Max Planck Institute Magdeburg

1. MAX-PLANCK-INSTITUT DYNAMIK KOMPLEXER TECHNISCHER SYSTEME MAGDEBURG November 5, 2010 Model Order Reduction for Parametric Systems Lihong Feng Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Max Planck Institute Magdeburg

2. Overview • A Glance at Model Order Reduction (MOR). • MOR for Parametric Systems (PMOR). • A Recycling Method for Accelerating the Process of PMOR. • Simulation Results. • Conclusions and Outlook. Model Order Reduction of Parametric Systems

3. A Glance at Model Order Reduction Original large model: Reduced small model: Step 1. Derive the reduced model. Step 2. Solve the reduced model, and get Step 3. Return back to the original unknown vector: Model Order Reduction (MOR) The error is very small. Model Order Reduction of Parametric Systems

4. Parametric Systems Non-parametric systems, e.g. Parametric systems, e.g. Model Order Reduction of Parametric Systems

5. Where do the Parameters Come From? • Process variation in manufacturing of (integrated circuits) ICs Copper interconnect pattern (IBM) Model Order Reduction of Parametric Systems

6. Where do the Parameters Come From? • Geometrical variations in Microelectromechanical systems (MEMS) design. Butterfly Gyroscope discretization PDEs ODEs Model Order Reduction of Parametric Systems

7. Where do the Parameters Come From? • Chemical engineering Simulated moving bed (SMB) chromatography Model Order Reduction of Parametric Systems

8. Where do the Parameters Come From? • The systems are of very large dimension. • An efficient technique to reduce the complexity is PMOR. In this talk: • PMOR is introduced. • A recycling method which speeds up the process of PMOR is • introduced. Model Order Reduction of Parametric Systems

9. Parametric MOR Original large model: Reduced small model: MOR based on moment matching: Unknown vector x in Laplace domain: Model Order Reduction of Parametric Systems

10. Parametric MOR Parametric systems, e.g. Define: span { coefficients till i=r } • The coefficients can be computed recursively and with numerical stability. Model Order Reduction of Parametric Systems

11. Parametric MOR The recursion developed in The projection matrix V can be computed by modified Gram-Schmidt process, which is numerically stable. Notice Model Order Reduction of Parametric Systems

12. A Recycling Method How to deal with the inverse matrix in each term? When LU is inefficient LU factorization of • Standard iterative solvers like: CG, GMRES(m), etc.. • A recycling method is developed to accelerate GMRES(m) . • Applied to PMOR in . Model Order Reduction of Parametric Systems

13. A Recycling Method Recycle the invariant subspace S: Implement GMRES(m) with initial guess Use an invariant subspace S of to accelerate the convergence rate. S First modify the initial guess: Then implement GMRES(m) with initial guess , use Sagain to further accelerate the convergence rate, update S if necessary. S First modify the initial guess: , … . Model Order Reduction of Parametric Systems

14. Simulation Results Efficiency of PMOR: Original transfer function Butterfly Gyroscope Reduced transfer function Beam thickness d Frequency (MHz) Beam thickness d Absolute error: Frequency (MHz) A system with 6 parameters Model Order Reduction of Parametric Systems

15. Simulation Results Efficiency of PMOR: Harmonic analysis of the system • Dimension of the original system: 17,931 • Dimension of the reduced system: 289 Model Order Reduction of Parametric Systems

16. Simulation Results Computational Complexity saved by recycling the invariantsubspace Recycling vs. standard solvers for solving a single linear system Totally 258 linear systems, 6708s (1.9 hours) are saved vs. . even more vs. . Model Order Reduction of Parametric Systems

17. Conclusions and Outlook • The reduced small parametric system works well. • The recycling method can speed up the process of PMOR. • The recycling method can also be used to solve • Efficient PMOR methods for more complicated parametric • systems are desired and challenging. Thank you! Model Order Reduction of Parametric Systems