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Circuit Simulation using Matrix
Exponential Method
Shih-Hung Weng, Quan Chen and Chung-Kuan Cheng
CSE Department, UC San Diego, CA 92130
Contact: [email protected]
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Outline
• Introduction
• Computation of Matrix Exponential Method
– Krylov Subspace Approximation
– Adaptive Time Step Control
• Experimental Results
• Conclusions
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Circuit Simulation
• Numerical integration
– Approximate with rational functions
– Explicit: simplified computation vs. small time steps
– Implicit: linear system derivation vs. large time steps
– Trade off between stability and performance
• Time step of both methods still suffer from accuracy
– Truncation error from low-order rational approximation
• Method beyond low-order approximation?
– Require: scalable and accurate for modern design
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Statement of Problem
• Linear circuit formulation
Cx(t )  Gx(t )  u(t )
• Let A=-C-1G, b=C-1u, the analytical solution is
x(t  h)  e Ah x(t )  e A (t  h )
t h

e  A b( )d
t
• Let input be piecewise linear
e

x(t  h)  x(t ) 
Ah
 I
A
 Ax(t )  b(t ) 
e


Ah
  Ah  I   b(t  h)  b(t )
A2
h
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Statement of Problem
• Integration Methods
– Explicit (Forward Euler): eAh => (I+Ah)
“Simpler” computation but smaller time
steps
– Implicit (Backward Euler): eAh => (I-Ah)-1
Direct matrix solver (LU Decomp) with
complexity O(n1.4) where n=#nodes
– Error derived from Taylor’s expansion
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Statement of Problem
voltage
• Integration Methods
Local Truncation Error
Low order approx.
tn
tn+1
time
Error of low order polynomial approximation
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Approach
• Parallel Processing: Avoid LU decomp matrix
solver
• Matrix Exponential Operator:
– Stability: Good
– Operation: Matrix vector multiplications
• Assumption
– C-1v exits and is easy to derive
– Regularization when C is singular
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Matrix Exponential Method
• Krylov subspace approximation
– Orthogonalization: Better conditions
– High order polynomial
• Adaptive time step control
– Dynamic order adjustment
– Optimal tuning of parameters
• Better convergence with coefficient 1/k! at kth
term
eA= I + A + ½ A2 + … + 1/k! Ak +…
(I-A)-1= I + A + A2 +…+ Ak +…
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Krylov Subspace Approximation (1/2)
• Krylov subspace
– K(A, v, m)={v, Av, A2v, …, Amv}
– Matrix vector multiplication
Av=-C-1(Gv)
AVm  Vm H m  H(m  1, m) v m 1emT
– Orthogonalization (Arnoldi
Process): Vm=[v1 v2 … vm]
• Matrix exponential operator
Hm
A
e v  v 2 Vme e1
– Size of Hm is about 10~30 while
size of A can be millions
– Ease of computation of eHm
• Posteriori Error Estimation
T Hm
– Evaluate without extra
err  v H(m  1, m) em e e1
overhead
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RC circuit of 500 nodes, random cap ranges 1e11~1e-16, h = 1e-13
Krylov Subspace Approximation (2/2)
• Matrix exponential method
e

x(t  h)  x(t ) 
Ah
 I
A
 Ax(t )  b(t ) 
e


Ah
  Ah  I   b(t  h)  b(t )
A2
h
v1
Krylov space
Approximation
x(t  h)  x(t )  v1 2 Vm1
e
Hm1h
 I
Hm1h
e1  v 2 2 Vm2
v2
e
Hm2 h
  Hm2 h  I  
(Hm2 h)
2
e1
• Error estimation for matrix exponential method
T
m
err1  v1 H m1 (m  1, m) e
e
Hm1 h
 I
H m1 h
e1
T
m
err2  v 2 H m2 (m  1, m) e
e
Hm2 h
  H m2 h  I  
 Hm2 h 
2
11
e1
Adaptive Time Step Control
• Strategy:
– Maximize step size with a given error budget
– Error are from Krylov space method and nonlinear
component
toll
tolnl
f (h)  err1  err2  h
, residual  h
T
T
• Step size adjustment
– Krylov subspace approximation
• Require only to scale Hm: αA→αHm
– Backward Euler
• (C+hG)-1 changes as h changes
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Experimental Results
• EXP (matrix exp.) and BE (Backward Euler) in
MATLAB
• Machine
– Linux Platform
– Xeon 3.0 GHz and 16GB memory
• Test cases
Circuit (L)
Description #nodes Circuit (NL)
Description
#nodes
D1
trans. Line
5.6K
D5
Inv. chain
82
D2
power grid
160K
D6
power amp
342
D3
power grid
1.6M
D7
16-bit adder
572
D4
power grid
4M
D8
ALU
10K
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Stability and Accuracy
• BE requires smaller time steps
• EXP can leap large steps
Test case: D2
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Performance at fixed time step sizes
• Reference: BE with small step size href
• EXP runs faster under the same error tol.
• D2: 20x
• D3: 4x
• D4: inf
• Scalable for large cases
• Case D4: BE runs out of memory (4M nodes)
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Adaptive Time Step – Linear Circuits
• Strategy:
– Enlarge by 1.25
– Shrink by 0.8
• Adaptive EXP
– Speedup by large step
– Efficient re-evaluation
• Adaptive BE
– Smaller step for
accuracy
– Slow down by resolving linear system
• 10X speedup for D2
Test case: D2
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Adaptive Time Step – Nonlinear
• Strategy:
– Enlarge by 1.25
– Shrink by 0.8
• Adaptive BE
– Multiple Newton
iterations for
convergence
Test case: D7
• Up to 7X speedup
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Summary
Method
Equation
Stability
(passive)
Matrix
inverse
Major
Oper.
Memory1
Adaptive
Parameters2
Cost 3
Adaption
Error
Implicit
Rational
order < 10
High
C+hG
LU
decomp
NC+G1.4
Time
Step h
High
Taylor
series
Poly.
Explicit
Polynom.
order < 10
Weak
C
Mat-vec
product
NC*
Time
Step h
Low
Taylor
series
C
Arnoldi
Process
NC*+
mN
Step h
Order m
Low
Matrix
exp.
Matrix
Exp.
1
Analytical
High
Nc* for C-1; 2 Variable order BDF is not considered
here; 3 Cost of re-evaluation for a new step size
Summary
• Matrix exponential method is scalable
– Stability: Good
– Accuracy: SPICE
• Krylov subspace approximation
– Reduce the complexity
• Preliminary results
– Up to 10X and 7X for linear and nonlinear, respectively
• Limitations of matrix exponential method
– Singularity of C
– Stiffness of C-1G
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Future Works
• Scalable Parallel Processing
– Integration
– Matrix Operations
• Applications
– Power Ground Network Analysis
– Substrate Noises
– Memory Analysis
– Tera Hertz Circuit Simulation
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Thank You!
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