• Presentations will take place at Heho 20, Room 1.27 on 19.06.2019 from 12pm until 5pm and on 05.07.2019 from 12pm until 3pm.
• Written reports are due on your day of presentation. 

Numerical simulation has found broad application in many different fields and has become an integral part of modern day science and technology. As the models used for simulation get more accurate and complex, the required computational power grows as well. In many cases the computational, time and storage constraints imposed by the limitations of available hardware or economic considerations, do not allow for usage of accurate models. This has led to the development of Model Order Reduction (MOR) techniques that attempt to solve complex systems in real time, while controlling the resulting approximation error. Reduced Basis Methods (RBM) is a special type of MOR that has proven to be very successful in various applications. In this seminar we will brush through MOR, focusing specifically on RBM. The seminar can serve as a good preparation for a subsequent master thesis.

Responsible

• Prof. Dr. Karsten Urban
• Dr. Silke Glas

Prerequisites

• Lectures in basic numerics.

Study programs (master)

• All mathematical master programs, Lehramt, CSE, mathematische Biometrie.

Schedule

• Topic assignment end of the winter semester 18/19. 
• First week of lecture “Reduced Basis Methods” will serve as an introduction to the topic.
• Seminar presentation tba, possibly weekly presentations starting in May/June.
• Check website for updates.

Topics 

1. W. Dahmen, Ch. Plesken and G Welper: “Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems”, 2014. 
2. P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova and P. Wojtaszczyk: “Convergence Rates for Greedy Algorithms in Reduced Basis Methods”, 2011. 
3. A. Nouy: “Low-Rank Tensor Methods for Model Order Reduction”, 2017. 
4. O. Balabanov and A. Nouy: “Randomized Linear Algebra for Model Reduction. Part I: Galerkin Methods and Error Estimation”, 2018.
5. W. Dahmen: “How to Best Sample a Solution Manifold?”, 2015. 
6. M. Bachmayr, A. Cohen and W. Dahmen: “Parametric PDEs: Sparse or Low-Rank Approximation?”, 2017. 
7. G. Welper: “Transformed Solution Interpolation with High Resolution Transforms”, 2019.
8. D. Rim and K. Mandli: “Model Reduction of a Parametrized Scalar Hyperbolic Conservation Law using Displacement Interpolation”, 2018. 
9. S. Ullmann and J. Lang: “Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Elliptic PDEs”, 2019. 

References

• Haasdonk, B. (2017). Reduced Basis Methods for Parametrized PDEs -- A Tutorial Introduction  for Stationary and Instationary Problems. In P. Benner, A. Cohen, M. Ohlberger, & K. Willcox, P. Benner, A. Cohen, M. Ohlberger, & K. Willcox (Eds.), Model Reduction and Approximation: Theory and Algorithms (pp. 65--136). SIAM, Philadelphia.
• Quarteroni, A., A. Manzoni and Negri, F. (2016). Reduced Basis Methods for Partial Differential Equations. Volume 92 of Unitext, Springer, Cham.  
• Hesthaven, J. S., G. Rozza and Stamm, B. (2016). Certified reduced basis methods for partial differential equations. Springer, Cham.