522 research outputs found
Localized model reduction for parameterized problems
In this contribution we present a survey of concepts in localized model order
reduction methods for parameterized partial differential equations. The key
concept of localized model order reduction is to construct local reduced spaces
that have only support on part of the domain and compute a global approximation
by a suitable coupling of the local spaces. In detail, we show how optimal
local approximation spaces can be constructed and approximated by random
sampling. An overview of possible conforming and non-conforming couplings of
the local spaces is provided and corresponding localized a posteriori error
estimates are derived. We introduce concepts of local basis enrichment, which
includes a discussion of adaptivity. Implementational aspects of localized
model reduction methods are addressed. Finally, we illustrate the presented
concepts for multiscale, linear elasticity and fluid-flow problems, providing
several numerical experiments.
This work has been accepted as a chapter in P. Benner, S. Grivet-Talocia, A.
Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Sileira. Handbook on Model Order
Reduction. Walter De Gruyter GmbH, Berlin, 2019+
Discontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation
The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time
Computational methods in cardiovascular mechanics
The introduction of computational models in cardiovascular sciences has been
progressively bringing new and unique tools for the investigation of the
physiopathology. Together with the dramatic improvement of imaging and
measuring devices on one side, and of computational architectures on the other
one, mathematical and numerical models have provided a new, clearly
noninvasive, approach for understanding not only basic mechanisms but also
patient-specific conditions, and for supporting the design and the development
of new therapeutic options. The terminology in silico is, nowadays, commonly
accepted for indicating this new source of knowledge added to traditional in
vitro and in vivo investigations. The advantages of in silico methodologies are
basically the low cost in terms of infrastructures and facilities, the reduced
invasiveness and, in general, the intrinsic predictive capabilities based on
the use of mathematical models. The disadvantages are generally identified in
the distance between the real cases and their virtual counterpart required by
the conceptual modeling that can be detrimental for the reliability of
numerical simulations.Comment: 54 pages, Book Chapte
Computing parametrized solutions for plasmonic nanogap structures
The interaction of electromagnetic waves with metallic nanostructures
generates resonant oscillations of the conduction-band electrons at the metal
surface. These resonances can lead to large enhancements of the incident field
and to the confinement of light to small regions, typically several orders of
magnitude smaller than the incident wavelength. The accurate prediction of
these resonances entails several challenges. Small geometric variations in the
plasmonic structure may lead to large variations in the electromagnetic field
responses. Furthermore, the material parameters that characterize the optical
behavior of metals at the nanoscale need to be determined experimentally and
are consequently subject to measurement errors. It then becomes essential that
any predictive tool for the simulation and design of plasmonic structures
accounts for fabrication tolerances and measurement uncertainties.
In this paper, we develop a reduced order modeling framework that is capable
of real-time accurate electromagnetic responses of plasmonic nanogap structures
for a wide range of geometry and material parameters. The main ingredients of
the proposed method are: (i) the hybridizable discontinuous Galerkin method to
numerically solve the equations governing electromagnetic wave propagation in
dielectric and metallic media, (ii) a reference domain formulation of the
time-harmonic Maxwell's equations to account for geometry variations; and (iii)
proper orthogonal decomposition and empirical interpolation techniques to
construct an efficient reduced model. To demonstrate effectiveness of the
models developed, we analyze geometry sensitivities and explore optimal designs
of a 3D periodic annular nanogap structure.Comment: 28 pages, 9 figures, 4 tables, 2 appendice
Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
In this paper, we discuss a general multiscale model reduction framework
based on multiscale finite element methods. We give a brief overview of related
multiscale methods. Due to page limitations, the overview focuses on a few
related methods and is not intended to be comprehensive. We present a general
adaptive multiscale model reduction framework, the Generalized Multiscale
Finite Element Method. Besides the method's basic outline, we discuss some
important ingredients needed for the method's success. We also discuss several
applications. The proposed method allows performing local model reduction in
the presence of high contrast and no scale separation
Energy preserving model order reduction of the nonlinear Schr\"odinger equation
An energy preserving reduced order model is developed for two dimensional
nonlinear Schr\"odinger equation (NLSE) with plane wave solutions and with an
external potential. The NLSE is discretized in space by the symmetric interior
penalty discontinuous Galerkin (SIPG) method. The resulting system of
Hamiltonian ordinary differential equations are integrated in time by the
energy preserving average vector field (AVF) method. The mass and energy
preserving reduced order model (ROM) is constructed by proper orthogonal
decomposition (POD) Galerkin projection. The nonlinearities are computed for
the ROM efficiently by discrete empirical interpolation method (DEIM) and
dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and
mass are shown for the full order model (FOM) and for the ROM which ensures the
long term stability of the solutions. Numerical simulations illustrate the
preservation of the energy and mass in the reduced order model for the two
dimensional NLSE with and without the external potential. The POD-DMD makes a
remarkable improvement in computational speed-up over the POD-DEIM. Both
methods approximate accurately the FOM, whereas POD-DEIM is more accurate than
the POD-DMD
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