Preprints
Afkham, B. M.; Bhatt, A.; Haasdonk, B. & Hesthaven, J. S.:
Symplectic Model-Reduction with a Weighted Inner Product,
2018.
@unpublished
{Afkham2018,
author = {Babak Maboudi Afkham and Ashish Bhatt and Bernard Haasdonk and Jan S. Hesthaven}
,
title = {Symplectic Model-Reduction with a Weighted Inner Product}
,
year = {2018}
,
note = {Submitted}
}
Wittwar, D. & Haasdonk, B.:
Greedy Algorithms for Matrix-Valued Kernels,
University of Stuttgart,
2018.
@techreport
{WH18,
author = {Wittwar, Dominik and Haasdonk, Bernard}
,
title = {Greedy Algorithms for Matrix-Valued Kernels}
,
year = {2018}
,
url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1773}
}
Abstract: We are interested in approximating vector-valued functions on a compact set $Omegasubset mathbb R^d$. We consider reproducing kernel Hilbert spaces of $mathbb R^m$-valued functions which each admit a unique matrix-valued reproducing kernel $k$. These spaces seem promising, when modelling correlations between the target function components. The approximation of a function is a linear combination of matrix-valued kernel evaluations multiplied with coefficient vectors. To guarantee a fast evaluation of the approximant the expansion size, i.e. the number of centers $n$ is desired to be small. We thus present three different greedy algorithms by which a suitable set of centers is chosen in an incremental fashion: First, the $P$-Greedy which requires no function evaluations, second and third, the $f$-Greedy and $f/P$-Greedy which require function evaluations but produce centers tailored to the target function. The efficiency of the approaches is investigated on some data from an artificial model.
Alla, A.; Haasdonk, B. & Schmidt, A.:
Feedback control of parametrized PDEs via model order reduction and dynamic programming principle,
University of Stuttgart,
2017.
@techreport
{AHS17,
author = {Alla, A. and Haasdonk, B. and Schmidt, A.}
,
title = {Feedback control of parametrized PDEs via model order reduction and dynamic programming principle}
,
year = {2017}
,
note = {Submitted}
,
url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1765}
}
Köppl, T.; Santin, G.; Haasdonk, B. & Helmig, R.:
Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and machine learning techniques,
University of Stuttgart,
2017.
@techreport
{KSHH2017,
author = {Köppl, Tobias and Santin, Gabriele and Haasdonk, Bernard and Helmig, Rainer}
,
title = {Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and machine learning techniques}
,
year = {2017}
,
url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1743}
}
Abstract: In this work, we consider two kinds of model reduction techniques to simulate blood flow through the largest systemic arteries, where a stenosis is located in a peripheral artery i.e. in an artery that is located far away from the heart. For our simulations we place the stenosis in one of the tibial arteries belonging to the right lower leg (right post tibial artery). The model reduction techniques that are used are on the one hand side dimensionally reduced models (1-D and 0-D models, the so-called mixed-dimension model) and on the other hand side surrogate models produced by kernel methods. Both methods are combined in such a way that the mixed-dimension models yield training data for the surrogate model, where the surrogate model is parametrised by the degree of narrowing of the peripheral stenosis. By means of a well-trained surrogate model, we show that simulation data can be reproduced with a satisfactory accuracy and that parameter optimisation problems can be solved in a very efficient way. Furthermore it is demonstrated that a surrogate model enables us to present after a very short simulation time the impact of a varying degree of stenosis on blood flow.
Köppel, M.; Franzelin, F.; Kröker, I.; Oladyshkin, S.; Santin, G.; Wittwar, D.; Barth, A.; Haasdonk, B.; Nowak, W.; Pflüger, D. & Rohde, C.:
Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario,
2017.
@techreport
{UQcomparison2017,
author = {Köppel, M. and Franzelin, F. and Kröker, I. and Oladyshkin, S. and Santin, G. and Wittwar, D. and Barth, A. and Haasdonk, B. and Nowak, W. and Pflüger, D. and Rohde, C.}
,
title = {Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario}
,
year = {2017}
,
url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1759}
}
Abstract: A variety of methods is available to quantify uncertainties arising within the modeling of flow and transport in carbon dioxide storage, but there is a lack of thorough comparisons. Usually, raw data from such storage sites can hardly be described by theoretical statistical distributions since only very limited data is available. Hence, exact information on distribution shapes for all uncertain parameters is very rare in realistic applications. We discuss and compare four different methods tested for data-driven uncertainty quantification based on a benchmark scenario of carbon dioxide storage. In the benchmark, for which we provide data and code, carbon dioxide is injected into a saline aquifer modeled by the nonlinear capillarity-free fractional flow formulation for two incompressible fluid phases, namely carbon dioxide and brine. To cover different aspects of uncertainty quantification, we incorporate various sources of uncertainty such as uncertainty of boundary conditions, of conceptual model definitions and of material properties. We consider recent versions of the following non-intrusive and intrusive uncertainty quantification methods: arbitary polynomial chaos, spatially adaptive sparse grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The performance of each approach is demonstrated assessing expectation value and standard deviation of the carbon dioxide saturation against a reference statistic based on Monte Carlo sampling. We compare the convergence of all methods reporting on accuracy with respect to the number of model runs and resolution. Finally we offer suggestions about the methods’ advantages and disadvantages that can guide the modeler for uncertainty quantification in carbon dioxide storage and beyond.
Schmidt, A. & Haasdonk, B.:
Data-driven surrogates of value functions and applications to feedback control for dynamical systems,
University of Stuttgart,
2017.
@techreport
{SH17b,
author = {Schmidt, A. and Haasdonk, B.}
,
title = {Data-driven surrogates of value functions and applications to feedback control for dynamical systems}
,
year = {2017}
,
note = {Submitted to MathMod 2018}
,
url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1742}
}
Wittwar, D.; Schmidt, A. & Haasdonk, B.:
Reduced Basis Approximation for the Discrete-time Parametric Algebraic Riccati Equation,
University of Stuttgart,
2017.
@techreport
{WSH17,
author = {Wittwar, D. and Schmidt, A. and Haasdonk, B.}
,
title = {Reduced Basis Approximation for the Discrete-time Parametric Algebraic Riccati Equation}
,
year = {2017}
}
Carlberg, K.; Brencher, L.; Haasdonk, B. & Barth, A.:
Data-driven time parallelism via forecasting,
2016.
@unpublished
{CBHB16,
author = {Carlberg, Kevin and Brencher, Lukas and Haasdonk, Bernard and Barth, Andrea}
,
title = {Data-driven time parallelism via forecasting}
,
year = {2016}
,
note = {submitted to SIAM J. of Sci. Comp.}
}
Fritzen, F.; Haasdonk, B.; Ryckelynck, D. & Schöps, S.:
An algorithmic comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a nonlinear thermal problem,
University of Stuttgart,
2016.
@techreport
{FHRS16,
author = {Felix Fritzen and Bernard Haasdonk and David Ryckelynck and Sebastian Schöps}
,
title = {An algorithmic comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a nonlinear thermal problem}
,
year = {2016}
,
url = {https://arxiv.org/abs/1610.05029}
}
Haasdonk, B.:
Reduced Basis Methods for Parametrized PDEs -- A Tutorial Introduction for Stationary and Instationary Problems,
IANS, University of Stuttgart, Germany,
2014.
@techreport
{Haasdonk2014a,
author = {B. Haasdonk}
,
title = {Reduced Basis Methods for Parametrized PDEs -- A Tutorial Introduction for Stationary and Instationary Problems}
,
year = {2014}
,
note = {Chapter in P. Benner, A. Cohen, M. Ohlberger and K. Willcox (eds.): "Model Reduction and Approximation: Theory and Algorithms", SIAM, Philadelphia, 2017}
,
url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=938}
}
Haasdonk, B.; Salomon, J. & Wohlmuth, B.:
A Reduced Basis Method for Parametrized Variational Inequalities,
University of Stuttgart,
2012.
@techreport
{HSW12preprint,
author = {B. Haasdonk and J. Salomon and B. Wohlmuth}
,
title = {A Reduced Basis Method for Parametrized Variational Inequalities}
,
year = {2012}
}
Wirtz, D. & Haasdonk, B.:
An Improved Vectorial Kernel Orthogonal Greedy Algorithm,
University of Stuttgart,
2012.
@techreport
{WH12c_pre,
author = {D. Wirtz and B. Haasdonk}
,
title = {An Improved Vectorial Kernel Orthogonal Greedy Algorithm}
,
year = {2012}
,
note = {In preparation}
,
url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=742}
}
Abstract: This work is concerned with derivation and analysis of a modifed vectorial kernel orthogonal greedy algorithm (VKOGA) for approximation of nonlinear vectorial functions. The algorithm pursues simultaneous approximation of all vector components over a shared linear subspace of the underlying function Hilbert space in a greedy fashion [14, 33] and inherits the selection principle of the f=P-Greedy algorithm [18]. For the considered algorithm we perform a limit analysis of the selection criteria for already included subspace basis functions. We show that the approximation gain is bounded globally and for the multivariate case the limit functions correspond to a directional Hermite interpolation. We further prove algebraic convergence similar to [13], improved by a dimension-dependent factor, and introduce a new a-posteriori error bound. Comparison to related variants of our algorithm are presented. Targeted applications of this algorithm are model reduction of multiscale models [40].
Wirtz, D.; Karajan, N. & Haasdonk, B.:
Model order reduction of multiscale models using kernel methods,
SRC SimTech, University of Stuttgart, Germany,
2012.
@techreport
{WHK13_pre,
author = {D. Wirtz and N. Karajan and B. Haasdonk}
,
title = {Model order reduction of multiscale models using kernel methods}
,
year = {2012}
,
note = {Submitted}
}
Wirtz, D.; Sorensen, D. & Haasdonk, B.:
A-posteriori error estimation for DEIM reduced nonlinear dynamical systems,
University of Stuttgart,
2012.
@techreport
{WSH12_pre,
author = {D. Wirtz and D.C. Sorensen and B. Haasdonk}
,
title = {A-posteriori error estimation for DEIM reduced nonlinear dynamical systems}
,
year = {2012}
,
note = {Submitted to SISC}
,
url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=733}
}
Abstract: In this work an effcient approach for a-posteriori error estimation for POD-DEIM reduced nonlinear dynamical systems is introduced. The considered nonlinear systems may also include time and parameter-affne linear terms as well as parametrically dependent inputs and outputs. The reduction process involves a Galerkin projection of the full system and approximation of the system's nonlinearity by the DEIM method [Chaturantabut & Sorensen (2010)]. The proposed a-posteriori error estimator can be effciently decomposed in an offine/online fashion and is obtained by a one dimensional auxiliary ODE during reduced simulations. Key elements for effcient online computation are partial similarity transformations and matrix DEIM approximations of the nonlinearity Jacobians. The theoretical results are illustrated by application to an unsteady Burgers equation and a cell apoptosis model.
Haasdonk, B.:
Reduzierte-Basis-Methoden, Vorlesungsskript SS 2011,
University of Stuttgart,
2011.
@techreport
{Haasdonk2011c,
author = {B. Haasdonk}
,
title = {Reduzierte-Basis-Methoden, Vorlesungsskript SS 2011}
,
year = {2011}
,
number = {2011-004}
}
Pekalska, E. & Haasdonk, B.:
Kernel Quadratic Discriminant Analysis with Positive and Indefinite Kernels,
University of Münster,
2008.
@techreport
{Pekalska2008a,
author = {E. Pekalska and B. Haasdonk}
,
title = {Kernel Quadratic Discriminant Analysis with Positive and Indefinite Kernels}
,
year = {2008}
,
number = {06/08}
}
Haasdonk, B.; Ohlberger, M. & Rozza, G.:
A Reduced Basis Method for Evolution Schemes with Parameter-Dependent Explicit Operators,
University of Münster,
2007.
@techreport
{HOR07,
author = {B. Haasdonk and M. Ohlberger and G. Rozza}
,
title = {A Reduced Basis Method for Evolution Schemes with Parameter-Dependent Explicit Operators}
,
year = {2007}
,
number = {09/07 - N, FB 10}
,
note = {Accepted by ETNA.}
}
Abstract: During the last decades, reduced basis (RB) methods have been developed to a wide methodology for model reduction of problems that are governed by parametrized partial differential equations ($P^2DE$s ). In particular equations of elliptic and parabolic type for linear, low polynomial or monotonic nonlinearities have been treated successfully by RB methods using finite element schemes. Due to the characteristic offline-online decomposition, the reduced models often become suitable for a multi-query or real-time setting, where simulation results, such as field-variables or output estimates, can be approximated reliably and rapidly for varying parameters. In the current study, we address a certain class of time-dependent evolution schemes with explicit discretization operators that are arbitrarily parameter dependent. We extend the RB-methodology to these cases by applying the empirical interpolation method to localized discretization operators. The main technical ingredients are: (i) generation of a collateral reduced basis modelling the effects of the discretization operator under parameter variations in the offline-phase and (ii) an online simulation scheme based on a numerical subgrid and localized evaluations of the evolution operator. We formulate an a-posteriori error estimator for quantification of the resulting reduced simulation error. Numerical experiments on a parametrized convection problem, discretized with a finite volume scheme, demonstrate the applicability of the model reduction technique. We obtain a parametrized reduced model, which enables parameter variation with fast simulation response. We quantify the computational gain with respect to the non-reduced model and investigate the error convergence.
Haasdonk, B. & Ohlberger, M.:
Reduced Basis Method for Finite Volume Approximations of Parametrized Evolution Equations,
University of Freiburg, Institute of Applied Mathematics,
2006.
@techreport
{Haasdonk2006b,
author = {B. Haasdonk and M. Ohlberger}
,
title = {Reduced Basis Method for Finite Volume Approximations of Parametrized Evolution Equations}
,
year = {2006}
,
number = {12/2006}
,
note = {Extended version of M2AN article.}
}
Haasdonk, B.; Poluru, B. & Teynor, A.:
Presto-Box 1.1 Library Documentation,
IIF-LMB, Universität Freiburg,
2003.
@techreport
{Haasdonk2003,
author = {Haasdonk, B. and Poluru, B.R. and Teynor, A.}
,
title = {Presto-Box 1.1 Library Documentation}
,
year = {2003}
,
number = {2/03}
}
Haasdonk, B.; Ohlberger, M.; Rumpf, M.; Schmidt, A. & Siebert, K.-G.:
h-p-Multiresolution Visualization of Adaptive Finite Element Simulations,
Mathematics Department, University of Freiburg,
2001.
@techreport
{Haasdonk2001c,
author = {B. Haasdonk and M. Ohlberger and M. Rumpf and A. Schmidt and K.-G. Siebert}
,
title = {h-p-Multiresolution Visualization of Adaptive Finite Element Simulations}
,
year = {2001}
,
number = {Preprint 01-26}
}
Geßner, T.; Haasdonk, B.; Lenz, M.; Metscher, M.; Neubauer, R.; Ohlberger, M.; Rosenbaum, W.; Rumpf, M.; Schwörer, R.; Spielberg, M. & Weikard, U.:
A Procedural Interface for Multiresolutional Visualization of General Numerical Data,
University of Bonn,
1999.
@techreport
{Gessner1999,
author = {T. Geßner and B. Haasdonk and M. Lenz and M. Metscher and R. Neubauer and M. Ohlberger and W. Rosenbaum and M. Rumpf and R. Schwörer and M. Spielberg and U. Weikard}
,
title = {A Procedural Interface for Multiresolutional Visualization of General Numerical Data}
,
year = {1999}
,
number = {28}
}
