EMPIRE: An Efficient and Compact MultipleParameterized Model Order Reduction
Method for Physical Optimization
Yiyu Shi, Lei He
Electrical Engineering Department,
UCLA http//:eda.ee.ucla.edu
This work is partially supported by NSF Career Award and a UC Micro
grant sponsored by Analog Devices, Intel and Mindspeed.
Outline

Background and motivation

EMPIRE algorithm

Experimental results

Conclusions
2
Parameterized MOR

Most physical design and optimization problems involve nonlinear
optimization

Decap allocation, shields insertion, thermal via planning, and structured P/G
clock network sizing
 Sensitivities are needed to linearize the nonlinear objection function

Parameterized model order reduction can generate macromodels with
all the parameters preserved

The moments of the parameters of the design (POD) are exactly the
sensitvities
 Moment matching
sensivity matching

Previous works

[Daniel:TCAD’04] extends PRIMA to handle parameterized systems


But can handle only a small number of parameters and match moments up to a
low order.
CORE [Li:ICCAD’05] uses explicit-and-implicit moment matching for
parameterized interconnect model reduction

Still cannot match the moments of a huge number of parameters to a very high
order
 Cannot match the moments of different parameters with different accuracy
3
Importance of High Order POD Moments

Output integral w.r.t. a randomly selected parameter (pitch width) for a
P/G mesh

The reduced model by CORE cannot match the original well when the
reduced order is less than 70.
4
Major Contribution of EMPIRE

It is an efficient yet accurate model order reduction method
for physical design with multiple parameters:

It uses implicit moment matching to match high order POD
moments,
 more
accurate than the explicit moment matching used in CORE;

It can match the moments of different PODs with different accuracy
according to their influence on the objective.

Experimental results show that compared with CORE and
[Daniel:TCAD’04], EMPIRE reduces error by 47.8X at a similar
runtime.
5
Outline

Background

EMPIRE algorithm

Experimental results

Conclusions
6
Framework of EMPIRE
7
Parameter Number Reduction

Canonical form of a general parameterized system
(E0 + E1s1 + E2s2 + … + Etst) x = Bu
y=LTx,

Define the significance of a parameter si as
SIG(si ) || Ei ||2 sˆi
Any value
in the
range of si

Theorem 1: SIG(si) is the perturbation magnitude of si to
the output.

Therefore, we can neglect the parameters that have
relative small SIG values and thus reduce the total number
of parameters.
8
Verification

Normalized output perturbation w.r.t. the 2-norm of the coefficient
matrix

With the increase of the norm, the perturbation increases.
9
Framework of EMPIRE
10
Projection Space Collapse

Find the original projection matrix Vˆ  [vˆ1 , vˆ2 ,...vˆ p ] by the
traditional algorithms

Calculate a new projection matrix V so that the weighted
distance between colspan(V) and colspan(Vˆ) is minimized,
i.e.,
q0
p
min . d (V ,Vˆ )  W  || vˆ  v ||
i 1

i
j 1
Three different methods



Nonlinear Programming (NP)
Sequential Least Square (sLS)
Sequential Barycenter Allocation (sBA)
NP
runtime
accuracy
sLS
sBA
i
j
2
Directly solve
optimization
problem
Iteratively solve
the optimization
problem
Use quadratic
approximation
11
Three different methods

Find a new projection matrix V so that the weighted
distance between colspan(V) and colspan(Vˆ) is minimized,
i.e.,
q0
p
min . d (V ,Vˆ )  W  || vˆ  v ||
i 1



j
2
Directly solve the optimization problem
Expensive but provides optimal solution
Can be used for small scale problem
Sequential Least Square (sLS)



j 1
i
Nonlinear Programming (NP)


i
Solve the optimization problem incrementally
Each time find one column vector in Vˆ that has the smallest
distance to the vectors in V orthogonalized by the ones already
found.
Sequential Barycenter Allocation (sBA)

Use the barycenter to approximate the optimal solution of sLS.
12
Verification

The weighted distance between two subspaces w.r.t the reduced order

With the increase of the reduced order, the distance decreases to zero.

sBA converges to zero the slowest.
 NP converges to zero the fastest
13
Framework of EMPIRE
14
Frequency Domain Moment Expansion & Projection

Frequency domain moments are critical to the waveform
accuracy
matching more moments!

Let
1
0
As  E Es
Coefficient matrix
corresponding to
frequency
variable
and
1
0
Rs  E B , then
V  [V , Rs , As Rs ,... A Rs ]

Projection
15
q
s

T
Ei  V EiV , i

B  V T B,

L  V T L.
Match up to q-th
order of frequency
domain moments
Outline

Background

EMPIRE algorithm

Experimental results

Conclusions
16
Experimental Settings

We use different sizes of extracted RLC meshes from
industrial applications.

All the algorithms are implemented in MATLAB

We use a Linux workstation (P4 2.66G CPU and 2G RAM)

We compare the runtime, time/frequency domain accuracy
and scalability of our hybrid algorithm with
[Daniel:TCAD’04] and CORE.
17
Waveform Comparison
(a)
(b)

P/G RC meshes with 10000 nodes and 5000 parameters (pitch width)

EMPIRE is identical to the original in both time domain (a) and
frequency domain (b) , more accurate compared with CORE and
[Daniel:TCAD’04].
18
Waveform Comparison

Output integral w.r.t. a randomly selected parameter (pitch width)

EMPIRE is close to the original, more accurate than CORE and
[Daniel:TCAD’04]
19
Scalability Comparison

Time domain waveform relative error w.r.t reduction size

EMPIRE has less error and coverges faster than CORE.
20
Runtime for EMPIRE

Runtime for EMPIRE w.r.t different original circuit size



A: NP (small scale problems)
B: sLS (medium scale problems)
C: SBA (large scale problems)
21
Runtime Comparison

Runtime comparison between the three methods on RC
meshes of different scales.

EMPIRE has a similar runtime compared with CORE, and is 18.3X
faster than [Daniel:TCAD’04] for model reduction time and 61.2X
faster for simulation time.

In addition, [Daniel:TCAD’04] cannot finish large examples.
22
Conclusions and future work

We have developed an efficient yet accurate model order
reduction method EMPIRE for physical design with multiple
parameters:

Compared with CORE, with a small reduction size, it uses implicit
moment matching to match high order POD moments,
 more

accurate than the explicit moment matching used in CORE;

It can match the moments of different PODs with different accuracy
according to their influence on the objective.

Experimental results show that compared with CORE and
[Daniel:TCAD’04], EMPIRE results in 47.8X improved accuracy at a
similar runtime.
We will extend the EMPIRE algorithm and apply it in real
physical design problems.
23
24
Thank You!