering in Layered Media

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 Rayleigh-Wood anomalies, 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.

math.NA↗

arXiv2020

Optimization methods for achieving high diffraction efficiency with perfect electric conducting gratings

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.

math.OC↗