Beyond low-order statistical response surfaces: latent variable regression for efficient, highly nonlinear fitting

ABSTRACT

The number and magnitude of process variation sources are increasing as we scale further into the nano regime. Today's most successful response surface methods limit us to low-order forms -- linear, quadratic -- to make the fitting tractable. Unfortunately, not all variation-al scenarios are well modeled with low-order surfaces. We show how to exploit latent variable regression ideas to support efficient extraction of arbitrarily nonlinear statistical response surfaces. An implementation of these ideas called SiLVR, applied to a range of analog and digital circuits, in technologies from 90 to 45nm, shows significant improvements in prediction, with errors reduced by up to 21X, with very reasonable runtime costs.

REFERENCES

• 1: M. Hane, T. Ikezawa, T. Ezaki, "Atomistic 3D process/device simulation considering gate line-edge roughness and poly-Si random crystal orientation effects", IEEE Internat. Electron Devices Meeting, 2003.
• 2: T. Ezaki, T. Ikezawa, M. Hane, "Investigation of realistic dopant fluctuation induced device characteristics variation for sub-100nm CMOS by using atomistic 3d process/device simulator", IEEE IEDM, 2002.
• 3: D. J. Frank et al., "Monte Carlo modeling of threshold variation due to dopant fluctuations", Symp. VLSI Tech., 1999.
• 4: Hongliang Chang , Sachin S. Sapatnekar, Statistical Timing Analysis Considering Spatial Correlations using a Single Pert-Like Traversal, Proceedings of the 2003 IEEE/ACM international conference on Computer-aided design, p.621, November 09-13, 2003 [doi>10.1109/ICCAD.2003.129]
• 5: Z. Wang, S. Director, "An efficient yield optimization method using a two step linear approximation of circuit performance", EDAC, 1994.
• 6: A. Dharchoudhury, S. M. Kang, "Worst-case analysis and optimization of VLSI circuit performances", IEEE Trams. CAD, 14(4), 1995.
• 7: Xin Li , Jiayong Le , L. T. Pileggi , A. Strojwas, Projection-based performance modeling for inter/intra-die variations, Proceedings of the 2005 IEEE/ACM International conference on Computer-aided design, p.721-727, November 06-10, 2005, San Jose, CA
• 8: Zhuo Feng , Peng Li, Performance-oriented statistical parameter reduction of parameterized systems via reduced rank regression, Proceedings of the 2006 IEEE/ACM international conference on Computer-aided design, November 05-09, 2006, San Jose, California [doi>10.1145/1233501.1233684]
• 9: A. J. Burnham, R. Viveros, J. F. MacGregor, "Frameworks for latent variable multivariate regression", J. of Chemometrics, 10, 1996.
• 10: S. Wold, M. Sjöström, L. Eriksson, "PLS-regression: a basic tool of chemometrics", Chemometrics and Intelligent Lab. Sys., 58, 2001.
• 11: P. T. Davies, M. K-S. Tso, "Procedures for reduced-rank regression", Applied Statistics, 31(3), 1982.
• 12: A-L. Boulesteix, K. Strimmer, "Partial least squares: a versatile tool for the analysis of high-dimensional genomic data", Brief. in Bioinf., 2006.
• 13: G. Baffi, E. B. Martin, A. J. Morris, "Non-linear projection to latent structures revisited (the neural network PLS algorithm)", Computers Chem. Engg., 23(9), 1999.
• 14: C. Malthouse, A. C. Tamhane, R. S. H. Mah, "Nonlinear partial least squares", Computers Chem. Engg., 21(8), 1997.
• 15: Brian D. Ripley , N. L. Hjort, Pattern Recognition and Neural Networks, Cambridge University Press, New York, NY, 1995
• 16: K. E. Muller, "Relationships between redundancy analysis, canonical correlation, and multivariate regression", Psychometrika, 46(2), 1981.
• 17: G. E. Noether, "Introduction to Statistics: The Nonparametric Way", Springer, 1990.
• 18: D. Nguyen, B. Widrow, "Improving the learning speed of 2-layer neural networks by choosing initial values of the adaptive weights", Int. Joint Conf. Neural Nets., 1990.
• 19: M. T. Hagan, M. G. Menhaj, "Training feedforward networks with the Marquardt algorithm", Trans. Neural Nets., 5(6), Nov. 1994.
• 20: D. P. Bertsekas, "Nonlinear programming", 2nd ed, Athena Scientific, 1995.
• 21: David J. C. MacKay, A practical Bayesian framework for backpropagation networks, Neural Computation, v.4 n.3, p.448-472, May 1992 [doi>10.1162/neco.1992.4.3.448]
• 22: F. D. Foresee, M. T. Hagan, "Gauss-Newton approximation to Bayesian learning", Int. Conf. Neural Nets., 1997.
• 23: Wei Zhao , Yu Cao, New Generation of Predictive Technology Model for Sub-45nm Design Exploration, Proceedings of the 7th International Symposium on Quality Electronic Design, p.585-590, March 27-29, 2006 [doi>10.1109/ISQED.2006.91]
• 24: D. Johns, K. Martin, "Analog Integrated Circuit Design", Wiley, 1996.
• 25: H. Banba et al., "A CMOS Bandgap Reference Circuit with Sub-I-V Operation", IEEE J. Solid State Cir., 34(5), pp. 670--674, 1999.

INDEX TERMS

Primary Classification:
  B. Hardware
  B.7 INTEGRATED CIRCUITS
      B.7.2 Design Aids

General Terms:
Algorithms, Design

Keywords:
DFM, dimensionality reduction, regression, response surface