arXiv subjects

Felix Schindler

Publications and source records attributed to Felix Schindler.

23 records · Page 2Linked to original sources

arXiv2016

pyMOR - Generic Algorithms and Interfaces for Model Order Reduction

Reduced basis methods are projection-based model order reduction techniques for reducing the computational complexity of solving parametrized partial differential equation problems. In this work we discuss the design of pyMOR, a freely available software library of model order reduction algorithms, in particular reduced basis methods, implemented with the Python programming language. As its main design feature, all reduction algorithms in pyMOR are implemented generically via operations on well-defined vector array, operator and discretization interface classes. This allows for an easy integration with existing open-source high-performance partial differential equation solvers without adding any model reduction specific code to these solvers. Besides an in-depth discussion of pyMOR's design philosophy and architecture, we present several benchmark results and numerical examples showing the feasibility of our approach.

cs.MS↗

arXiv2016

Model Reduction for Multiscale Lithium-Ion Battery Simulation

In this contribution we are concerned with efficient model reduction for multiscale problems arising in lithium-ion battery modeling with spatially resolved porous electrodes. We present new results on the application of the reduced basis method to the resulting instationary 3D battery model that involves strong non-linearities due to Buttler-Volmer kinetics. Empirical operator interpolation is used to efficiently deal with this issue. Furthermore, we present the localized reduced basis multiscale method for parabolic problems applied to a thermal model of batteries with resolved porous electrodes. Numerical experiments are given that demonstrate the reduction capabilities of the presented approaches for these real world applications.

math.NA↗

arXiv2016

Extending DUNE: The dune-xt modules

We present our effort to extend and complement the core modules of the Distributed and Unified Numerics Environment DUNE (http://dune-project.org) by a well tested and structured collection of utilities and concepts. We describe key elements of our four modules dune-xt-common, dune-xt-grid, dune-xt-la and dune-xt-functions, which aim at further enabling the programming of generic algorithms within DUNE as well as adding an extra layer of usability and convenience.

cs.MS↗

arXiv2015

Error control for the localized reduced basis multi-scale method with adaptive on-line enrichment

In this contribution we consider localized, robust and efficient a-posteriori error estimation of the localized reduced basis multi-scale (LRBMS) method for parametric elliptic problems with possibly heterogeneous diffusion coefficient. The numerical treatment of such parametric multi-scale problems are characterized by a high computational complexity, arising from the multi-scale character of the underlying differential equation and the additional parameter dependence. The LRBMS method can be seen as a combination of numerical multi-scale methods and model reduction using reduced basis (RB) methods to efficiently reduce the computational complexity with respect to the multi-scale as well as the parametric aspect of the problem, simultaneously. In contrast to the classical residual based error estimators currently used in RB methods, we are considering error estimators that are based on conservative flux reconstruction and provide an efficient and rigorous bound on the full error with respect to the weak solution. In addition, the resulting error estimator is localized and can thus be used in the on-line phase to adaptively enrich the solution space locally where needed. The resulting certified LRBMS method with adaptive on-line enrichment thus guarantees the quality of the reduced solution during the on-line phase, given any (possibly insufficient) reduced basis that was generated during the offline phase. Numerical experiments are given to demonstrate the applicability of the resulting algorithm with online enrichment to single phase flow in heterogeneous media.

math.NA↗

arXiv2014

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 $μ$ 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.

math.NA↗