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 $\varepsilon$ and the online parameter $\mu$ 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 $m$, 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