adjoint transient sensitivity analysis in circuit simulation zoran ilievski 22 nd nov, 2006 comson...
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Adjoint Transient Sensitivity Analysis in Circuit
Simulation
Zoran Ilievski
22nd Nov, 2006COMSON RTN
2
Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
3
Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
4
COupled Multiscale Simulation and Optimization
in Nanoelectronics
COMSON
• Eindhoven
• Wuppertal
• Bucharest
• Calabria
• Catania
• Infineon
• NXP / Philips
• STMicroelectronics
5
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
6
COMSONTU/e responsibilities – Model Order
Reduction
• Parameter analysis (BUC)
• Nonlinear circuits (WUP)
• Implementation (WUP, BUC)
WP3:Coordination by NXP
Partners
7
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
8
Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
9
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
10
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.
11
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
12
),(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
13
Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
14
• 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)
15
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
16
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
17
• 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
18
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:
19
Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
20
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
21
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
22
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
23
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
24
Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
25
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
26
Overview
• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work
27
)(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
28
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.
29
• 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
30
• 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
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