@article {FHRS18,
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} ,
journal = {Math. Comput. Appl. 2018} ,
year = {2018} ,
volume = {23} ,
number = {1} ,
url = {http://www.mdpi.com/2297-8747/23/1/8/pdf} ,
doi = {10.3390/mca23010008} }

@inbook {HS2017a,
author = {Haasdonk, Bernard and Santin, Gabriele} ,
editor = {Keiper, Winfried and Milde, Anja and Volkwein, Stefan} ,
title = {Greedy Kernel Approximation for Sparse Surrogate Modeling} ,
booktitle = {Reduced-Order Modeling (ROM) for Simulation and Optimization: Powerful Algorithms as Key Enablers for Scientific Computing} ,
publisher = {Springer International Publishing} ,
year = {2018} ,
pages = {21--45} ,
url = {https://doi.org/10.1007/978-3-319-75319-5_2} ,
doi = {10.1007/978-3-319-75319-5_2} }

Abstract: Modern simulation scenarios frequently require multi-query or real-time responses of simulation models for statistical analysis, optimization, or process control. However, the underlying simulation models may be very time-consuming rendering the simulation task difficult or infeasible. This motivates the need for rapidly computable surrogate models. We address the case of surrogate modeling of functions from vectorial input to vectorial output spaces. These appear, for instance, in simulation of coupled models or in the case of approximating general input--output maps. We review some recent methods and theoretical results in the field of greedy kernel approximation schemes. In particular, we recall the vectorial kernel orthogonal greedy algorithm (VKOGA) for approximating vector-valued functions. We collect some recent convergence statements that provide sound foundation for these algorithms, in particular quasi-optimal convergence rates in case of kernels inducing Sobolev spaces. We provide some initial experiments that can be obtained with non-symmetric greedy kernel approximation schemes. The results indicate better stability and overall more accurate models in situations where the input data locations are not equally distributed.

@incollection {Haasdonk2017,
author = {B. Haasdonk} ,
editor = {P. Benner and A. Cohen and M. Ohlberger and K. Willcox} ,
title = {Reduced Basis Methods for Parametrized PDEs -- A Tutorial Introduction for Stationary and Instationary Problems} ,
booktitle = {Model Reduction and Approximation: Theory and Algorithms} ,
publisher = {SIAM, Philadelphia} ,
year = {2017} ,
pages = {65--136} ,
url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=938} }

@article {Martini2015,
author = {Martini, I. and Rozza, G. and Haasdonk, B.} ,
title = {Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models} ,
journal = {Journal of Scientific Computing} ,
year = {2017} ,
url = {http://link.springer.com/article/10.1007/s10915-017-0430-y} ,
doi = {10.1007/s10915-017-0430-y} }

@article {SH16b,
author = {G. Santin and B. Haasdonk} ,
title = {Convergence rate of the data-independent P-greedy algorithm in kernel-based approximation} ,
journal = {Dolomites Research Notes on Approximation} ,
year = {2017} ,
volume = {10} ,
pages = {68--78} ,
url = {http://www.emis.de/journals/DRNA/9-2.html} }

Abstract: Kernel-based methods provide flexible and accurate algorithms for the reconstruction of functions from meshless samples. A major question in the use of such methods is the influence of the samples’ locations on the behavior of the approximation, and feasible optimal strategies are not known for general problems. Nevertheless, efficient and greedy point-selection strategies are known. This paper gives a proof of the convergence rate of the data-independent $P$-greedy algorithm, based on the application of the convergence theory for greedy algorithms in reduced basis methods. The resulting rate of convergence is shown to be quasi-optimal in the case of kernels generating Sobolev spaces. As a consequence, this convergence rate proves that, for kernels of Sobolev spaces, the points selected by the algorithm are asymptotically uniformly distributed, as conjectured in the paper where the algorithm has been introduced

@article {SH17,
author = {Schmidt, Andreas and Haasdonk, Bernard} ,
title = {Reduced basis approximation of large scale parametric algebraic Riccati equations} ,
journal = {ESAIM: Control, Optimisation and Calculus of Variations} ,
publisher = {EDP Sciences} ,
year = {2017} ,
url = {http://dx.doi.org/10.1051/cocv/2017011} ,
doi = {10.1051/cocv/2017011} }

@article {Dihlmann2014a,
author = {Dihlmann, M. A. and Haasdonk, B.} ,
title = {Certified PDE-constrained parameter optimization using reduced basis surrogate models for evolution problems} ,
journal = {COAP, Computational Optimization and Applications} ,
year = {2015} ,
volume = {60} ,
number = {3} ,
pages = {753--787} ,
doi = {DOI: 10.1007/s10589-014-9697-1} }

@article {Kaulmann2015,
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 = {Internat. J. Numer. Methods Engrg.} ,
year = {2015} ,
volume = {102} ,
pages = {1018--1040} ,
doi = {DOI: 10.1002/nme.4773} }

@article {Wirtz2015a,
author = {D. Wirtz and N. Karajan and B. Haasdonk} ,
title = {Surrogate Modelling of multiscale models using kernel methods} ,
journal = {International Journal of Numerical Methods in Engineering} ,
year = {2015} ,
volume = {101} ,
number = {1} ,
pages = {1--28} ,
doi = {10.1002/nme.4767} }

@article {H13,
author = {Haasdonk, B.} ,
title = {Convergence Rates of the POD--Greedy Method} ,
journal = {ESAIM: Mathematical Modelling and Numerical Analysis} ,
publisher = {EDP Sciences} ,
year = {2013} ,
volume = {47} ,
number = {3} ,
pages = {859--873} ,
url = {http://dx.doi.org/10.1051/m2an/2012045} ,
doi = {10.1051/m2an/2012045} }

Abstract: Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approximating solution sets of parametrized partial differential equations. Recently, a-priori convergence rate statements for this algorithm have been given (Buffa et al 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD-Greedy algorithm (Haasdonk and Ohlberger 2008). In this algorithm, each greedy step is invoking a temporal compression step by performing a proper orthogonal decomposition (POD). Using a suitable coefficient representation of the POD-Greedy algorithm, we show that the existing convergence rate results of the Greedy algorithm can be extended. In particular, exponential or algebraic convergence rates of the Kolmogorov n-widths are maintained by the POD-Greedy algorithm.

@article {Haasdonk2013,
author = {B. Haasdonk and K. Urban and B. Wieland} ,
title = {Reduced basis methods for parametrized partial differential equations with stochastic influences using the Karhunen Loeve expansion} ,
journal = {SIAM/ASA J. Unc. Quant.} ,
year = {2013} ,
volume = {1} ,
pages = {79-105} }

@article {Wirtz2013,
author = {Daniel Wirtz and Bernard Haasdonk} ,
title = {A Vectorial Kernel Orthogonal Greedy Algorithm} ,
journal = {Dolomites Res. Notes Approx.} ,
year = {2013} ,
volume = {6} ,
pages = {83-100} ,
note = {Proceedings of DWCAA12} ,
url = {http://drna.padovauniversitypress.it/system/files/papers/WirtzHaasdonk-2013-VKO.pdf} }

Abstract: This work is concerned with derivation and analysis of a modified vectorial kernel orthogonal greedy algorithm (VKOGA) for approximation of nonlinear vectorial functions. The algorithm pursues sim- ultaneous approximation of all vector components over a shared linear subspace of the underlying function Hilbert space in a greedy fashion [16, 37] and inherits the selection principle of the f /P-Greedy algorithm [20]. 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 [14], 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 [42].

@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.

@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} }

@article {HDO11,
author = {Haasdonk, B. and Dihlmann, M. and Ohlberger, M.} ,
title = {A Training Set and Multiple Basis Generation Approach for Parametrized Model Reduction Based on Adaptive Grids in Parameter Space} ,
journal = {Mathematical and Computer Modelling of Dynamical Systems} ,
year = {2011} ,
volume = {17} ,
pages = {423--442} }

@article {Haasdonk2011b,
author = {Haasdonk, Bernard and Ohlberger, Mario} ,
title = {Efficient reduced models and it a posteriori error estimation for parametrized dynamical systems by offline/online decomposition} ,
journal = {Math. Comput. Model. Dyn. Syst.} ,
year = {2011} ,
volume = {17} ,
number = {2} ,
pages = {145--161} ,
url = {http://dx.doi.org/10.1080/13873954.2010.514703} ,
doi = {10.1080/13873954.2010.514703} }

@incollection {Haasdonk2009,
author = {Haasdonk, B. and Ohlberger, M.} ,
title = {Reduced basis method for explicit finite volume approximations of nonlinear conservation laws} ,
booktitle = {Hyperbolic problems: theory, numerics and applications} ,
publisher = {Amer. Math. Soc.} ,
year = {2009} ,
volume = {67} ,
pages = {605--614} }

@article {HO08,
author = {B. Haasdonk and M. Ohlberger} ,
title = {Reduced basis method for finite volume approximations of parametrized linear evolution equations} ,
journal = {ESAIM: M2AN} ,
year = {2008} ,
volume = {42} ,
number = {2} ,
pages = {277--302} ,
url = {http://dx.doi.org/10.1051/m2an:2008001} ,
doi = {10.1051/m2an:2008001} }

@article {Haasdonk2007,
author = {Haasdonk, B. and Burkhardt, H.} ,
title = {Invariant Kernels for Pattern Analysis and Machine Learning} ,
journal = {Machine Learning} ,
year = {2007} ,
volume = {68} ,
pages = {35--61} ,
doi = {DOI 10.1007/s10994-007-5009-7} }

@article {H05,
author = {B. Haasdonk} ,
title = {Feature Space Interpretation of SVMs with Indefinite Kernels} ,
journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence} ,
publisher = {IEEE Computer Society} ,
year = {2005} ,
volume = {27} ,
number = {4} ,
pages = {482-492} ,
doi = {http://doi.ieeecomputersociety.org/10.1109/TPAMI.2005.78} }

@inproceedings {Bahlmann2002,
author = {C. Bahlmann and B. Haasdonk and H. Burkhardt} ,
title = {On-line Handwriting Recognition with Support Vector Machines - A Kernel Approach} ,
booktitle = {Proc. of the 8th International Workshop on Frontiers in Handwriting Recognition} ,
publisher = {IEEE Computer Society} ,
year = {2002} ,
pages = {49--54} }

Abstract: In this contribution we describe a novel classification approach for on-line handwriting recognition. The technique combines dynamic time warping (DTW) and support vector machines (SVMs) by establishing a new SVM kernel. We call this kernel Gaussian DTW (GDTW) kernel. This kernel approach has a main advantage over common HMM techniques. It does not assume a model for the generative class conditional densities. Instead, it directly addresses the problem of discrimination by creating class boundaries and thus is less sensitive to modeling assumptions. By incorporating DTW in the kernel function, general classification problems with variable-sized sequential data can be handled. In this respect the proposed method can be straightforwardly applied to all classification problems, where DTW gives a reasonable distance measure, e.g.~speech recognition or genome processing. We show experiments with this kernel approach on the UNIPEN handwriting data, achieving results comparable to an HMM-based technique.

@article {Haasdonk2001a,
author = {Haasdonk, B. and Kröner, D. and Rohde, C.} ,
title = {Convergence of a staggered Lax-Friedrichs scheme for nonlinear conservation laws on unstructured two-dimensional grids} ,
journal = {Numer. Math.} ,
year = {2001} ,
volume = {88} ,
number = {3} ,
pages = {459--484} ,
url = {http://dx.doi.org/10.1007/s211-001-8011-x} ,
doi = {10.1007/s211-001-8011-x} }