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1
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
2
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
3
Outline
Review of Model Order Reduction Grimme’s projection theorem PRIMA
Block Structure-preserving Model Order Reduction
Experiment Results Conclusions and Future Work
Background
State 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 a n tS ta te M a trix
B lo ck S tru c tu re -p reserv in gM o d e l 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 B
G C B
G G C C B
1 1 0 0
0 0
0 0m m
v v
V 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 1
1 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 v
BSMOR G PRIMA G v G v
v G v v G v
mmm
m
GG
GG
G
1
111
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Properties (III)
11
11
11 11 1
1
[ ,..., ]
( ) 0 0
0 0
0 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 Y
Y 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
0
1010 0 00
0
0
0m mmm m
T Tm
V IY C
V IY C
IC 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
20
Reduction Time
Under the same error bound, BSMOR has 20X smaller reduction time than PRIMA Fewer iterations are needed by BSMOR
21
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