arXiv subjects

Stephan Rave

Publications and source records attributed to Stephan Rave.

29 records · Page 2Linked to original sources

arXiv2018

Hierarchical Approximate Proper Orthogonal Decomposition

Proper Orthogonal Decomposition (POD) is a widely used technique for the construction of low-dimensional approximation spaces from high-dimensional input data. For large-scale applications and an increasing amount of input data vectors, however, computing the POD often becomes prohibitively expensive. This work presents a generic, easy-to-implement approach to compute an approximate POD based on arbitrary tree hierarchies of worker nodes, where each worker computes a POD of only a small amount of input vectors. The tree hierarchy can be freely adapted to optimally suit the available computational resources. In particular, this hierarchical approximate POD (HAPOD) allows for both simple parallelization with low communication overhead, as well as live sequential POD computation under restricted memory capacities. Rigorous error estimates ensure the reliability of our approach, and extensive numerical examples underline its performance.

math.NA↗

arXiv2016

True Error Control for the Localized Reduced Basis Method for Parabolic Problems

We present an abstract framework for a posteriori error estimation for approximations of scalar parabolic evolution equations, based on elliptic reconstruction techniques [10, 9, 3, 5]. In addition to its original application (to derive error estimates on the discretization error), we extend the scope of this framework to derive offline/online decomposable a posteriori estimates on the model reduction error in the context of Reduced Basis (RB) methods. In addition, we present offline/online decomposable a posteriori error estimates on the full approximation error (including discretization as well as model reduction error) in the context of the localized RB method [14]. Hence, this work generalizes the localized RB method with true error certification to parabolic problems. Numerical experiments are given to demonstrate the applicability of the approach.

math.NA↗

arXiv2016

ArbiLoMod: Local Solution Spaces by Random Training in Electrodynamics

The simulation method ArbiLoMod has the goal to provide users of Finite Element based simulation software with quick re-simulation after localized changes to the model under consideration. It generates a Reduced Order Model (ROM) for the full model without ever solving the full model. To this end, a localized variant of the Reduced Basis method is employed, solving only small localized problems in the generation of the reduced basis. The key to quick re-simulation lies in recycling most of the localized basis vectors after a localized model change. In this publication, ArbiLoMod's local training algorithm is analyzed numerically for the non-coercive problem of time harmonic Maxwell's equations in 2D, formulated in H(curl).

math.NA↗

arXiv2016

ArbiLoMod, a Simulation Technique Designed for Arbitrary Local Modifications

Engineers manually optimizing a structure using Finite Element based simulation software often employ an iterative approach where in each iteration they change the structure slightly and resimulate. Standard Finite Element based simulation software is usually not well suited for this workflow, as it restarts in each iteration, even for tiny changes. In settings with complex local microstructure, where a fine mesh is required to capture the geometric detail, localized model reduction can improve this workflow. To this end, we introduce ArbiLoMod, a method which allows fast recomputation after arbitrary local modifications. It employs a domain decomposition and a localized form of the Reduced Basis Method for model order reduction. It assumes that the reduced basis on many of the unchanged domains can be reused after a localized change. The reduced model is adapted when necessary, steered by a localized error indicator. The global error introduced by the model order reduction is controlled by a robust and efficient localized a posteriori error estimator, certifying the quality of the result. We demonstrate ArbiLoMod for a coercive, parameterized example with changing structure.

math.NA↗

arXiv2016

Localized Reduced Basis Approximation of a Nonlinear Finite Volume Battery Model with Resolved Electrode Geometry

In this contribution we present first results towards localized model order reduction for spatially resolved, three-dimensional lithium-ionbattery models. We introduce a localized reduced basis scheme based on non-conforming local approximation spaces stemming from a finite volume discretizationof the analytical model and localized empirical operator interpolation for the approximation of the model's nonlinearities. Numerical examples are provided indicating the feasibility of our approach.

math.NA↗

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

Reduced Basis Methods: Success, Limitations and Future Challenges

Parametric model order reduction using reduced basis methods can be an effective tool for obtaining quickly solvable reduced order models of parametrized partial differential equation problems. With speedups that can reach several orders of magnitude, reduced basis methods enable high fidelity real-time simulations of complex systems and dramatically reduce the computational costs in many-query applications. In this contribution we analyze the methodology, mainly focussing on the theoretical aspects of the approach. In particular we discuss what is known about the convergence properties of these methods: when they succeed and when they are bound to fail. Moreover, we highlight some recent approaches employing nonlinear approximation techniques which aim to overcome the current limitations of reduced basis methods.

math.NA↗

arXiv2014

A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations

The Reduced Basis (RB) method is a well established method for the model order reduction of problems formulated as parametrized partial differential equations. One crucial requirement for the application of RB schemes is the availability of an a posteriori error estimator to reliably estimate the error introduced by the reduction process. However, straightforward implementations of standard residual based estimators show poor numerical stability, rendering them unusable if high accuracy is required. In this work we propose a new algorithm based on representing the residual with respect to a dedicated orthonormal basis, which is both easy to implement and requires little additional computational overhead. A numerical example is given to demonstrate the performance of the proposed algorithm.

math.NA↗

arXiv2014

A Model Reduction Framework for Efficient Simulation of Li-Ion Batteries

In order to achieve a better understanding of degradation processes in lithium-ion batteries, the modelling of cell dynamics at the mircometer scale is an important focus of current mathematical research. These models lead to large-dimensional, highly nonlinear finite volume discretizations which, due to their complexity, cannot be solved at cell scale on current hardware. Model order reduction strategies are therefore necessary to reduce the computational complexity while retaining the features of the model. The application of such strategies to specialized high performance solvers asks for new software designs allowing flexible control of the solvers by the reduction algorithms. In this contribution we discuss the reduction of microscale battery models with the reduced basis method and report on our new software approach on integrating the model order reduction software pyMOR with third-party solvers. Finally, we present numerical results for the reduction of a 3D microscale battery model with porous electrode geometry.

math.NA↗

arXiv2013

The method of freezing as a new tool for nonlinear reduced basis approximation of parameterized evolution equations

We present a new method for the nonlinear approximation of the solution manifolds of parameterized nonlinear evolution problems, in particular in hyperbolic regimes with moving discontinuities. Given the action of a Lie group on the solution space, the original problem is reformulated as a partial differential algebraic equation system by decomposing the solution into a group component and a spatial shape component, imposing appropriate algebraic constraints on the decomposition. The system is then projected onto a reduced basis space. We show that efficient online evaluation of the scheme is possible and study a numerical example showing its strongly improved performance in comparison to a scheme without freezing.

math.NA↗