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BSMOR: Block Structure-preserving Model Order Reduction

http//:eda.ee.ucla.edu

Hao Yu, Lei HeElectrical Engineering Dept., UCLA

Sheldon S.D. TanElectrical Engineering Dept., UCR

This project is founded NSF and UC-Micro fund from Analog Devices

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Motivation Deep submicron design needs to consider a large

number of linear elements Interconnect, Substrate, P/G grid, and Package

Accurate extraction leads to the explosion of data storage and runtime

Need efficient macro-model

Nonlinear Elements

Linear Elements

Nonlinear Elements

Reduced Model

Model Order Reduction

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Outline

Review of Model Order Reduction Grimme’s projection theorem PRIMA

Block Structure-preserving Model Order Reduction

Experiment Results Conclusions and Future Work

BackgroundState variable Input

Output

( ) ( ) ( )

( ) ( )

Gx s sCx s Bu s

y s Lx s

MNA Matrix

Krylov subspace

2span , , ,x R AR A R

1 10 0( ) , ( )A G s C C R G s C B

2 1, ; , , ,

( / )

q

p

K A R q span R AR A R A R

q ceil n n

The qth-order Krylov subspaceThe qth-order Krylov subspace

n: dimension of the spanned spacen: dimension of the spanned spacennpp: number of ports: number of ports

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IfIf

Grimme’s Projection Theorem

);,(},...,{ 1 qRAKvvspanV q

AVVAVRRLVL

R)A(I-sL(s):H

RA)H(s):L(I-s

TT

-

-

ˆ,ˆ,ˆ

ˆˆˆˆ

H(s) of moments qfirst themacthes Vby (s)H projected The

1

1

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PRIMA

To improve matching accuracy Apply Arnoldi orthnormalization to obtain independent basis

To preserve passivity Project G and C respectively in form of congruence transformation

Tq qV V I

^ ^ ^ ^, , ,T TL VL B VB G V GV C V CV

TqV

A qVA

NxNNxN

nxnnxn

NxnNxn

nxNnxN

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Limitation of PRIMA

Flat-projection loses the substructure information of the original state matrices Original state matrices are sparse, but reduced state matrices

are dense It becomes inefficient to match poles for structured state

matrices

It can not handle large number of ports efficiently Accuracy degrades as port number increases Reduced macro-model in form of flat port matrix is too large and

dense to analyze

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Our Contribution Basic Idea

Explore the substructure information by partitioning the state matrices

Partition the projection matrix accordingly and construct a new block-structured projection matrix

Properties Reduced model matches mq poles for the block diagonal state

matrices Reduced model matches q dominant poles exactly and (m-1)q

poles approximately for general state matrices with an additional block diagonalization procedure

Reduced state matrices are sparse Reduced model can be further decomposed into blocks, each

with a small number of ports

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BSMOR Flow

B lo c k D ia g o na liza tio n

B lo ck -d ia g o n a l-d o m in an tS ta te M a trix

B lo ck S tru c tu re -p reserv in gM o d el R ed u ctio n

B lo c k S truc ture -pre s e rve dM a c ro -m o de l

B o r de r e d-bl o c k-di ag o nal D e c o m po s i t i o nC l us te r i ng o f P o r ts

D e c o m po se d B lo c kM a c ro -m o de l

InputS ta te M a trix

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Outline

Review of Model Order Reduction Block Structure-preserving Model Order

Reduction BSMOR Method Properties of BSMOR Bordered-block-diagonal Decomposition

Experiment Results Conclusions and Future Work

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BSMOR Method Given m blocks within G, C and B matrices (block-

diagonal-dominant)11 1 11 1 1

1 1

, ,m m

m mm m mm m

G G C C BG C B

G G C C B

1 1 0 00 00 0m m

v vV V

v v

Construct a new projection matrix V-tilde with the block structure accordingly based on V from PRIMA

Block Structure-preserved Projection

BVBVCVCVGVG ~~,~~~,~~~

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Properties (I)

q-moment matching

Passivity preservation

matches the first q moments of :

( , ; )q

q q

H s H s

K A R q V V

( ) 0, ( ) 0

Tq q

T T T Tq q q q

V V I

V G G V V C C V

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Properties (II)

Block structure-preserving Results in a sparse reduced matrices (but not PRIMA) Enable further block-ports decomposition (but not PRIMA)

1 11 1 1 1

1 11 1 1 1 1

ˆ: , :

T T T Tm m m m

T Tq q i ij j

i jT T T Tm m m

v G v v G vBSMOR G PRIMA G v G v

v G v v G v

mmm

m

GG

GGG

1

111

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Properties (III)

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11

11 11 1

1

[ ,..., ]

( ) 0 00 00 0 ( )

mm

Tq q

mm

T

Tm mm m

A V AV

diag A A

v G C v

v G C v

11If ( ) ( )

matches poles of mmNull eigen A eigen A

A mq A

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

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BBDC Analysis

11 1

1

( )m

m mm

Y YY s

Y Y

Resulted MIMO macro-model has preserved block structure but has dense couplings between blocks Each block is now represented by a subset of ports

Enable multi-level partioned solution by branch-tearing [Wu:TCAS’76]

Represent it into bordered-block-diagonal form with a global coupling block (with branch addmaince Y00)

1 111 10

01

010 0 00

0

00

m mmm mT T

m

V IY C

V IY CIC C Y

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Outline

Review of Model Order Reduction Block Structure-preserving Model Order

Reduction Experiment Results Conclusions and Future Work

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Sparsity Preservation

16x16-BSMOR shows 72% and 93% sparsity for G and C matrices of a 256x256 RC-mesh Matrices reduced by PRIMA are fully dense

Beforereduction

Afterreduction

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m x q Pole Matching

For a non-uniform mesh composed by 32 sub-meshes 8x8-BSMOR exactly matches 8 poles and closely matches additional 56

poles PRIMA only matches 8 poles

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Frequency Response

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

improved accuracy

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Reduction Time

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

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Simulation Time

The dense macro-model by PRIMA leads to a similar runtime growth as the original model

Level-(1,2) BBDC has a much slower growth (up to 30X simulation time reduction)

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Conclusions and Future Work

BSMOR achieves higher model reduction efficiency and accuracy by leveraging and preserving the structure information of the input state matrices

Reduced block can be further hierarchically analyzed that further boosts the efficiency

How to find the best way to do the block diagonalization

How to apply BSMOR to system that has strong inductive couplings

Updates available at http://eda.ee.ucla.edu