- General information
- Subprojects
- SP 1 (Braunschweig)
- Partners
- Activities

Mathematics for Innovations in Industry and Services

--MoreSim4Nano--

- Subproject leader:
**Prof. Dr. Matthias Bollhöfer, TU Braunschweig**

This subproject deals with problems arising from electromagnetism and investigates model order reduction based on Balanced Truncation [49, 71, 72]. These problems are typically described by Maxwell's equations in three spatial dimensions and along other applications, e.g., the numerical simulation of cross-talk between neighboring wires can be modeled with Maxwell's equations. The full simulation of these effects as part of the whole circuit simulation would be far too expensive. Here we discuss spatial discretizations using FEM (Nédélec elements), boundary elements (BEM) as well as the finite integration technique (FIT, cf. SP2). Balanced Truncation is applied to the resulting descriptor system after semi-discretization in space. For the computation of the balanced reduction it is necessary to solve generalized Lyapunov equations, where the system matrix arises from the discretized Maxwell's equations together with associated mass matrices. Beside the projections onto the space of (discrete) divergence-free functions the structure of the resulting system is being analyzed (in cooperation with SP4). This analysis should yield whether these projections and possibly further projections are required for BT to properly reveal the major properties of the original equation. We expect that results for similar problems, as discussed in [95], carry over to Maxwell's equations. For the first order formulation of the transient Maxwell's equations the shift parameters required for the ADI method [69] could be computed as the square roots of the optimal shift parameters [102] for the discrete*curl curl*operator. Alternatively MOR techniques for the reduced second order problem are being investigated. The resulting systems refer to shifted Maxwell systems. Depending on the size of the shift, on one hand fast FEM multigrid solvers [63, 64] as well as the fast 3D Maxwell solver of SP2 could be applied, which is also used by the industry partner CST. For larger shifts the systems tend to become more and more diagonal dominant, which allows for the use of algebraic Multilevel-ILU solver as an attractive alternative, e.g., using the software package ILUPACK [21, 20]. The Multilevel-ILU solver can employ similar shift techniques as used for Helmholtz equations [19] and will be adapted to the underlying problem. The Balanced Truncation method requires in every step the application of projectors to the space of finite eigenvalues of the underlying descriptor system. As an alternative it is planned (in cooperation with SP4) to discuss projection-free methods by applying BT to suitable constrained problems. A possible approach was presented in [54]. Finally MOR techniques will be adapted to modern computer architectures (multicore processors) in order to develop an adapted MOR library.

Milestones:

- [Months 1-12] Analysis of spectral projectors and usage for MOR.

- [Months 1-6] Benchmark EM problems will be discretized using FEM (Nédélec elements) and BEM in order to provide them to all other projects. Hierarchical structures will be used for BEM (e.g., like in HLIBootnote{www.hlib.org}). MOR will first be performed using algebraic solvers. Later we will switch to multilevel methods.
- [Months 7-12] Model problems from SP2, mainly discretized using FIT will be discussed. Based on these problems MOR will be applied using the 3D Maxwell solver of SP2.
- [Months 13-24] Efficient solvers (including the spectral projections and the ADI iteration) will be developed. We will compare the approach from SP1 with methods being used in SP2, SP3 and SP6. The comparisons are expected to reveal advantages and disadvantages of each individual approach. A combination of Balanced Truncation and Krylov subspace methods could be a suitable combination to be considered.
- [Months 25-36] An alternative approach based on projection-free methods (SP4) will be analyzed. Finally the codes will be adapted for multicore architectures.

- [Months 1-12] Analysis of spectral projectors and usage for MOR.