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# Staff Priv.-Doz. Dr. Christian Rieger

### Contact Information

Address:
Institut für Numerische Simulation
Endenicher Allee 19b
53115 Bonn
Phone: +49 228 73-69813
Office: EA19b 3.005
E-Mail: ed tod nnob-inu tod sni ta regeira tod b@foo tod de

beurlaubt für das WS18/19, Aachen:
http://www.mathcces.rwth-aachen.de/5people/rieger/start

### Teaching

#### Summer semester 2019

• Mathematik Vorkurs BA Block seminar.

### Publications

1. A representer theorem for deep kernel learning. B. Bohn, M. Griebel, and C. Rieger. Journal of Machine Learning Research, 20(64):1–32, 2019. Also available as INS Preprint No. 1714.
2. Sampling inequalities for anisotropic tensor product grids. C. Rieger and H. Wendland. IMA Journal of Numerical Analysis, 2019. Online first. Also available as INS Preprint No. 1805.
3. $\varepsilon$-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs. D. Dũng, M. Griebel, V. N. Huy, and C. Rieger. Journal of Complexity, 46:66–89, 2018. Also available as INS Preprint No. 1703.
4. Regularized kernel-based reconstruction in generalized Besov spaces. M. Griebel, C. Rieger, and B. Zwicknagl. Foundations of Computational Mathematics, 18(2):459–508, 2018. Also available as INS Preprint No. 1517.
5. Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations. M. Griebel, C. Rieger, and P. Zaspel. Accepted by International Journal for Uncertainty Quantification. Also available as INS Preprint No. 1813, 2018.
6. An inverse theorem for compact Lipschitz regions in $R^d$ using localized kernel bases. T. Hangelbroek, F. J. Narcowich, C. Rieger, and J. D. Ward. Mathematics of Computation, 87:1949–1989, 2018.
7. Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds, pages 517–543. T. Hangelbroek, F. J. Narcowich, C. Rieger, and J. D. Ward. Springer International Publishing, Cham, 2018.
8. Kernel-based reconstructions for parametric PDEs. R. Kempf, H. Wendland, and C. Rieger. Accepted for publication in: Meshfree Methods for Partial Differential Equations IX, Springer, Lecture Notes in Computational Science, editors: M. Griebel and M. A. Schweitzer. Also available as INS Preprint No. 1804., 2018.
9. Incremental kernel based approximations for Bayesian inverse problems. C. Rieger. Available as INS Preprint No. 1807., 2018.
10. Iterated Landweber method for radial basis functions interpolation with finite accuracy. C. Rieger. Available as INS Preprint No. 1806., 2018.
11. Effects of a gamma DSD with variable shape parameter on polarimetric radar moments. K. Schinagl, C. Rieger, C. Simmer, S. Trömel, and P. Friederichs. 2018 19th International Radar Symposium (IRS), pages 1–10, 2018.
12. Reproducing kernel Hilbert spaces for parametric partial differential equations. M. Griebel and C. Rieger. SIAM/ASA J. Uncertainty Quantification, 5:111–137, 2017. also available as INS Preprint No. 1511.
13. Upwind schemes for scalar advection-dominated problems in the discrete exterior calculus. M. Griebel, C. Rieger, and A. Schier. In D. Bothe and A. Reusken, editors, Transport Processes at Fluidic Interfaces, pages 145–175. Springer International Publishing, 2017.
14. Sampling inequalities for sparse grids. C. Rieger and H. Wendland. Numerische Mathematik, 2017. Also available as INS preprint no. 1609.
15. Spectral Approximation in Reproducing Kernel Hilbert Spaces. C. Rieger. Habilitation, Institute for Numerical Simulation, University of Bonn, 2016.
16. Multiscale approximation and reproducing kernel Hilbert space methods. M. Griebel, C. Rieger, and B. Zwicknagl. SIAM Journal on Numerical Analysis, 53(2):852–873, 2015. Also available as INS Preprint No. 1312.
17. Improved exponential convergence rates by oversampling near the boundary. C. Rieger and B. Zwicknagl. Constructive Approximation, 39(2):323–341, 2014.
18. An inverse theorem on bounded domains for meshless methods using localized bases. T. Hangelbroek, F. J. Narcowich, C. Rieger, and J. D. Ward. ArXiv e-prints, 2014. Preprint.
19. Sampling inequalities and support vector machines for Galerkin type data. C. Rieger. In Meshfree Methods for Partial Differential Equations V, volume 79 of Lecture Notes in Computational Science and Engineering, pages 51–63. Springer, New York, 2011.
20. Sampling and stability. C. Rieger, R. Schaback, and B. Zwicknagl. In Mathematical Methods for Curves and Surfaces, volume 5862 of Lecture Notes in Computer Science, pages 347–369. Springer, New York, 2010.
21. Sampling inequalities for infinitely smooth functions, with applications to interpolation and machine learning. C. Rieger and B. Zwicknagl. Advances in Computational Mathematics, 32(1):103–129, 2010.
22. Deterministic error analysis of support vector regression and related regularized kernel methods. C. Rieger and B. Zwicknagl. Journal of Machine Learning Research, 10:2115–2132, 2009. PDF: http://jmlr.csail.mit.edu/papers/v10/rieger09a.html.
23. Approximate interpolation with applications to selecting smoothing parameters. H. Wendland and C. Rieger. Numerische Mathematik, 101:729–748, 2005.