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

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. 24.11.2018 Model Order Reduction of Parametric Systems

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. 24.11.2018 Model Order Reduction of Parametric Systems

Parametric Systems Non-parametric systems, e.g. 24.11.2018 Model Order Reduction of Parametric Systems

Copper interconnect pattern (IBM) Where do the Parameters Come From? Process variation in manufacturing of (integrated circuits) ICs Copper interconnect pattern (IBM) 24.11.2018 Model Order Reduction of Parametric Systems

Where do the Parameters Come From? Geometrical variations in Microelectromechanical systems (MEMS) design. Butterfly Gyroscope discretization PDEs ODEs 24.11.2018 Model Order Reduction of Parametric Systems

Simulated moving bed (SMB) Where do the Parameters Come From? Chemical engineering Simulated moving bed (SMB) chromatography 24.11.2018 Model Order Reduction of Parametric Systems

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. 24.11.2018 Model Order Reduction of Parametric Systems

Parametric MOR MOR based on moment matching: Original large model: Reduced small model: MOR based on moment matching: Unknown vector x in Laplace domain: 24.11.2018 Model Order Reduction of Parametric Systems

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. 24.11.2018 Model Order Reduction of Parametric Systems

Parametric MOR The recursion developed in The projection matrix V can be computed by modified Gram-Schmidt process, which is numerically stable. Notice 24.11.2018 Model Order Reduction of Parametric Systems

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 . 24.11.2018 Model Order Reduction of Parametric Systems

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: , … . 24.11.2018 Model Order Reduction of Parametric Systems

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 24.11.2018 Model Order Reduction of Parametric Systems

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 24.11.2018 Model Order Reduction of Parametric Systems

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 24.11.2018 16

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! 24.11.2018 Model Order Reduction of Parametric Systems