Model Order Reduction of Model Order Reduction of Parameterized - PowerPoint PPT Presentation

Model Order Reduction of Model Order Reduction of Parameterized Interconnect Networks Parameterized Interconnect Networks via a Two- -Directional Directional Arnoldi Arnoldi Process Process via a Two Yung-Ta Li joint work with Zhaojun Bai, Yangfeng Su, Xuan Zeng RANMEP, Jan 5, 2008 1

Problem statement Problem statement Parameterized linear interconnect networks where Problem: find a reduced-order system where 2

Application: variational variational RLC circuit RLC circuit Application: Complex Structure of Interconnects spacing parameters L. Daniel et al., IEEE TCAD 2004 Image source: OIIC J. Philips, ICCAD 2005 MNA (Modified Nodal Analysis) 3

Application: micromachined micromachined disk resonator disk resonator Application: Electrode Disk resonator thickness Silicon wafer PML region thickness Cutaway schematic (left) and SEM picture (right) of a micromachined disk resonator. Through modulated electrostatic attraction between a disk and a surrounding ring of electrodes, the disk is driven into mechanical resonance. Because the disk is anchored to a silicon wafer, energy leaks from the disk to the substrate, where radiates away as elastic waves. To study this energy loss, David Bindel has constructed finite element models in which the substrate is modeled by a perfectly matched absorbing layer. Resonance poles are approximated by eigenvalues of a large, sparse complex-symmetric matrix pencil. For more details, see D. S. Bindel and S. Govindjee, "Anchor Loss Simulation in Resonators," International Journal for Numerical Methods in Engineering , vol 64, issue 6. Resonator micrograph courtesy of Emmanuel Quévy. 4

Outline Outline 1. Transfer function and multiparameter moments 2. MOR via subspace projection - projection subspace and moment-matching 3. Projection matrix computation - “2D’’ Krylov subspace and Arnoldi process 4. Numerical examples - affine model of one geometric parameter - affine model of multiple geometric parameters - polynomial type model 5. Concluding remarks 5

Transfer function State space model: One geometric parameter for clarity of presentation Transfer function: : and where for the frequency 6

Multiparameter moments and 2D recursion moments and 2D recursion Multiparameter Power series expansion: Multiparameter moments: Moment generating vectors: 7

Outline Outline 1. Transfer function and multiparameter moments 2. MOR via subspace projection - projection subspace and moment matching 3. Projection matrix computation - “2D’’ Krylov subspace and Arnoldi process 4. Numerical examples - affine model of one geometric parameter - affine model of multiple geometric parameters - polynomial type model 5. Concluding remarks 8

MOR via subspace projection MOR via subspace projection • A proper projection subspace: • Orthogonal projection: • System matrices of the reduced-order system: 9

MOR via subspace projection MOR via subspace projection Goals: 1. Keep the affined form in the state space equations 1. Keep the affined form in the state space equations 2. Preserve the stability and the passivity 2. Preserve the stability and the passivity 3. Maximize the number of matched moments 3. Maximize the number of matched moments Goal 1 is guaranteed via orthogonal projection Goal 1 is guaranteed via orthogonal projection Goal 2 can be achieved if the original system complies Goal 2 can be achieved if the original system complies with a certain passive form with a certain passive form The number of matched moments is The number of matched moments is decided by the projection subspace decided by the projection subspace 10

Projection subspace and moment- -matching matching Projection subspace and moment Projection subspace: (1) for (2) 11

Outline Outline 1. Transfer function and multiparameter moments 2. MOR via subspace projection - projection subspace and moment matching 3. Projection matrix computation - “2D’’ Krylov subspace and Arnoldi process 4. Numerical examples - affine model of one geometric parameter - affine model of multiple geometric parameters - polynomial type model 5. Concluding remarks 12

Krylov subspace subspace Krylov [ 1 ] [ 1 ] [ 1 ] [ 1 ] [ 1 ] [ 1 ] r r r r r r [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] (1,i)th Krylov subspace: Use Arnoldi process to compute an orthonormal basis 13

Krylov subspace subspace Krylov [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] r r r r r r [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] (2,i)th Krylov subspace : : How to efficiently compute an orthonormal basis ? 14

Efficiently compute Efficiently compute [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] r r r r r r [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Computed by 2D Arnoldi process 15

Projection subspace Projection subspace 2D Krylov 2D Krylov subspace subspace Define We have the linear recurrence where 16

2D Krylov Krylov subspace and 2D subspace and 2D Arnoldi Arnoldi decomposition decomposition 2D • (j,i)th Krylov subspace: • two-directional Arnoldi decomposition: where Computed by 2D Arnoldi process • two properties: be used to construct projection matrix vectors in the projection subspace 17

Generate V by 2D Arnoldi Arnoldi process process Generate V by 2D Use V to construct reduced-order models by orthogonal projection PIMTAP ( Parameterized Interconnect Macromodeling via a Two-directional Arnoldi Process) 18

Outline Outline 1. Transfer function and multiparameter moments 2. MOR via subspace projection - projection subspace and moment-matching 3. Projection matrix computation - “2D’’ Krylov subspace and Arnoldi process 4. Numerical examples - affine model of one geometric parameter - affine model of multiple geometric parameters - polynomial type model 5. Concluding remarks 19

RLC circuit with one parameter RLC network: 8-bit bus with 2 shield lines variations on and • Structure-preserving MOR method : SPRIM [Freund, ICCAD 2004] 20

RLC circuit : Relative error : Relative error Compare PIMTAP and CORE [X Li et al., ICCAD 2005] CORE (p,q)=(40,1) 0 10 −2 10 CORE (p,q)=(80,1) order of ROM=81 −4 10 ^ | / | h | −6 10 Relative error: | h − h PIMTAP (p,q)=(40,1) −8 10 order of the ROM=76 −10 10 −12 10 −14 CORE (p,q)=(40,1) 10 CORE (p,q)=(80,1) PIMTAP (p,q)=(40,1) −16 10 0 1 2 3 4 5 6 7 8 9 10 Frequency (GHz) 21

RLC circuit : Numerical stability : Numerical stability 11 original CORE (p,q)=(80,2) PIMTAP (p,q)=(40,2) 10 CORE (p,q)=(80,2) 9 |h(s)| 8 PIMTAP (p,q)=(40,2) 7 6 5 0 1 2 3 4 5 6 7 8 9 10 Frequency (GHz) 22

RLC circuit : PIMTAP for q=1,2,3,4 : PIMTAP for q=1,2,3,4 10 − q=1,2 2 bus8b: λ = 0.06 −2 Relative error: abs( Horiginal − Hreduced )/ abs( Horiginal ) 10 10 − −4 4 10 q=3,4 −6 10 −8 10 −10 10 −12 10 PIMTAP (p,q)=(40,1) −14 PIMTAP (p,q)=(40,2) 10 PIMTAP (p,q)=(40,3) PIMTAP (p,q)=(40,4) −16 10 0 2 4 6 8 10 Frequency (GHz) 23

Parametric thermal model [ Parametric thermal model [Rudnyi Rudnyi et al. 2005] et al. 2005] Thermal model with parameters and Power series expansion of the transfer function on satisfies the recursion: 24

Parametric thermal model Parametric thermal model Stack the vectors via the following ordering: (0,0,0) (1,0,0) � (0,1,0) � (0,0,1) (2,0,0) � (1,1,0) � (1,0,1) � (0,2,0) � (0,1,1) � (0,0,2) � � The sequence : (0,0,0) � (1,0,0) � (0,1,0) � (0,0,1) � (2,0,0) � …… � (0,0,2) 25

Polynomial type model Polynomial type model State space model: Transfer function: Moment generating vectors: 26

Projection subspace Projection subspace 2D Krylov 2D Krylov subspace subspace Define We have the linear recurrence where 27

Outline Outline 1. Transfer function and multiparameter moments 2. MOR via subspace projection - projection subspace and moment matching 3. Projection matrix computation - “2D’’ Krylov subspace and Arnoldi process 4. Numerical examples - affine model of one geometric parameter - affine model of multiple geometric parameters - polynomial type model 5. Concluding remarks 28

Concluding remarks: Concluding remarks: 1. PIMTAP is a moment matching based approach 2. PIMTAP is designed for systems with a low- dimensional parameter space 3. Systems with a high-dimensional parameter space are dealt by parameter reduction and PMOR 4. A rigorous mathematical definition of projection subspaces is given for the design of Pad’e-like approximation of the transfer function 5. The orthonormal basis of the projection subspace is computed adaptively via a novel 2D Arnoldi process 29

Current projects: Oblique projection Current projects: 30

Oblique projection 31