1. 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 1

2. Outline Introduction Computation of Matrix Exponential Method – Krylov Subspace Approximation – Adaptive Time Step Control Experimental Results Conclusions 2

3. 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 3

4. Statement of Problem Linear circuit formulation Let A=-C -1 G, b=C -1 u, the analytical solution is Let input be piecewise linear 4

5. Statement of Problem Integration Methods – Explicit (Forward Euler): e Ah => (I+Ah) “Simpler” computation but smaller time steps – Implicit (Backward Euler): e Ah => (I-Ah) -1 Direct matrix solver (LU Decomp) with complexity O(n 1.4 ) where n=#nodes – Error derived from Taylor’s expansion 5

6. Statement of Problem Integration Methods Error of low order polynomial approximation 6 voltage time tntn t n+1 Low order approx. Local Truncation Error

7. Approach Parallel Processing: Avoid LU decomp matrix solver Matrix Exponential Operator: – Stability: Good – Operation: Matrix vector multiplications Assumption – C -1 v exits and is easy to derive – Regularization when C is singular 7

8. 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 e A = I + A + ½ A 2 + … + 1/k! A k +… (I-A) -1 = I + A + A 2 +…+ A k +… 8

9. Krylov Subspace Approximation (1/2) Krylov subspace – K(A, v, m)={v, Av, A 2 v, …, A m v} – Matrix vector multiplication Av=-C -1 (Gv) – Orthogonalization (Arnoldi Process): V m =[v 1 v 2 … v m ] Matrix exponential operator – Size of H m is about 10~30 while size of A can be millions – Ease of computation of e Hm Posteriori Error Estimation – Evaluate without extra overhead 9

10. RC circuit of 500 nodes, random cap ranges 1e- 11~1e-16, h = 1e-13

11. Krylov Subspace Approximation (2/2) Matrix exponential method Error estimation for matrix exponential method 11 Krylov space Approximation v1v1 v2v2

12. 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 H m : α A →α H m – Backward Euler (C+hG) -1 changes as h changes 12

13. Experimental Results EXP (matrix exp.) and BE (Backward Euler) in MATLAB Machine – Linux Platform – Xeon 3.0 GHz and 16GB memory Test cases 13 Circuit (L)Description#nodesCircuit (NL)Description#nodes D1trans. Line5.6KD5Inv. chain82 D2power grid160KD6power amp342 D3power grid1.6MD716-bit adder572 D4power grid4MD8ALU10K

14. 14 Test case: D2 BE requires smaller time steps EXP can leap large steps Stability and Accuracy

15. Performance at fixed time step sizes 15 Reference: BE with small step size h ref 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)

16. 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 16 Test case: D2

17. Adaptive Time Step – Nonlinear Strategy: – Enlarge by 1.25 – Shrink by 0.8 Adaptive BE – Multiple Newton iterations for convergence Up to 7X speedup 17 Test case: D7

18. MethodEquation Stability (passive) Matrix inverse Major Oper. Memory 1 Adaptive Parameters 2 Cost 3 Adaption Error Implicit Rational order < 10 HighC+hG LU decomp N C+G 1.4 Time Step h High Taylor series Poly. Explicit Polynom. order < 10 WeakC Mat-vec product NC*NC* Time Step h Low Taylor series Matrix Exp. AnalyticalHighC Arnoldi Process N C * + mN Step h Order m Low Matrix exp. 1 N c * for C -1 ; 2 Variable order BDF is not considered here; 3 Cost of re-evaluation for a new step size Summary

19. 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 -1 G 19

20. Future Works Scalable Parallel Processing – Integration – Matrix Operations Applications – Power Ground Network Analysis – Substrate Noises – Memory Analysis – Tera Hertz Circuit Simulation 20

21. Thank You! 21