Sven Kaulmann

@article {Kaulmann2014wa,
author = {Kaulmann, Sven and Flemisch, Bernd and Haasdonk, Bernard and Lie, Knut-Andreas and Ohlberger, Mario} ,
title = {The Localized Reduced Basis Multiscale method for two-phase flows in porous media} ,
journal = {arXiv.org} ,
year = {2014} ,
url = {http://arxiv.org/abs/1405.2810v1} }

Abstract: In this work, we propose a novel model order reduction approach for two-phase flow in porous media by introducing a formulation in which the mobility, which realizes the coupling between phase saturations and phase pressures, is regarded as a parameter to the pressure equation. Using this formulation, we introduce the Localized Reduced Basis Multiscale method to obtain a low-dimensional surrogate of the high-dimensional pressure equation. By applying ideas from model order reduction for parametrized partial differential equations, we are able to split the computational effort for solving the pressure equation into a costly offline step that is performed only once and an inexpensive online step that is carried out in every time step of the two-phase flow simulation, which is thereby largely accelerated. Usage of elements from numerical multiscale methods allows us to displace the computational intensity between the offline and online step to reach an ideal runtime at acceptable error increase for the two-phase flow simulation.

@inproceedings {Kaulmann2013tv,
author = {Kaulmann, Sven and Haasdonk, Bernard} ,
editor = {Moitinho de Almeida, José Paulo Baptista and Diez, Pedro and Tiago, Carlos and Parés, Núria} ,
title = {Online Greedy Reduced Basis Construction Using Dictionaries} ,
booktitle = {VI International Conference on Adaptive Modeling and Simulation (ADMOS 2013)} ,
year = {2013} ,
pages = {365--376} ,
url = {http://www.lacan.upc.edu/admos2013/Proceedings.html} }

Abstract: The Reduced Basis method is a means for model order reduction for parametrized partial differential equations. In the last decades it has found broad application for problems with multi-query or real-time character. While the method has shown to be performing well for numerous different fields of applications, problems with high parameter dimension or high sensitivity with respect to the parameter still pose major challenges. In our contribution, we present a new basis generation algorithm that is particularly fit to these kinds of problems: Instead of building the reduced basis during the offline phase we build a large dictionary of basis vector candidates and compute a small parameter-adapted basis from that dictionary with a Greedy procedure during the online phase.

@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} }

@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.

@mastersthesis {Kaulmann2011cr,
author = {Kaulmann, Sven} ,
title = {A Localized Reduced Basis Approach for Heterogenous Multiscale Problems} ,
publisher = {Westfälische Wilhelms Universität Münster} ,
school = {Westfälische Wilhelms Universität Münster} ,
year = {2011} }

@article {Kaulmann2011,
author = {Kaulmann, Sven and Ohlberger, Mario and Haasdonk, Bernard} ,
title = {A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems} ,
journal = {Comptes Rendus Mathematique} ,
year = {2011} ,
volume = {349} ,
number = {23-24} ,
pages = {1233--1238} ,
url = {http://www.sciencedirect.com/science/article/pii/S1631073X11003074} ,
doi = {10.1016/j.crma.2011.10.024} }

Abstract: Inspired by the reduced basis approach and modern numerical multiscale methods, we present a new framework for an efficient treatment of heterogeneous multiscale problems. The new approach is based on the idea of considering heterogeneous multiscale problems as parametrized partial differential equations where the parameters are smooth functions. We then construct, in an offline phase, a suitable localized reduced basis that is used in an online phase to efficiently compute approximations of the multiscale problem by means of a discontinuous Galerkin method on a coarse grid. We present our approach for elliptic multiscale problems and discuss an a posteriori error estimate that can be used in the construction process of the localized reduced basis. Numerical experiments are given to demonstrate the efficiency of the new approach.