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.

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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

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