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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: ckcheng@ucsd.edu. Outline. Introduction Computation of Matrix Exponential Method Krylov Subspace Approximation Adaptive Time Step Control

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circuit simulation using matrix exponential method

Circuit Simulation using Matrix Exponential Method

Shih-Hung Weng, Quan Chen and Chung-Kuan Cheng

CSE Department, UC San Diego, CA 92130

Contact: ckcheng@ucsd.edu

outline
Outline
  • Introduction
  • Computation of Matrix Exponential Method
    • Krylov Subspace Approximation
    • Adaptive Time Step Control
  • Experimental Results
  • Conclusions
circuit simulation
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
statement of problem
Statement of Problem
  • Linear circuit formulation
  • Let A=-C-1G, b=C-1u, the analytical solution is
  • Let input be piecewise linear
statement of problem1
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
statement of problem2
Statement of Problem
  • Integration Methods

Error of low order polynomial approximation

Local Truncation Error

Low order approx.

voltage

time

tn

tn+1

approach
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
matrix exponential method
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 +…

krylov subspace approximation 1 2
Krylov Subspace Approximation (1/2)
  • Krylov subspace
    • K(A, v,m)={v, Av, A2v, …, Amv}
    • Matrix vector multiplication Av=-C-1(Gv)
    • Orthogonalization (Arnoldi Process): Vm=[v1 v2 … vm]
  • Matrix exponential operator
    • Size of Hm is about 10~30 while size of A can be millions
    • Ease of computation of eHm
  • Posteriori Error Estimation
    • Evaluate without extra overhead
krylov subspace approximation 2 2
Krylov Subspace Approximation (2/2)
  • Matrix exponential method
  • Error estimation for matrix exponential method

v1

v2

Krylov space Approximation

adaptive time step control
Adaptive Time Step Control
  • Strategy:
    • Maximize step size with a given error budget
    • Error are from Krylov space method and nonlinear component
  • Step size adjustment
    • Krylov subspace approximation
      • Require only to scale Hm: αA→αHm
    • Backward Euler
      • (C+hG)-1 changes as h changes
experimental results
Experimental Results
  • EXP (matrix exp.) and BE (Backward Euler) in MATLAB
  • Machine
    • Linux Platform
    • Xeon 3.0 GHz and 16GB memory
  • Test cases
stability and accuracy
Stability and Accuracy
  • BE requires smaller time steps
  • EXP can leap large steps

Test case: D2

performance at fixed time step sizes
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)
adaptive time step linear circuits
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 re-solving linear system
  • 10X speedup for D2

Test case: D2

adaptive time step nonlinear
Adaptive Time Step – Nonlinear
  • Strategy:
    • Enlarge by 1.25
    • Shrink by 0.8
  • Adaptive BE
    • Multiple Newton iterations for convergence
  • Up to 7X speedup

Test case: D7

slide18

Summary

1 Nc* for C-1; 2 Variable order BDF is not considered here; 3 Cost of re-evaluation for a new step size

summary
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
future works
Future Works
  • Scalable Parallel Processing
    • Integration
    • Matrix Operations
  • Applications
    • Power Ground Network Analysis
    • Substrate Noises
    • Memory Analysis
    • Tera Hertz Circuit Simulation