1. Model Order Reduction for Parametric Systems
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-INSTITUT
DYNAMIK KOMPLEXER
TECHNISCHER SYSTEME
MAGDEBURG
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. Model Order Reduction (MOR)
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.
Model Order Reduction of Parametric Systems

5. Copper interconnect pattern (IBM)
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. Simulated moving bed (SMB)
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 MOR based on moment matching: 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. span { coefficients till i=r }
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 S again 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: Absolute error:
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
CPU time
Original
days
Reduced
297s
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 invariant subspace
Recycling vs. standard solvers for solving a single linear system
Average CPU
time
Recycling
161s
187s
5226s
Average MV-products
Recycling
561
764
24952
Totally 258 linear systems, 6708s (1.9 hours) are saved vs
even more vs
Model Order Reduction of Parametric Systems 16

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