Interconnect Focus Center

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Model Reduction for Nonlinear and Parameterized
Systems
J. White Slides thanks to D. Luca, M. Reichelt,
M. Rewienski, D. Vasilyev
Interconnect Focus Center
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A Multitechnology Phase-Locked Loop
CNT FET
Micromachined Resonator
Bachtold, et al., Science, Nov. 2001
www.discera.com
Phase Detector
Loop Filter
VCO
Divider
Opto-electrical transducers
Kimerling Group

• Evaluating the New Technology
• What is system performance (capture, lock,
  noise, etc)?
• What is the impact of modifying technology
  parameters?
• How tight must manufacturing tolerances be?

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CAD for Diverse IC Technology

• Initial Assessment
• What is possible with a combination of
  technology?
• Will new technology improve SYSTEM performance?
• Requires a rough optimization step!
• System Performance optimization
• Assess intra and inter technology trade-offs .
• What is the impact of fabrication decisions?
• Automate Analysis and Synthesis/Optimization
• Manufacturability/Yield optimization
• Optimize design considering variations!

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Need to Assess and Optimize System Performance

• Hierarchical Simulation
• Encapsulate the physics.
• Automatically move between hierarchical levels.
• Approach must apply given diverse technology.
• Hooks for Synthesis/Optimization
• Compute Performance Sensitivities to
• Fabrication decisions
• Layout modifications
• Architectural Changes.
• Manufacturability/Yield
• Optimize design considering variations!

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Goal Optimize Technology for the Application
CNT FET
Micromachined Resonator
Bachtold, et al., Science, Nov. 2001
www.discera.com
Phase Detector
Loop Filter
VCO
Divider
Opto-electrical transducers
Kimerling Group

• Need to simulate ENTIRE system with dynamically
  accurate models for ALL the components
• Capture Simulation will require thousands of
  oscillator cycles

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Multiphysics Simulation Approach

• Circuit
• Ordinary differential equation solver
• Carbon Nanotube Transistor
• Molecular Dynamics or Atomistic Simulation
• Microresonator
• Coupled 3-D Electro-Elasto-Fluidic Simulation
• Optical Transducers
• 3-D Coupled Device-Optics Simulator
• Interconnect Substrate
• 3-D Full-Wave Simulation

Capture Simulation of thousands of cycles will
never finish! Must Generate Macromodels
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Macromodel Generation Now Done By Hand
Will Never Keep Up With Diverse Technology
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The Numerical Macromodeling Paradigm
Generate a Reduced-Order Model Directly from 3-D
Geometry and Physics
Automatic
Lundstrom et al.
Low order state-space model which captures input
(u)/output(y) behavior
Complicated Geometry, Coupled Physics, possibly
even statistical
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Whats Needed For Numerical Macromodeling
1) Fast Coupled Domain 3-D Solvers

• Fluids, EM Fields, mechanics, Transport
• Must handle ENTIRE Devices!

2) Model-Order Reduction

• Start with a Meshed 3-D Structure (gt100,000
  DOFs)
• Or Start with molecular positions
• Automatic generation of low-order model (lt100
  DOFs)

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Where Are We Now?
Linear, Few Port Problem is Getting there.

• Fast 3-D E-M Solvers
• Multipole, Hierarchical SVD, Precorrected-FFT,
  Wavelets
• Efficient MOR
• Krylov, Krylov-TBR, Projection methods
• Still Issues
• Passivity
• Performance for Distributed Systems

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State-Space Description

• Original Dynamical System - Single Input/Output
• Reduced Dynamical System q ltlt N, but I/O preserved

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Projection Framework
Change of variables
Equation Testing
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Forming the Reduced Matrix
qxq

NxN

• No explicit A need, Only Matrix-vector products

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Picking U and V

• Use Eigenvectors
• Use Time Series Data
• Compute
• Use the SVD to pick q lt k important vectors
• Use Frequency Domain Data
• Compute
• Use the SVD to pick q lt k important vectors
• Use Krylov Subspace Vectors?
• Use Singular Vectors of System Grammians?

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Moment Matching Theorem
If
And
Then
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Point Matching Versus Moment matching
Point matching can be very inaccurate in between
points
Moment (derivatives) matching accurate around
expansion point, but inaccurate on wide
frequency band
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A Moment Matching Perserving Alternative
First Invert A before applying reduction
Form reduced model by projecting inverse of A
The Projection Theorem Still Holds!!
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A inverse system
N100
q1
q2
Exact
Matches q moments
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A system
N100
q1
q3
Exact
Matches q moments
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Easy to Model Even Complicated Frequency Behavior

• Krylov subspace methods (red)
• Excellent match over a narrow range of
  frequencies
• SVD of Hankel Operator (TBR) (blue)
• Minimizes worst case frequency domain error
• Recently developed fast algorithms (CFADI).

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Interconnect in Timing Analysis Long Time
Behavior is important
Interconnect
Gate
Gate
load model
Gate model
Interconnect model
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Standard Approach Match Moments
Stamp Into Matrices
Reduce Using Prima
n
q
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Standard Approach Continued
Reduce Using Prima
Solve the reduced system
Small System delay easily estimated
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Drives and Loads keep Changing
Rdrive a function of load, low rank perturbation
Gate model
Rdrive
Generates same model as rereducing with updated G
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Coupling to Floating Line
Nomial extraction uses dummy load
Rank-one update extracts dummy load
Rank-one update Avoids Singular G matrix!
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Motivation Example RF micro-inductor

• How are the substrate eddy currents affecting the
  quality factor of the inductor?
• How are the displacement currents affecting the
  resonance of the inductor?
• Need to capture all 2nd order effects

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Model Order Reduction for LINEARLY Parameterized
Systems

• Given a large parameterized linear system

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Interpolation Approaches Generalize
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Nonlinear MOR Representation Problem

• Nonlinear dynamical systems

• Projection of the nonlinear operator f(x)

x
f(.)
f(x)
V space
V space
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Problems with MOR for nonlinear
to

• Substitute

• Using VTf(Vz) is too expensive!

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Volterra Approach
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Trajectory Piecewise Linear approximation of f.
Training trajectory
x0
x2
x1

wi(x) is zero outside circle
xn
Simulating trajectory
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Projection and TPWL approximation yields
efficient f r
q x 1
Air
Ai
q
V

Air
q
n
n
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TPWL approximation of f. Extraction algorithm

• Compute A1
• Obtain W1 and V1 using linear reduction for A1
• Simulate training input, collect and reduce
  linearizations Air W1TAiV1 f r
  (xi)W1Tf(xi)

Initial system position
x0
x2
x1

xn
Training trajectory
Non-reduced state space
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Example problem
RLC line
Linearized system has nonsymmetric, indefinite
Jacobian
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Numerical results nonlinear RLC transmission
line
System response for input current i(t)
(sin(2p/10)1)/2

• Input

training input
testing input
Voltage at node 1 V
Time s
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Key issue choosing projection
Krylov-subspace methods
Balanced-truncation methods
Result projection matrices W and V
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Numerical results RLC transmission line
TBR-based TPWL beat Krylov-based 4-th order
TBR TPWL reaches the limit of TPWL representation
Error in transient
yr y2
Order of the reduced model
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Micromachined device example
FD model
non-symmetric indefinite Jacobian
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TPWL-TBR results MEMS switch example
Errors in transient
Unstable!
Odd order models unstable! Even order models
beat Krylov
yr y2
Why???
Order of reduced system
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Eigenvalue behavior of linearized models
Eigenvalues of reduced Jacobians, q8
Eigenvalues of reduced Jacobians, q7
TBR is adding complex-conjugate pair
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Explanation of even-odd effect Problem
statement
Consider two LTI systems
Initial ( )
Perturbed ( )
TBR reduction
TBR reduction

Projection basis V
Projection basis V
Define our problem How perturbation in the
initial system affects projection basis?
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Hankel singular values, MEMS beam example
This is the key to the problem. Singular values
are arranged in pairs!
of the Hankel singular value
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Explaining even-odd behavior
The closer Hankel singular values lie to each
other, the more corresponding eigenvectors of V
tend to intermix!

• Analysis implies simple recipe for using TBR
• Pick reduced order to insure
• Remaining Hankel singular values are small enough
• The last kept and first removed Hankel Singular
  Values are well separated
• Helps insure that all linearizations stably
  reduced

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Many Methods Under Investigation

• Projection Methods
• Data Mining
• Support Vector Machines
• Nonlinear Generalizations of Controllability and
  Observability
• Finite-State Automata
• Sophisticated Sampling and Fitting

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Massively Coupled Effects

• Digital Narrow Signal Range 20db
• Effective to Screen Small couplings
• Analog Wide Signal Dynamic Range 80db
• Small couplings must be retained
• Analog Block 1000s of interacting interconnect
  lines
• Millions of Coupling terms

Massively Coupled Problem!
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Still to Come Massively Coupled Interconnect
Analysis
Courtesy of Harris Semiconductor

• Need to draw a box and extract everything
• Including all the small couplings
• Extracted Result must be efficient in a simulator
• Will try to use SVD based methods plus model
  order reduction
• SVD for the geometric coupling
• MOR for the frequency dependence

Still Massively Coupled Problem-- But New
Approaches!
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Impact of Reliable nonlinear MOR

• Automatic Compact Model Generation

PDEs D-D, Schrod, Etc.
Q-V, I-V equations

• Multiscale modeling?

PDEs D-D, Schrod, Etc.
Atomic-level

• New device/technology models

Valve