An Algorithmic Framework of Large-Scale Circuit Simulation Using Exponential Integrators

Transcription

1 An Algorithmic Framework of Large-Scale Circuit Simulation Using Exponential Integrators Hao Zhuang 1, Wenjian Yu 2, Ilgweon Kang 1, Xinan Wang 1, and Chung-Kuan Cheng 1 1. University of California, San Diego 2. Tsinghua University

2 Outline Motivation & Contributions Background of time-domain circuit simulation Our algorithmic framework Exponential integrators Invert Krylov subspace method Experimental results Conclusions & future directions 2

3 SPICE Motivation critical to wide ranges of IC Modern IC billions of transistors complex interconnects Requirement: new structures e.g., FinFET, 3D strong coupled post-layout effects capability & accuracy Simulation runtime long/infinite From Dick Sites, Datacenter Computers modern challenges in CPU design Google Inc & Intel i7 From Synopsys Inc. Issue 3, 2012 Technology Update FinFET: The Promises and the Challenges 3

4 Contributions Exponential Integration Stable, Explicit No Newton-Raphson Target of matrix factorization: conductance matrix G ONLY Less expensive Handling tasks (even when traditional schemes FAIL) large-scale, strong coupled, post-layout A promising framework 4

5 Basic & BENR as An Example (1) Differential Equations BE: Backward Euler conductance (/incidence) compents time step capacitance (/inductance) components input nonlinear devices dynamics 5

6 Basic & BENR as An Example (2) NR: Newton-Raphson Jacobian matrix BENR: Backward Euler + Newton-Raphson iterations 6

7 Basic & BENR as An Example (3) NR: Newton-Raphson Jacobian matrix BENR: Backward Euler + Newton-Raphson iterations capacitance matrix 7

8 Matrix Exponential Method Our previous attempt [Weng12] where 8

9 Matrix Exponential Method Our previous attempt [Weng12] where It also uses NR The Jacobian matrix capacitance matrix 9

10 Matrices from a Post-Layout Case C G C, G matrices from FreeCPU [Zhang, Yu TCAD 2013] nnz: non-zero terms 10

11 Matrices from a Post-Layout Case C G C, G matrices L U lu(c) 11

12 Matrices from a Post-Layout Case C G L U C, G matrices lu( C h + G) 12

13 Matrices from a Post-Layout Case C, G matrices L and U of lu( C h + G) L U L and U of lu(c) lu(g) 13

14 Matrices from a Post-Layout Case Traditional methods are all challenged by C, when C is complicated, L and U of lu( C h + G) In this example, lu(g) contains less nnz (~10%) & less complicated nnz distributions L and U of lu(g) 14

15 Two techniques: Our proposed framework ER: Exponential Rosenbrock Formulation Invert Krylov subspace to compute e J v Computational advantages Simple matrix factorization target: exploit the feature of lu(g) Stable explicit method to solve circuit system 15

16 ER: Exponential Rosenbrock Start from dx t = g(x, u, t) dt The next time step solution [Hochbruck, et. al. SIAM09] x k+1 = x k + h k φ 1 h k J k g(x k, u, t k ) + h k 2 φ 2 h k J k b k where J k = g/ x, b k = g/ t φ 1 h k J k = (e h kj k I n )/h k J k φ 2 h k J k = (e h kj k I n )/h k 2 J k 2 I n /h k J k Exponential Integrators: Proved to be Stable, Explicit, High-Order Accuracy for ODE 16

17 Chain rule: ER in Circuit Simulation where dq x t dx dx t dt = Bu t f(x) dq x t dx = C x t = C k, J k = C k 1 G k, g k = J k + C k 1 F k + Bu t, b k = C k 1 Bu t k+1 Bu t k h k We have ALL the components to obtain x k+1 x k+1 (h k ) = x k + h k φ 1 h k J k g(x, u, t) + h k 2 φ 2 h k J k b k 17

18 Local Nonlinear Error Control The local nonlinear error estimator [Caliari09] e rr x k+1, x k = φ 1 h k J k C 1 k ΔF k where ΔF k = F x k+1 F(x k ) ER-C: ER with Correction Term Reuse ΔF k to improve the accuracy by padding the extra term D k = γh k φ 2 h k J k C k 1 ΔF k The further corrected solution is x k+1,c = x k+1 D k 18

19 Krylov Method for MEVP e J v e J v: Matrix Exponential and Vector Product (MEVP) via standard Krylov subspace [Weng12] K m J, v span v, Jv, J 2 v,, J m 1 v Arnoldi process and Matrix reduction: JV m = V m H m + h m+1,m v m+1 e m T MEVP is computed by e J v v 2 V m e H me 1 Explicit feature: time stepping only by scaling H m with h, e hj v v 2 V m e hh me 1 19

20 Standard Krylov subspace (a) Standard Krylov Basis [Weng12] K m J, v span v, Jv, J 2 v,, J m 1 v like these eigenvalues Im 0 Re spectrum of J = C 1 G Eigenvalues of J: small magnitude of Re Eigenvalues of J: large magnitude of Re 20

21 Standard Krylov subspace (a) Standard Krylov Basis [Weng12] K m J, v span v, Jv, J 2 v,, J m 1 v spectrum of J = C 1 G Im 0 Re these eigenvalues defines the major dynamical behavior demand more bases to characterize Eigenvalues of J: small magnitude of Re Eigenvalues of J: large magnitude of Re 21

22 Invert Krylov subspace Invert Krylov Basis [Zhuang, et. al. DAC14] K m J 1, v span v, J 1 v, J 2 v,, J m+1 v Invert Krylov subspace method captures important eigenvalues in the original spectrum Im Im 0 Re 0 Re spectrum of J spectrum of J 1 Eigenvalues of J: small magnitude of Re Eigenvalues of J: large magnitude of Re 22

23 Simple Matrix Fct. Taget Invert Krylov Subspace approach transfers J = C 1 G J 1 = G 1 C At each iteration, we generate invert Krylov subspace V m = v 1, v 2,, v m by solving Gw = Cv i 1 23

24 Overall Framework No Newton-Raphson Build upon exponential integrators explicit method for DAE solver adjust error by step size control ER-C: further improve the solution 24

25 Experimental Results Implemented in MATLAB2013a & C/C++ (GCC 4.7.3) Opensource BSIM3 device model with C MATLAB Executable (MEX) external interface between device evaluation and matrix solvers Linux workstation Intel CPU i7 3.4GHZ 32GB memory. Utilize single thread mode. 25

26 Accuracy 26

27 Runtime Performance #Dev.: the number of devices. nnzc & nnzg: the number of non-zero elements in linear C and G. #step: the number of steps for transient simulation; For each time step, #NR a : the average NR iterations #m a : the average dimension of invert Krylov subspace RT(s): the runtime. SP: the runtime speedup Test circuits 27

28 Conclusions and Future Directions Accelerate SPICE-like time-domain simulation framework Exponential Integrators Stable Explicit method MEVP w/ invert Krylov Subspace & Less expensive matrix factorizations. Handling tasks even when traditional methods fail. Future directions: parallelism, can be accelerated further by multicore/many-core computing systems. many derivatives & tools can be built upon. 28

29 Thanks and Q&A 29