/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

The term photonic crystals appears because of the analogy between electron waves in crystals and the light waves in artificial periodic dielectric structures. During the recent years the investigation of one-, two-and three-dimensional periodic structures has attracted a widespread attention of the world optics community because of great potentiality of such structures in advanced applied optical fields. The interest in periodic structures has been stimulated by the fast development of semiconductor technology that now allows the fabrication of artificial structures, whose period is comparable with the wavelength of light in the visible and infrared ranges.

http://ieeexplore.ieee.org/iel5/6354/16969/00781867.pdf

Photonic crystals

A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

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Nonlinear Model Reduction via Discrete Empirical Interpolation

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

/pdf/a-survey-of-projection-based-model-reduction-methods-for-3yjqbxk305.pdf

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.

/pdf/reduced-basis-approximation-and-a-posteriori-error-1csv7ta4m6.pdf

Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations

We present an overview of recent developments of HDG methods for numerically solving partial differential equations in fluid mechanics.

Hybridizable Discontinuous Galerkin Methods

We experimentally show that terahertz (THz) waves confined in sub-10 nm metallic gaps can detect refractive index changes caused by only a 1 nm thick (∼λ/106) dielectric overlayer. We use atomic layer lithography to fabricate a wafer-scale array of annular nanogaps. Using THz time-domain spectroscopy in conjunction with atomic layer deposition, we measure spectral shifts of a THz resonance peak with increasing Al2O3 film thickness in 1 nm intervals. Because of the enormous mismatch in length scales between THz waves and sub-10 nm gaps, conventional modeling techniques cannot readily be used to analyze our results. We employ an advanced finite-element-modeling (FEM) technique, Hybridizable Discontinuous Galerkin (HDG) scheme, for full three-dimensional modeling of the resonant transmission of THz waves through an annular gap that is 2 nm in width and 32 μm in diameter. Our multiscale 3D FEM technique and atomic layer lithography will enable a series of new investigations in THz nanophotonics that has not b...

Nanogap-Enhanced Terahertz Sensing of 1 nm Thick (λ/106) Dielectric Films

The hybridized Discontinuous Galerkin method for Implicit Large-Eddy Simulation of transitional turbulent flows

We propose a hybridizable discontinuous Galerkin (HDG) method to numerically solve the Oseen equations which can be seen as the linearized version of the incompressible Navier-Stokes equations. We use same polynomial degree to approximate the velocity, its gradient and the pressure. With a special projection and postprocessing, we obtain optimal convergence for the velocity gradient and pressure and superconvergence for the velocity. Numerical results supporting our theoretical results are provided.

Analysis of HDG Methods for Oseen Equations

http://www.optimization-online.org/DB_FILE/2009/07/2341.pdf

Ngoc Cuong Nguyen

Papers

Hybridizable Discontinuous Galerkin Methods

Nanogap-Enhanced Terahertz Sensing of 1 nm Thick (λ/106) Dielectric Films

The hybridized Discontinuous Galerkin method for Implicit Large-Eddy Simulation of transitional turbulent flows

Analysis of HDG Methods for Oseen Equations

Bandgap optimization of two-dimensional photonic crystals using semidefinite programming and subspace methods