Numerical Analysis:
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Project leader: Alexander Ostermann
Numerical Analysis Group
The Numerical Analysis Group at the Department of Mathematics pursues the following research topics:
- Time integration of dynamical systems, generated by partial differential equations;
- Discrete dynamics and long term behavior of numerical algorithms;
- Numerical analysis of exponential integrators and operator splitting methods;
- Time stepping in engineering problems and sensitivity analysis;
- Nonlinear optimization.
Within the framework of the DK-plus doctoral school, emphasis will be laid on the construction, analysis and implementation of advanced methods in time integration of (partial) differential equations. In particular the following two projects are proposed:
Students will benefit from a variety of international scientific and industrial collaborations. Part of the work can be carried out at one of the following institutions:
- M. Hochbruck (Karlsruhe Institute of Technology)
- Ch. Lubich (Univ. Tübingen)
- E. Hansen (Univ. Lund)
- M. Caliari (Univ. Verona)
- E. Emmrich (Univ. Bielefeld)
Numerical Analysis:
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The efficient time integration of evolution equations requires particular methods. For quite a long time, implicit and linearly implicit methods had been the methods of choice. These methods, however, make use of a continued solution of large systems of (non)linear equations. This can be the bottleneck in computations.
In recent years, exponential integrators have become an attractive alternative in many situations. In contrast to classical methods, they do not require the solution of large linear systems. Instead they make explicit use of the matrix exponential and related matrix functions.
At the moment, our group is implementing a mesh free exponential integrator for nonlinear reaction advection diffusion problems. Exponential integrators for highly nonlinear problems will be designed with the help of the Gröbner-Alekseev formula.
An ideal candidate for this project should have a strong background in numerical analysis and some knowledge in high performance computing.
Selected references:
M. Hochbruck, A. Ostermann: Exponential Integrators. Acta Numerica 2010 (to appear)
M. Hochbruck, M. Hönig, A. Ostermann: A convergence analysis of the exponential Euler iteration for nonlinear ill-posed problems. Inverse Problems 25, 075009 (2009)
M. Hochbruck, A. Ostermann, J. Schweitzer: Exponential Rosenbrock-type methods. SIAM J. Numer. Anal. 47, 786-803 (2009)
M. Caliari, A. Ostermann: Implementation of exponential Rosenbrock-type integrators. Applied Numerical Math. 59, 568-581 (2009)
Numerical Analysis:
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Splitting methods are widely used in practice for the time integration of evolution equations. Their error behavior, however, is still far from being well understood. As the classical error analysis based on the Baker-Campbell-Hausdorff formula is often not applicable to partial differential equations, new approaches have to be developed.
Quite recently, our research group has established new tools for analyzing the convergence behavior of splitting methods in the framework of strongly continuous and analytic semigroups. The next goal will be to obtain a full understanding of the encountered order reduction due to boundary conditions.
An ideal candidate for this project should have a strong background in both functional analysis and numerical analysis.
Selected references:
E. Hansen, A. Ostermann: High order splitting methods for analytic semigroups exist. BIT 49, 527-542 (2009)
E. Hansen, A. Ostermann: Exponential splitting for unbounded operators. Math. Comp. 78, 1485-1496 (2009)
E. Hansen, A. Ostermann: Dimension splitting for evolution equations. Numer. Math. 108, 557-570 (2008)