Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON...
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![Page 1: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/1.jpg)
Adjoint Transient Sensitivity Analysis in Circuit
Simulation
Zoran Ilievski
22nd Nov, 2006COMSON RTN
![Page 2: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/2.jpg)
2
Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
![Page 3: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/3.jpg)
3
Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
![Page 4: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/4.jpg)
4
COupled Multiscale Simulation and Optimization
in Nanoelectronics
COMSON
• Eindhoven
• Wuppertal
• Bucharest
• Calabria
• Catania
• Infineon
• NXP / Philips
• STMicroelectronics
![Page 5: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/5.jpg)
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COMSONMain
Objectives• Develop as Software, a Demonstrator Platform
• Coupled simulation of devices, interconnects,
circuits, EM fields and thermal effects
• WP1 - Mathematical Modelling and Analysis • WP2 - Demonstrator Platform• WP3 - Model Order Reduction• WP4 - Optimisation• WP5 - E-learning
Structure
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COMSONTU/e responsibilities – Model Order
Reduction
• Parameter analysis (BUC)
• Nonlinear circuits (WUP)
• Implementation (WUP, BUC)
WP3:Coordination by NXP
Partners
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Circuit Equations)())(())(( tt
dt
dt sxqxj
),()),,(()),,(( psppxqppxj ttdt
dt
x(t) = States j(x(t)) = Current
q(x(t)) = Charge s(x(t)) = Signal
p = some parameter
![Page 8: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/8.jpg)
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Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
![Page 9: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/9.jpg)
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Transient Sensitivity Analysis - TSA
A
LR
d
AKC P0
• Each circuit has an intended functionality
• Circuits are made in silicon (SoC), can be represented as a circuit diagram
• It could react to small changes in parameter values
• For example, width-length of a resistor
![Page 10: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/10.jpg)
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TSA – Typical Problems in which parameter variation plays a role
Power dissipation
Threshold value
• Important to manage available power to realize a functionality
• Change in parameters will affect this.
• Timing in circuits is VERY important.
• Change in parameter could cause a desired functionality to perform with a delay.
![Page 11: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/11.jpg)
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Observation Function
T
dtt0
)),,(()),(( ppxFppxG
p
pxpx
),(
),(ˆt
t
Example (power dissipation):• F, Power
• G, Energy
Sensitivity to a parameter p:
T
dtt
tt
d
d0
)),,((),(ˆ
)),,(()),((
p
ppxFpx
x
ppxF
p
ppxG
Inner Product
![Page 12: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/12.jpg)
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),(ˆ)),,((
pxx
ppxFt
t
Inner Product Cost
F
P
N
p)p),,F(x(
p
)x(
t
t
Inner product is calculated at each time point, very expensive.
The inner product
FPNNPF,minO 2 + costs for ),(ˆ px t
The cost
![Page 13: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/13.jpg)
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Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
![Page 14: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/14.jpg)
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• Alternative to previous method
• Enables a cheap and clever calculation of the inner product.
• Introduction of a function λ*(t)
• λ*(t) has a related DAE that needs to be solved.
Backward Adjoint Method - BAM
(* = complex transpose)
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BAM – Inner Product Cost Reduction
T
0
**
*
*T
0
*
dp
p),ds()(λ
p
p)p),,q(x(
dt
)(dλ
p
p)p),,j(x()(λ
p
p)p),q(x(t,p)(t,xC)(λpxC
dt
dλ)G(λ
dtt
tttt
t
tdttt
t
T
0
*
ˆ),(ˆ)(
Then partial integrating
),()),,(()),,(( psppxqppxj ttdt
dt
0),()),,(()),,((
)(0
*
dt
d
td
d
td
d
td
dt
dt
T
p
ps
p
ppxj
p
ppxq
Multiply circuit equations by λ*(t) diff. w.r.t p
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T
0
**
*
*T
0
*
dp
p),ds()(λ
p
p)p),,q(x(
dt
)(dλ
p
p)p),,j(x()(λ
p
p)p),q(x(t,p)(t,xC)(λpxC
dt
dλ)G(λ
dtt
tttt
t
tdttt
t
T
0
*
ˆ),(ˆ)(
BAM – Inner Product Cost Reduction
Let λ(t) satisfy the following adjoint equation
*** )),,((
)()(
x
ppxFGC
tt
dt
td
T
dtt
tt
d
d0
)),,((),(ˆ
)),,(()),((
p
ppxFpx
x
ppxF
p
ppxG
![Page 17: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/17.jpg)
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• is avoided by setting the initial condition (valid for DAE index =1)
• is easily found from the steady state DC condition, cheapest calculation
• Backward integration of adjoint equation enables that calculation of
),(ˆ px T
0)( T
),0(ˆ px
BAM – Inner Product Cost Reduction
*** )),,((
)()(
x
ppxFGC
tt
dt
td
)(t
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BAM – Inner Product Cost Reduction
dtd
tdt
t
dt
td
d
d
T
p
pxj
p
ps
p
ppxF
p
pxq
p
pxqx
x
pxq
p
ppxG
),(),()(
)),,((),()(
),(ˆ),(
)0()),((
**
0
00
0*
Calculate the following:)(t )(txdt
td )(
Substitution in the above will give the sensitivity.
Step 1:
Step 2:
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Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
![Page 20: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/20.jpg)
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BAM – Steps and ImplementationForward step: Calculation of x(t)
Ttttt xxx ,...,,0
Euler-Backward application to circuit equations
p),s(p)p),,j(x()q(x)q(xΔ
1 n1n ttt
• Newton-Raphson iteration gives with Newton matrix:
• Time step control
1nx
GCt
Y
1
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BAM – Steps and ImplementationBackward step: Calculation of )(t
*** )),,((
)()(
x
ppxFGC
tt
dt
td
• Backward differential formulation
• Ideally choose same step size as forward integral, eliminates interpolation errors.
• C and G matrices are time dependent, if original circuit is linear they retain the same value at each time t. Only need to calculate them once.
x
qC
x
jG
0Tt
GCt
Y
1
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BAM – Steps and Implementation
*** )),,((
)()(
x
ppxFGC
tt
dt
td
Validation of anddt
td )()(t
Estimation of dttd )(
Δt
)tλ(t
dt
)tdλ 010
(
0t
t2Δ
)tλ(t
dt
)tdλ 1r1rr
(
Tt 0
Δt
)tλ(t
dt
)tdλ n1n1n (
Tt
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BAM – Steps and Implementation
dtd
tdt
t
dt
td
d
d
T
p
pxj
p
ps
p
ppxF
p
pxq
p
pxqx
x
pxq
p
ppxG
),(),()(
)),,((),()(
),(ˆ),(
)0()),((
**
0
00
0*
Substitution now give the sensitivities
Application of trapezoidal rule for the integral
![Page 24: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/24.jpg)
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Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
![Page 25: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/25.jpg)
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Results 1= 2=0.025 ohm meters
A1=A2=0.0001 meters^2
L1=0.02 meters
Ap1=1*10^-6 meters^2
d1=1*10^-6 metersCalculating for L2=0.02,0.021,0.022m
and observing the 2nd resistor
L2 dG/dp 1*10^-10
0.02 -0.235
0.021 -0.247
0.022 -0.256
![Page 26: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/26.jpg)
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Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
![Page 27: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/27.jpg)
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)(tCalculation of
23 FNONO
Conclusions and further workHow fast is the adjoint method?
FPNNPF,minO 2
Original cost of inner product
Calculation of modified integral
FPPNO
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Conclusions and further work• If the circuit description is linear, BAM is immediately
attractive, only one LU decomposition is carried out.
• Remaining main burden is the O(N^3) LU decomposition in the backward integration for non-linear circuits where the time-varying C+G matrices force an LU decomposition at each time step.
• Noticing the output r = F x P << N this suggest a model order reduction approach could be taken for non-linear circuits.
![Page 29: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/29.jpg)
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• In MOR we are looking for a matrix such that
• Which we can then use to reduce C & G
• Which MOR method could we use?
RNV xVx ~
Conclusions and further work
*** )),,((
)()(
x
ppxFGC
ttVV
dt
tdVV TT
![Page 30: Adjoint Transient Sensitivity Analysis in Circuit Simulation Zoran Ilievski 22 nd Nov, 2006 COMSON RTN.](https://reader035.fdocuments.us/reader035/viewer/2022062314/56649e0d5503460f94af641a/html5/thumbnails/30.jpg)
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• Proper orthogonal decomposition is ideal
• POD requires snap shots of the output of a system, which we have.
Conclusions and further work
0tx 1tx 2tx