PiCAP: A Parallel and Incremental Capacitance Extraction Considering Stochastic Process Variation

1. PiCAP: A Parallel and Incremental Capacitance Extraction Considering Stochastic Process Variation Fang Gong1, Hao Yu2, and Lei He1 1Electrical Engineering Dept., UCLA 2Berkeley Design Automation Presented by Fang Gong

2. Outline • Background and Motivation • Algorithms • Experimental Results • Conclusion and Future Work

3. Outline • Background and Motivation • Algorithms • Experimental Results • Conclusion and Future Work

4. Process Variation and Cap Extraction From [Kang and Gupta] • Process variation leads to capacitance variation • OPC lithography and CMP polishing • Capacitance variation affects circuit performance • Delay variation and analog mismatch

5. Background of BEM Based Cap Extraction Source Panel j • Capacitance extraction in FastCap • Procedures • Discretize metal surface into panels • Form linear system by collocation • Results in dense potential coeffs • Solve by iterative GMRES • Fast Multipole method (FMM) to evaluate Matrix Vector Product (MVP) • Preconditioned GMRES iteration with guided convergence Observe Panel i • Difficulties for stochastic capacitance extraction • How to consider variations in FMM? • How to consider different variations in precondition?

6. Motivation of Our Work • Existing works • Stochastic integral by low-rank approximation • Zhu, Z. and White, J. “FastSies: a fast stochastic integral equation solver for modeling the rough surface effect”. In Proceedings of IEEE/ACM ICCAD 2005. • Pros: Rigorous formulation • Cons: Random integral is slow for full-chip extraction • Stochastic orthogonal polynomial (SOP) expansion • Cui, J., and etc. “Variational capacitance modeling using orthogonal polynomial method”. In Proceedings of the 18th ACM Great Lakes Symposium on VLSI 2008. • Pros: An efficient non-Monte-Carlo approach • Cons: SOP expansion results in an augmented and dense linear system • Objective of our work • Fast multi-pole method (FMM) with nearly O(n) performance with a furtherparallel improvement. • Pre-conditioner should be updated incrementally for different variation.

7. Outline • Background and Motivation • Algorithms • Experimental Results • Conclusion and Future Work

8. Flow of piCAP Geometry InfoProcess Variation Build Spectrum preconditioner Geometric Moments Solve with GMRES Incrementally update preconditioner Evaluate the MVP (Pxq) with FMM in parallel Potential Coefficient Calculate Cij with the charge distribution. Represent Pij with stochastic geometric moments Use parallel FMM to evaluate MVP of Pxq Obtain capacitance (mean and variance) with incrementally preconditioned GMRES

9. observer cube d0 source-panel r0 source-cube Stochastic Geometric Moment • Consider two independent variation sources: panel distance (d) and panel width (w)‏ • Multipole expansion along x-y-z coordinates: multipole moments and local moments • Mi and Li show an explicit dependence on geometry parameters, and are called geometric moments.

10. Stochastic Potential Coefficient Expansion • Stochastic Potential Coefficient • Relate geometric parameters to random variables • Let be random variable for panel width w, and be random variable for panel distance d. • Geometric moments Mp and Lp are: • Now, the potential coefficient is

11. Stochastic Potential Coefficient Expansion • Stochastic Potential Coefficient • Relate geometric parameters to random variables • Let be random variable for panel width w, and be random variable for panel distance d. • Geometric moments Mp and Lp are: • Now, the potential coefficient is • n-order stochastic orthogonal polynomial expansion of P • Accordingly, m-th order (m = 2n + n(n − 1)) expansion of charge is:

12. Augmented System • Recap: SOP expansion leads to a large and dense system equation Geometry InfoProcess Variation Build Spectrum preconditioner Geometric Moments Solve with GMRES Incrementally update preconditioner Evaluate the MVP (Pxq) with FMM in parallel Potential Coefficient Calculate Cij with the charge distribution.

13. Parallel Fast Multipole Method--upward • Overview of Parallel fast-multipole method (FMM)‏ • group panels in cubes, andbuild hierarchical tree for cubes • We use 8-degree trees in implementation, but use 2-degree trees for illustration here. • A parallel FMM distributes cubes to different processors • Upward Pass Level 0 Level 1 Level 2 M2M Level 3 • Update parent’s moments by summing the moments of its children—called M2M operation • starting from bottom level, it calculates stochastic geometric moments • M2M operations can be performed in parallel at different nodes

14. Parallel Fast Multipole Method--Downwards • Downward Pass M2L Level 0 L2L Level 1 Level 2 M2M Level 3 M2M, L2L are local operations, while M2L is global operation. How to reduce communication traffic due to global operation? • Sum L2L results with near-field potential for all panels at bottom level and return Pxq • At the top level, calculation of potential between two cubes—called M2L operation. • potential is further distributed down to children from their parent in parallel—called L2L operation. • Calculate near-field potential directly in parallel

15. Cube 1 Cube under calculation Reduction of traffic between processors Cube 0 Cube k Cube 0 Cube 1 … Cube k … Dependency List • Global data dependence exists in M2L operation at the top level • Pre-fetch moments: distributes its moments to all cubes on its dependency list before the calculations. • As such, it can hide communication time.

16. Flow of piCAP Geometry InfoProcess Variation Build Spectrum preconditioner Geometric Moments Solve with GMRES Incrementally update preconditioner Evaluate the MVP (Pxq) with FMM in parallel Potential Coefficient Calculate Cij with the charge distribution. Use spectrum pre-conditioner to accelerate convergence Incrementally update the pre-conditioner for different variation.

17. Deflated Spectral Iteration • Why need spectral preconditioner • GMRES needs too many iterations to achieve convergence. • Spectral preconditioner shifts the spectrum of system matrix to improve the iteration convergence • Deflated spectral iteration • k (k=1 power iteration) partial eigen-pairs • Spectrum preconditioner • Why need incremental precondition • Variation can significantly change spectral distribution • Building each pre-conditioner for different variations is expensive • Simultaneously considering all variations increases the complexity of our model.

18. Incremental Precondition • For updated system , the update for the i-th eigen vector is: • is the subspace composed of • is the updated spectrum • Updated pre-conditioner W’ is Inverse operation only involves diagonal matrix DK Consider different variations by updating the nominal preconditioner partially.

19. Outline • Background and Motivation • Algorithm • Experimental Results • Conclusion and Future Work

20. Accuracy Comparison 2 panels, d0 = 7.07μm, w0 = 1μm, d1 = 20%d0 MC (3000)‏ piCAP Cij (fF)‏ -0.3113 -0.3056 Time(s)‏ 10.786037 0.008486 2 panels, d0 = 11.31μm, w0 = 1μm, d1 = 10%d0 MC (3000)‏ piCAP Cij (fF)‏ -0.3861 -0.3824 Time(s)‏ 10.7763 0.007764 2 panels, d0 = 4.24μm, w0 = 1μm, d1 = 20%d0, w1 = 20%h0 MC (3000)‏ piCAP Cij (fF)‏ -0.2498 -0.2514 Time(s)‏ 11.17167 0.008684 • Setup: two panels with random variation for distance d and width w • Result: Stochastic Geometric Moments have high accuracy with average error of 1.8%, and can be up to ~1000X faster than MC

21. Runtime for parallel FMM #wire 20 40 80 160 #panels 12360 10320 11040 12480 1 proc. 0.737515/1.0 0.541515/1.0 0.605635/1.0 0.96831/1.0 2 proc. 0.440821/1.7X 0.426389/1.4X 0.352113/1.7X 0.572964/1.7X 3 proc. 0.36704/2.0X 0.274881/2.0X 0.301311/2.0X 0.489045/2.0X 4 proc. 0.273408/2.7X 0.19012/2.9X 0.204606/3.0X 0.340954/2.8X • Setup • Two-layer example with 20 conductors. • Other: 40, 80, 160 conductors • Evaluate Pxq (MVP) with 10% perturbationon panel distance • Result • All examples can have about 3X speedup with 4 processors

22. Efficiency of spectral preconditioner # panel # variable diagonal prec. spectral prec. # iter Time(s)‏ # iter Time(s)‏ plate 256 768 29 24.59 11 8.625 cubic 864 2592 32 49.59 11 19.394 bus 1272 3816 41 72.58 15 29.21 • Setup: Three test structures: single plate, 2x2 bus, cubic • Result • Compare diagonal precondition with spectrum precondition • Spectrum precondition accelerates convergence of GMRES (3X).

23. Speedup byIncremental Precondition discretizationw-t-l #panel #variable Total Runtime (s)‏ Non-incremental incremental 3x3x7 2040 6120 419.438 81.375 3x3x15 3960 11880 3375.205 208.266 3x3x24 6120 18360 - 504.202 3x3x50 12360 37080 - 3637.391 • Setup • Test on two-layer 20 conductor example • Incremental update of nominal pre-conditioner for different variation sources • Compare with non-incremental one • Result: Up to 15X speedup over non-incremental results, and only incremental one can finish all large examples.

24. Conclusion and Future Work • Introduce stochastic geometric moments • Develop a parallel FMM to evaluate the matrix-vector product with process variation • Develop a spectral pre-conditioner incrementally to consider different variations • Future Work: extend our parallel and incremental solver to solve other IC-variation related stochastic analysis

25. Thanks PiCAP: A Parallel and Incremental Capacitance Extraction Considering Stochastic Process VariationFang Gong, Hao Yu and Lei He