146 research outputs found
A-posteriori error estimates for the localized reduced basis multi-scale method
We present a localized a-posteriori error estimate for the localized reduced
basis multi-scale (LRBMS) method [Albrecht, Haasdonk, Kaulmann, Ohlberger
(2012): The localized reduced basis multiscale method]. The LRBMS is a
combination of numerical multi-scale methods and model reduction using reduced
basis methods to efficiently reduce the computational complexity of parametric
multi-scale problems with respect to the multi-scale parameter
and the online parameter simultaneously. We formulate the LRBMS based on
a generalization of the SWIPDG discretization presented in [Ern, Stephansen,
Vohralik (2010): Guaranteed and robust discontinuous Galerkin a posteriori
error estimates for convection-diffusion-reaction problems] on a coarse
partition of the domain that allows for any suitable discretization on the fine
triangulation inside each coarse grid element. The estimator is based on the
idea of a conforming reconstruction of the discrete diffusive flux, that can be
computed using local information only. It is offline/online decomposable and
can thus be efficiently used in the context of model reduction
Model Reduction for Complex Hyperbolic Networks
We recently introduced the joint gramian for combined state and parameter
reduction [C. Himpe and M. Ohlberger. Cross-Gramian Based Combined State and
Parameter Reduction for Large-Scale Control Systems. arXiv:1302.0634, 2013],
which is applied in this work to reduce a parametrized linear time-varying
control system modeling a hyperbolic network. The reduction encompasses the
dimension of nodes and parameters of the underlying control system. Networks
with a hyperbolic structure have many applications as models for large-scale
systems. A prominent example is the brain, for which a network structure of the
various regions is often assumed to model propagation of information. Networks
with many nodes, and parametrized, uncertain or even unknown connectivity
require many and individually computationally costly simulations. The presented
model order reduction enables vast simulations of surrogate networks exhibiting
almost the same dynamics with a small error compared to full order model.Comment: preprin
Localized Orthogonal Decomposition for two-scale Helmholtz-type problems
In this paper, we present a Localized Orthogonal Decomposition (LOD) in
Petrov-Galerkin formulation for a two-scale Helmholtz-type problem. The
two-scale problem is, for instance, motivated from the homogenization of the
Helmholtz equation with high contrast, studied together with a corresponding
multiscale method in (Ohlberger, Verf\"urth. A new Heterogeneous Multiscale
Method for the Helmholtz equation with high contrast, arXiv:1605.03400, 2016).
There, an unavoidable resolution condition on the mesh sizes in terms of the
wave number has been observed, which is known as "pollution effect" in the
finite element literature. Following ideas of (Gallistl, Peterseim. Comput.
Methods Appl. Mech. Engrg. 295:1-17, 2015), we use standard finite element
functions for the trial space, whereas the test functions are enriched by
solutions of subscale problems (solved on a finer grid) on local patches.
Provided that the oversampling parameter , which indicates the size of the
patches, is coupled logarithmically to the wave number, we obtain a
quasi-optimal method under a reasonable resolution of a few degrees of freedom
per wave length, thus overcoming the pollution effect. In the two-scale
setting, the main challenges for the LOD lie in the coupling of the function
spaces and in the periodic boundary conditions.Comment: 20 page
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