1. 1 BSMOR: Block Structure-preserving Model Order Reduction http//:eda.ee.ucla.edu Hao Yu, Lei He Electrical Engineering Dept., UCLA Sheldon S.D. Tan Electrical Engineering Dept., UCR This project is founded NSF and UC-Micro fund from Analog Devices

2. 2 Motivation n Deep submicron design needs to consider a large number of linear elements l Interconnect, Substrate, P/G grid, and Package n Accurate extraction leads to the explosion of data storage and runtime n Need efficient macro-model Nonlinear Elements Linear Elements Nonlinear Elements Reduced Model Model Order Reduction

3. 3 Outline n Review of Model Order Reduction l Grimme’s projection theorem l PRIMA n Block Structure-preserving Model Order Reduction n Experiment Results n Conclusions and Future Work

4. Background State variable Input Output n MNA Matrix n Krylov subspace The qth-order Krylov subspace n: dimension of the spanned space n p : number of ports

5. 5 If Grimme’s Projection Theorem

6. 6 PRIMA n To improve matching accuracy l Apply Arnoldi orthnormalization to obtain independent basis n To preserve passivity l Project G and C respectively in form of congruence transformation NxN nxn Nxn nxN

7. 7 Limitation of PRIMA n Flat-projection loses the substructure information of the original state matrices l Original state matrices are sparse, but reduced state matrices are dense l It becomes inefficient to match poles for structured state matrices n It can not handle large number of ports efficiently l Accuracy degrades as port number increases l Reduced macro-model in form of flat port matrix is too large and dense to analyze

8. 8 Our Contribution n Basic Idea l Explore the substructure information by partitioning the state matrices l Partition the projection matrix accordingly and construct a new block-structured projection matrix n Properties l Reduced model matches mq poles for the block diagonal state matrices l Reduced model matches q dominant poles exactly and (m-1)q poles approximately for general state matrices with an additional block diagonalization procedure l Reduced state matrices are sparse l Reduced model can be further decomposed into blocks, each with a small number of ports

9. 9 BSMOR Flow

10. 10 Outline n Review of Model Order Reduction n Block Structure-preserving Model Order Reduction l BSMOR Method l Properties of BSMOR l Bordered-block-diagonal Decomposition n Experiment Results n Conclusions and Future Work

11. 11 BSMOR Method n Given m blocks within G, C and B matrices (block- diagonal-dominant) n Construct a new projection matrix V-tilde with the block structure accordingly based on V from PRIMA n Block Structure-preserved Projection

12. 12 Properties (I) n q-moment matching n Passivity preservation

13. 13 Properties (II) n Block structure-preserving l Results in a sparse reduced matrices (but not PRIMA) l Enable further block-ports decomposition (but not PRIMA)

14. 14 Properties (III) Theorem: the reduced model matches mq poles, if G and C matrices are block diagonal Proof: the resulted Heisenberg matrix A-tilde is block diagonal

15. 15 BBDC Analysis n Resulted MIMO macro-model has preserved block structure but has dense couplings between blocks l Each block is now represented by a subset of ports n Enable multi-level partioned solution by branch-tearing [Wu:TCAS’76] n Represent it into bordered-block-diagonal form with a global coupling block (with branch addmaince Y 00 )

16. 16 Outline n Review of Model Order Reduction n Block Structure-preserving Model Order Reduction n Experiment Results n Conclusions and Future Work

17. 17 Sparsity Preservation n 16x16-BSMOR shows 72% and 93% sparsity for G and C matrices of a 256x256 RC-mesh n Matrices reduced by PRIMA are fully dense Before reduction After reduction

18. 18 m x q Pole Matching n For a non-uniform mesh composed by 32 sub-meshes n 8x8-BSMOR exactly matches 8 poles and closely matches additional 56 poles n PRIMA only matches 8 poles

19. 19 Frequency Response n Comparison of 2x2-BSMOR, 8x8-BSMOR and PRIMA l 8x8-BSMOR has best accuracy in all iterations of block Arnoldi procedures l Increasing block number leads to more matched poles and hence improved accuracy

20. 20 Reduction Time n Under the same error bound, BSMOR has 20X smaller reduction time than PRIMA l Fewer iterations are needed by BSMOR

21. 21 Simulation Time n The dense macro-model by PRIMA leads to a similar runtime growth as the original model n Level-(1,2) BBDC has a much slower growth (up to 30X simulation time reduction)

22. 22 Conclusions and Future Work n BSMOR achieves higher model reduction efficiency and accuracy by leveraging and preserving the structure information of the input state matrices n Reduced block can be further hierarchically analyzed that further boosts the efficiency n How to find the best way to do the block diagonalization n How to apply BSMOR to system that has strong inductive couplings Updates available at http://eda.ee.ucla.edu