Circuit Simulation using Matrix Exponential Method

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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

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
• Experimental Results
• Conclusions
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
• Linear circuit formulation
• Let A=-C-1G, b=C-1u, the analytical solution is
• Let input be piecewise linear
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 Problem
• Integration Methods

Error of low order polynomial approximation

Local Truncation Error

Low order approx.

voltage

time

tn

tn+1

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
• 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
• 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)
• Matrix exponential method
• Error estimation for matrix exponential method

v1

v2

Krylov space Approximation

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
• EXP (matrix exp.) and BE (Backward Euler) in MATLAB
• Machine
• Linux Platform
• Xeon 3.0 GHz and 16GB memory
• Test cases
Stability and Accuracy
• BE requires smaller time steps
• EXP can leap large steps

Test case: D2

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
• Strategy:
• Enlarge by 1.25
• Shrink by 0.8
• Speedup by large step
• Efficient re-evaluation
• Smaller step for accuracy
• Slow down by re-solving linear system
• 10X speedup for D2

Test case: D2

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

Test case: D7

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
• 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
• Scalable Parallel Processing
• Integration
• Matrix Operations
• Applications
• Power Ground Network Analysis
• Substrate Noises
• Memory Analysis
• Tera Hertz Circuit Simulation