1. DAC, July 2006 Model Order Reduction of Linear Networks with Massive Ports via Frequency-Dependent Port Packing Peng Li and Weiping Shi Department of ECE Texas A&M University {pli, wshi}@ece.tamu.edu

2. 2 Outline Motivation Motivation Related work Related work Proposed techniques Proposed techniques –Tangential interpolation –Frequency-dependent port packing –Implementation issues Experimental results Experimental results Conclusion Conclusion

3. 3 Interconnect Model Order Reduction Model order reduction (MOR) has become a driving force for interconnect analysis Model order reduction (MOR) has become a driving force for interconnect analysis large linear network large state space x1x1 x2x2 x3x3 x4x4 x5x5 xmxm … … ROM small linear network small state space

4. 4 Prior Work A large body of MOR techniques have been developed in the last few decades A large body of MOR techniques have been developed in the last few decades Moment matching/Krylov projection Moment matching/Krylov projection –[AWE, TCAD90] [PVL, TCAD95] [Silveira et al/ICCAD96] [Grimme/97], [PRIMA,TCAD98] [SPRIM, ICCAD04] –Most widely used due to low cost Truncated balanced realization Truncated balanced realization –[Heydari et al/ICCAD01] [Phillips et al/DAC02] –Theoretical error bounds exist Others Others

5. 5 Networks with Massive I/O Ports Passively I/O coupled networks are very common Passively I/O coupled networks are very common –Power grids, wide buses and substrate networks Very costly to analyze directly and deserve reduction more Very costly to analyze directly and deserve reduction more Standard MOR algorithms are difficult to apply Standard MOR algorithms are difficult to apply –Brute-force application can even result in models which are more complex than originals

6. 6 Multi-Port Model Order Reduction Krylov subspace projection Krylov subspace projection –Popular but keyed to number of ports Projection matrix: V Orthonormalize

7. 7 Multi-Port Model Order Reduction nxn qxn qxq SIMO MOR: nxq nxn QxQ MIMO MOR: … nxQ Qxn

8. 8 Related Multi-Port Modeling Work SVD-based parasitic extraction SVD-based parasitic extraction –Inductance extraction: [IES3, Kapur and Long/ICCAD97] substrate modeling: [Kanapka and White/ICCAD01] –Compress dense matrices using SVD Involves no dynamical system reduction aspect Involves no dynamical system reduction aspect Extended Krylov subspace method Extended Krylov subspace method –[Wang and Nguyen, DAC00] –Include moment expansions of inputs as part of Krylov subspace projection –Inputs to the system need to be known a priori Extension of TBR methods Extension of TBR methods –[Silveria and Phillips, DAC04] –Exploits input information in TBR

9. 9 Related Multi-Port Modeling Work SVD-based multi-port model generation SVD-based multi-port model generation –[SVDMOR, Feldmann DATE04] Extract correlation between matrix transfer functions at ports using SVD Extract correlation between matrix transfer functions at ports using SVD Create compressed network with fewer I/Os and apply existing techniques for reduction Create compressed network with fewer I/Os and apply existing techniques for reduction –[RecMOR, Feldmann and Liu ICCAD04] Hierarchical extension of SVDMOR Hierarchical extension of SVDMOR Generate larger but sparser models Generate larger but sparser models –[P. Liu et al, ICCAD05] Combines SVD and clustering algorithm Combines SVD and clustering algorithm

10. 10 Our Approach Multiport Circuit Packing (McPack) Multiport Circuit Packing (McPack) –Exploit system redundancy observed as input/output ports Similar to the ideas in SVDMOR/RecMOR Similar to the ideas in SVDMOR/RecMOR –Pack a large number of I/O ports by utilizing frequency- dependent correlations Exploit such correlation in cheap projection-based frameworks Exploit such correlation in cheap projection-based frameworks –Base on a new MOR formulism to do packing systematically Tangential interpolation Tangential interpolation –Model reduction also preserves passivity

11. 11 Tangential Interpolation Tangential interpolation emphasizes on a sub-space of input/output characteristics for model reduction Tangential interpolation emphasizes on a sub-space of input/output characteristics for model reduction –[Gallivan et al, SIAM J. Matrix Analysis and Applications 2004] Left tangential interpolation Left tangential interpolation interpolates @ if

12. 12 Tangential Interpolation Right tangential interpolation Right tangential interpolation Interpolation guides MOR Interpolation guides MOR –P(s)/Q(s) and Z p point to the I/O behavior of interest interpolates @ if

13. 13 Subspace Projection Standard Krylov subspace projection Standard Krylov subspace projection Projection for (right) tangential interpolation Projection for (right) tangential interpolation Interpolation matrix polynomial

14. 14 Subspace Projection Projection for (right) tangential interpolation Projection for (right) tangential interpolation

15. 15 Subspace Projection V V A A R R A2A2 A2A2 R R A q-1 R R … m Q(s) Q Q m r Q Q V’

16. 16 Find Q(s) Find Q(s) Use frequency-dependent correlation observed at the I/O ports Use frequency-dependent correlation observed at the I/O ports Port Correlation

17. 17 Port Correlation Find frequency dependent correlation using SVD Find frequency dependent correlation using SVD U r (s) represents the dominant row/column directions in H(s) U r (s) represents the dominant row/column directions in H(s) –Choose U r (s) as the interpolation matrix polynomial –Use moments to make it simple to compute Reciprocity Use this as the interpolation matrix

18. 18 Port Correlation Expand transfer function Expand transfer function Perform SVD Perform SVD Get the interpolation matrix polynomial Get the interpolation matrix polynomial

19. 19 Implementation Vectors in the projection matrix must be computed explicitly Vectors in the projection matrix must be computed explicitly –May cause numerical instability for a high-order moment matching Use different orders for moment matching and interpolation Use different orders for moment matching and interpolation

20. 20 Implementation Limit the explicit computation to the first g blocks Limit the explicit computation to the first g blocks Fall into a Krylov subspace Apply orthogonalization

21. 21 Experimental Results Coupled two RC lines Coupled two RC lines –2,593 elements, 1000 nodes, 98 current source inputs PRIMA PRIMA –95 th order reduced model –Compute 98 zero-th order moments then SVD them to 95 –Model generation time: 0.45s (Matlab) SVDMOR SVDMOR –6 th order reduced model –Expand TF at DC –Model generation time: 0.42s McPack McPack –6 th order reduced model –Expand TF at DC –Use a degree-2 Q(s) –Model generation time: 0.54s

22. 22 Experimental Results Two transimpedances Two transimpedances Z12,20Z41,20

23. 23 Experimental Results RC mesh RC mesh –1,600 nodes, 3,378 RC elements, 169 current sources distributed in the mesh Compute three 210 th order reduced models Compute three 210 th order reduced models PRIMA PRIMA –Compute 169 zero-th order moments + 41 vectors (169 1 st order moments SVD’ed to 41 vectors) –Model generation time: 2.39s (Matlab) SVDMOR SVDMOR –Uses a rank-70 approximation of DC transfer function –Model generation time: 2.09s McPack McPack –Uses a frequency-dependent rank-70 approximation Q(s) is of degree-2 –Model generation time: 3.73s

24. 24 Experimental Results Two transimpedances Two transimpedances Z10,10Z45,10

25. 25 Experimental Results Densely coupled 64-bit(128-port) RLC bus Densely coupled 64-bit(128-port) RLC bus –1,344 nodes, 1,280 resistors/inductors, 5,310 mutual inductors, 16,840 coupling capacitors PRIMA PRIMA –228 th order reduction model –128 zero-th order moments @ DC + 100 packed zero-th order moments @ 20GHz –Model generation time: 5.7s SVDMOR SVDMOR –204 th order reduced model –Expand @DC and 20GHz and use 68 SVD’ed vectors to approximate input/output matrices –Model generation time: 7.4s McPack McPack –200 th order reduced order model –Multi-point interpolation @DC and 20GHz –Model generation time: 10.6s

26. 26 Experimental Results Compute the 128-port Y-parameter model Compute the 128-port Y-parameter model Then apply a voltage input to the 20 th bit line Then apply a voltage input to the 20 th bit line Near end of the 20 th bitfar end of the 21 st bit

27. 27 Conclusion Networks with a large number of ports are extremely difficult to reduce Networks with a large number of ports are extremely difficult to reduce An algorithm based on frequency-dependent port packing is presented An algorithm based on frequency-dependent port packing is presented Port correlation is extracted by performing SVD on transfer function moments Port correlation is extracted by performing SVD on transfer function moments Model order reduction is achieved via tangential interpolation Model order reduction is achieved via tangential interpolation Improvement on accuracy has been observed for several circuits examples with many of I/O ports Improvement on accuracy has been observed for several circuits examples with many of I/O ports