Publications 2012

@inproceedings {Albrecht2012lq,
author = {Albrecht, Felix and Haasdonk, Bernard and Kaulmann, Sven and Ohlberger, Mario} ,
editor = {Handloviv cová, Angela and Minarechová, Zuzana and v Sevv coviv c, Daniel} ,
title = {The Localized Reduced Basis Multiscale Method} ,
booktitle = {Algoritmy 2012} ,
publisher = {Publishing House of STU} ,
year = {2012} ,
pages = {393--403} ,
url = {http://www.iam.fmph.uniba.sk/algoritmy2012/} }

Abstract: In this paper we introduce the Localized Reduced Basis Multiscale (LRBMS) method for parameter dependent heterogeneous elliptic multiscale problems. The LRBMS method brings together ideas from both Reduced Basis methods to efficiently solve parametrized problems and from multiscale methods in order to deal with complex heterogeneities and large domains. Experiments on 2D and real world 3D data demonstrate the performance of the approach.

@inproceedings {Dihlmann2012,
author = {Dihlmann, M. AND Kaulmann, S. AND Haasdonk, B.} ,
title = {Online Reduced Basis Construction Procedure for Model Reduction of Parametrized Evolution Systems} ,
booktitle = {Proc. MATHMOD 2012 - 7th Vienna International Conference on Mathematical Modelling} ,
year = {2012} }

@inproceedings {Drohmann2012a,
author = {M. Drohmann and B. Haasdonk and M. Ohlberger} ,
title = {Reduced Basis Model Reduction of Parametrized Two-phase Flow in Porous Media} ,
booktitle = {Proc. MATHMOD 2012 - 7th Vienna International Conference on Mathematical Modelling} ,
year = {2012} ,
url = {http://www.sciencedirect.com/science/article/pii/S1474667016307613} ,
doi = {https://doi.org/10.3182/20120215-3-AT-3016.00128} }

@article {Drohmann2012,
author = {Drohmann, M. and Haasdonk, B. and Ohlberger, M.} ,
title = {Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation} ,
journal = {SIAM J. Sci. Comput.} ,
year = {2012} ,
volume = {34} ,
number = {2} ,
pages = {A937-A969} ,
doi = {10.1137/10081157X} }

Abstract: We present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations. Our approach is based on empirical interpolation of nonlinear differential operators and their Fréchet derivatives. Efficient offline/online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals. An a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically. We introduce a new algorithm, the PODEI-greedy algorithm, which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way. The approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations. We show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions. In case of symmetries of the problem, the approach realizes an automatic and intuitive space-compression or even space-dimensionality reduction. We perform empirical investigations of the error convergence and run-times. In all cases we obtain a good run-time acceleration.

@incollection {Drohmann2012d,
author = {Drohmann, Martin and Haasdonk, Bernard and Ohlberger, Mario} ,
editor = {Dedner, Andreas and Flemisch, Bernd and Klöfkorn, Robert} ,
title = {A Software Framework for Reduced Basis Methods Using DUNE-RB and RBMATLAB} ,
booktitle = {Advances in DUNE: Proceedings of the DUNE User Meeting, Held in October 6th-8th 2010 in Stuttgart, Germany} ,
publisher = {Springer} ,
year = {2012} ,
url = {http://www.springer.com/engineering/computational+intelligence+and+complexity/book/978-3-642-28588-2} }

Abstract: Many applications from science and engineering are based on parametrized evolution equations and depend on time-consuming parameter studies or need to ensure critical constraints on the simulation time. For both settings, model order reduction by the reduced basis approach is a suitable means to reduce computational time. The method is based on a projection of an underlying high dimensional nu- merical scheme onto a low dimensional function space. In this contribution, a new software framework is introduced that allows fast development of reduced schemes for a large class of discretizations of evolution equations implemented in DUNE. The approach provides a strict separation of low-dimensional and high-dimensional computations, each implemented by its own software package RBMATLAB, respectively DUNE-RB. The functionality of the framework is exemplified for a finite volume approximation of an instationary linear convection-diffusion problem.

@article {Haasdonk2012a,
author = {B. Haasdonk and J. Salomon and B. Wohlmuth} ,
title = {A Reduced Basis Method for Parametrized Variational Inequalities} ,
journal = {SIAM Journal on Numerical Analysis} ,
year = {2012} ,
volume = {50} ,
number = {5} ,
pages = {2656--2676} }

@inproceedings {Haasdonk2012,
author = {B. Haasdonk and J. Salomon and B. Wohlmuth} ,
title = {A Reduced Basis Method for the Simulation of American Options} ,
booktitle = {ENUMATH 2011 Proceedings} ,
year = {2012} ,
url = {http://arxiv.org/pdf/1201.3289v1} }

@article {Ruiner2012,
author = {Ruiner, T. AND Fehr, J. AND Haasdonk, B. AND Eberhard, P.} ,
title = {A-posteriori error estimation for second order mechanical systems} ,
journal = {Acta Mechanica Sinica} ,
year = {2012} ,
volume = {28(3)} ,
pages = {854-862} }

@article {Waldherr2012,
author = {S. Waldherr and B. Haasdonk} ,
title = {Efficient Parametric Analysis of the Chemical Master Equation through Model Order Reduction} ,
journal = {BMC Systems Biology} ,
year = {2012} ,
volume = {6} ,
pages = {81} ,
url = {http://www.biomedcentral.com/1752-0509/6/81} }

@article {WH12,
author = {D. Wirtz and B. Haasdonk} ,
title = {Efficient a-posteriori error estimation for nonlinear kernel-based reduced systems} ,
journal = {Systems and Control Letters} ,
year = {2012} ,
volume = {61} ,
number = {1} ,
pages = {203 - 211} ,
url = {http://www.sciencedirect.com/science/article/pii/S0167691111002672} ,
doi = {10.1016/j.sysconle.2011.10.012} }

Abstract: In this paper, we consider the topic of model reduction for nonlinear dynamical systems based on kernel expansions. Our approach allows for a full offline/online decomposition and efficient online computation of the reduced model. In particular, we derive an a-posteriori state-space error estimator for the reduction error. A key ingredient is a local Lipschitz constant estimation that enables rigorous a-posteriori error estimation. The computation of the error estimator is realized by solving an auxiliary differential equation during online simulations. Estimation iterations can be performed that allow a balancing between estimation sharpness and computation time. Numerical experiments demonstrate the estimation improvement over different estimator versions and the rigor and effectiveness of the error bounds.

@inproceedings {Wirtz2012a,
author = {Wirtz, Daniel and Haasdonk, Bernard} ,
title = {A-posteriori error estimation for parameterized kernel-based systems} ,
booktitle = {Proc. MATHMOD 2012 - 7th Vienna International Conference on Mathematical Modelling} ,
year = {2012} ,
url = {http://www.ifac-papersonline.net/} }

Abstract: This work is concerned with derivation of fully offine/online decomposable effcient aposteriori error estimators for reduced parameterized nonlinear kernel-based systems. The dynamical systems under consideration consist of a nonlinear, time- and parameter-dependent kernel expansion representing the system's inner dynamics as well as time- and parameter-affne inputs, initial conditions and outputs. The estimators are established for a reduction technique originally proposed in [7] and are an extension of the estimators derived in [11] to the fully time-dependent, parameterized setting. Key features for the effcient error estimation are to use local Lipschitz constants provided by a certain class of kernels and an iterative scheme to balance computation cost against estimation sharpness. Together with the affnely time/parameter-dependent system components a full offine/online decomposition for both the reduction process and the error estimators is possible. Some experimental results for synthetic systems illustrate the effcient evaluation of the derived error estimators for different parameters.