Adjoint Transient Sensitivity Analysis in Circuit Simulation

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Adjoint Transient Sensitivity Analysis in Circuit
Simulation

• Zoran Ilievski
• 22nd Nov, 2006
• COMSON RTN

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Overview

• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work

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Overview

• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work

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COMSON

• COupled Multiscale Simulation and Optimization in
  Nanoelectronics

• Eindhoven
• Wuppertal
• Bucharest
• Calabria
• Catania

• Infineon
• NXP / Philips
• STMicroelectronics

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COMSON

• 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

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COMSON
WP3Coordination by NXP

• Parameter analysis (BUC)
• Nonlinear circuits (WUP)
• Implementation (WUP, BUC)

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Circuit Equations
x(t) States j(x(t))
Current q(x(t)) Charge s(x(t)) Signal
p some parameter
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Overview

• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work

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Transient Sensitivity Analysis - TSA

• 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

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TSA Typical Problems in which parameter
variation plays a role

• 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.

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Observation Function

• F, Power
• G, Energy

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Inner Product Cost
Inner product is calculated at each time point,
very expensive.
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Overview

• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work

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Backward Adjoint Method - BAM

• Alternative to previous method
• Enables a cheap and clever calculation of the
  inner product.
• Introduction of a function ?(t)
• ?(t) has a related DAE that needs to be solved.

( complex transpose)
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BAM Inner Product Cost Reduction
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BAM Inner Product Cost Reduction
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BAM Inner Product Cost Reduction

• 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

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BAM Inner Product Cost Reduction
Calculate the following
Substitution in the above will give the
sensitivity.
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Overview

• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work

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BAM Steps and Implementation
Forward step Calculation of x(t)
Euler-Backward application to circuit equations

• Newton-Raphson iteration gives with Newton
  matrix
• Time step control

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BAM Steps and Implementation

• 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.

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BAM Steps and Implementation
Estimation of
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BAM Steps and Implementation
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Overview

• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work

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Results
? 1 ? 20.025 ohm meters A1A20.0001
meters2 L10.02 meters Ap1110-6
meters2 d1110-6 meters
Calculating for L20.02,0.021,0.022m and
observing the 2nd resistor
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Overview

• Introduction (circuit analysis, COMSON)
• Transient Sensitivity, the direct approach
• The backward adjoint method
• Implementation
• Results
• Conclusions and further work

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Conclusions and further work
How fast is the adjoint method?
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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(N3) LU
  decomposition in the backward integration for
  non-linear circuits where the time-varying CG
  matrices force an LU decomposition at each time
  step.
• Noticing the output r F x P ltlt N this suggest a
  model order reduction approach could be taken for
  non-linear circuits.

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Conclusions and further work

• 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?

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Conclusions and further work

• Proper orthogonal decomposition is ideal
• POD requires snap shots of the output of a
  system, which we have.