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Tsinghua Science and Technology 2023, 28(3): 566-579 https://doi.org/10.26599/TST.2022.9010024
Open Access | Issue | Published: 13 December 2022

An Early-Stop Adjoint Transient Sensitivity Analysis Method for Objective Functions Associated with Many Time Points

Show Author's Information Wenfei Hu1,  Sen Yin1,  Liang Wang2,  Chunxue Liu2,  Zhikai Wang1,  Zuochang Ye1( ),  Yan Wang1,3,4( )
School of Integrated Circuits, Tsinghua University, Beijing 100084, China
Beijing Microelectronics Technology Institute (BMTI), Beijing 100076, China
Beijing Innovation Center for Future Chips, Tsinghua University, Beijing 100084, China
Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China
Keywords:
transient analysis, adjoint sensitivity, Fourier transformation, many time point sensitivity (MTPS), early-stop
Cite this article:
Hu W, Yin S, Wang L, et al. An Early-Stop Adjoint Transient Sensitivity Analysis Method for Objective Functions Associated with Many Time Points. Tsinghua Science and Technology, 2023, 28(3): 566-579. https://doi.org/10.26599/TST.2022.9010024
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Abstract Full text About this article

Transient sensitivity analysis aims to obtain the gradients of objective functions (circuit performance) with respect to design or variation parameters in a simulator, which can be widely used in yield analysis and circuit optimization, among others. However, the traditional method has a computational complexity of O⁢(N2) for objective functions containing circuit states at N time points. The computational complexity is too expensive for large N, especially in time-frequency transform. This paper proposes a many-time-point sensitivity method to reduce the computational complexity to O⁢(N) in multiparameter many-time-point cases. The paper demonstrates a derivation process that improves efficiency by weighting the transfer chain and multiplexing the backpropagation process. We also proposed an early-stop method to improve efficiency further under the premise of ensuring accuracy. The algorithm enables sensitivity calculation of performances involving thousands of time points, such as signal-to-noise and distortion ratio and total harmonic distortion, with significant speed improvements.


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About this article

An Early-Stop Adjoint Transient Sensitivity Analysis Method for Objective Functions Associated with Many Time Points

Show Author's information Wenfei Hu1, Sen Yin1, Liang Wang2, Chunxue Liu2, Zhikai Wang1, Zuochang Ye1( ), Yan Wang1,3,4( )
School of Integrated Circuits, Tsinghua University, Beijing 100084, China
Beijing Microelectronics Technology Institute (BMTI), Beijing 100076, China
Beijing Innovation Center for Future Chips, Tsinghua University, Beijing 100084, China
Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China

Abstract

Transient sensitivity analysis aims to obtain the gradients of objective functions (circuit performance) with respect to design or variation parameters in a simulator, which can be widely used in yield analysis and circuit optimization, among others. However, the traditional method has a computational complexity of O⁢(N2) for objective functions containing circuit states at N time points. The computational complexity is too expensive for large N, especially in time-frequency transform. This paper proposes a many-time-point sensitivity method to reduce the computational complexity to O⁢(N) in multiparameter many-time-point cases. The paper demonstrates a derivation process that improves efficiency by weighting the transfer chain and multiplexing the backpropagation process. We also proposed an early-stop method to improve efficiency further under the premise of ensuring accuracy. The algorithm enables sensitivity calculation of performances involving thousands of time points, such as signal-to-noise and distortion ratio and total harmonic distortion, with significant speed improvements.

Keywords: transient analysis, adjoint sensitivity, Fourier transformation, many time point sensitivity (MTPS), early-stop

References(40)

[1]
S. H. Wu, Y. S. Shu, A. Y. C. Chiou, W. H. Huang, Z. X. Chen, and H. Y. Hsieh, 9.1 a current-sensing front-end realized by a continuous-time incremental ADC with 12b SAR quantizer and reset-then-open resistive DAC achieving 140dB DR and 8ppm INL at 4kS/s, in Proc. 2020 IEEE Int. Solid-State Circuits Conf., San Francisco, CA, USA, 2020, pp. 154–156.
DOI Google Scholar
[2]
M. H. Naderi, C. Park, S. Prakash, M. Kinyua, E. G. Soenen, and J. Silva-Martinez, A 27.7 FJ/conv-step 500 MS/s 12-bit pipelined ADC employing a sub-ADC forecasting technique and low-power class AB slew boosted amplifiers, IEEE Trans. Circ. Syst. I Regul. Pap., vol. 66, no. 9, pp. 3352–3364, 2019.
DOI Google Scholar
[3]
T. C. Hung, F. W. Liao, and T. H. Kuo, A 12-bit time-interleaved 400-MS/s pipelined ADC with split-ADC digital background calibration in 4,000 conversions/channel, IEEE Trans. Circ. Syst. II Express Briefs, vol. 66, no. 11, pp. 1810–1814, 2019.
DOI Google Scholar
[4]
D. H. Kim, S. H. Yang, K. S. Kim, and K. R. Cho, Design of a linear CMOS OTA with wide input voltage range, in Proc. 2006 Int. Symp. on Communications and Information Technologies, Bangkok, Thailand, 2006, pp. 360–363.
DOI Google Scholar
[5]
T. Haine, J. Segers, D. Flandre, and D. Bol, Gradient importance sampling: An efficient statistical extraction methodology of high-sigma SRAM dynamic characteristics, in Proc. 2018 Design, Automation & Test in Europe Conf. & Exhibition, Dresden, Germany, 2018, pp. 195–200.
DOI Google Scholar
[6]
C. J. Gu and J. Roychowdhury, An efficient, fully nonlinear, variability-aware non-Monte-Carlo yield estimation procedure with applications to SRAM cells and ring oscillators, in Proc. 2008 Asia and South Pacific Design Automation Conf., Seoul, Republic of Korea, 2008, pp. 754–761.
Google Scholar
[7]
W. F. Hu, Z. K. Wang, S. Yin, Z. C. Ye, and Y. Wang, Sensitivity importance sampling yield analysis and optimization for high sigma failure rate estimation, in Proc. 2021 58th ACM/IEEE Design Automation Conf., San Francisco, CA, USA, 2021, pp. 895–900.
DOI Google Scholar
[8]
B. Agrawal, F. Liu, and S. Nassif, Circuit optimization using scale based sensitivities, in Proc. IEEE Custom Integrated Circuits Conf. 2006, Jose, CA, USA, 2006, pp. 635–638.
DOI Google Scholar
[9]
J. M Corres, F. J Arregui, and I. R. Matías, Sensitivity optimization of tapered optical fiber humidity sensors by means of tuning the thickness of nanostructured sensitive coatings, Sens. Actuators B Chem., vol. 122, no. 2, pp. 442–449, 2007.
DOI Google Scholar
[10]
O. Lavan, Adjoint sensitivity analysis and optimization of transient problems using the mixed lagrangian formalism as a time integration scheme, Struct. Multidisc. Optim., vol. 61, no. 2, pp. 619–634, 2020.
DOI Google Scholar
[11]
I. Stevanovic and C. C. McAndrew, Quadratic backward propagation of variance for nonlinear statistical circuit modeling, IEEE Trans. Comput. Aided Des. Integr. Circ. Syst., vol. 28, no. 9, pp. 1428–1432, 2009.
DOI Google Scholar
[12]
B. Gu, K. Gullapalli, Y. Zhang, and S. Sundareswaran, Faster statistical cell characterization using adjoint sensitivity analysis, in Proc. 2008 IEEE Custom Integrated Circuits Conf., San Jose, CA, USA, 2008, pp. 229–232.
DOI Google Scholar
[13]
N. B. Hamida and B. Kaminska, Multiple fault analog circuit testing by sensitivity analysis, Analog Integr. Circ. Sig. Process., vol. 4, no. 3, pp. 231–243, 1993.
DOI Google Scholar
[14]
M. B Giles and N. A Pierce, An introduction to the adjoint approach to design, Flow Turbul. Combust., vol. 65, no. 3, pp. 393–415, 2000.
DOI Google Scholar
[15]
J. B. Liu, Q. X. Zhang, K. H. Mo, X. Y. Xiang, J. Li, D. B. Cheng, R. Gao, B. S. Liu, K. Y. Chen, and G. J. Wei, An efficient adversarial example generation algorithm based on an accelerated gradient iterative fast gradient, Comput. Stand. Interfaces, vol. 82, p. 103612, 2022.
DOI Google Scholar
[16]
M. Al-Shedivat, T. Bansal, Y. Burda, I. Sutskever, I. Mordatch, and P. Abbeel, Continuous adaptation via meta-learning in nonstationary and competitive environments, presented at the 6th Int. Conf. on Learning Representations, Vancouver, Canada, 2018.
Google Scholar
[17]
H. Masood, A. Zafar, M. U. Ali, T. Hussain, M. A. Khan, U. Tariq, and R. Damaševičius, Tracking of a fixed-shape moving object based on the gradient descent method, Sensors, vol. 22, no. 3, p. 1098, 2022.
DOI Google Scholar
[18]
J. P. Fishburn and A. E. Dunlop, TILOS: A posynomial programming approach to transistor sizing, in Proc. Int. Conf. on Computer Aided Design, Delft, The Netherlands, 1985, pp. 236–238.
Google Scholar
[19]
R. M. Errico, What is an adjoint model? Bull. Am. Meteor. Soc., vol. 78, no. 11, pp. 2577–2592, 1997.
DOI
[20]
S. Director and R. Rohrer, The generalized adjoint network and network sensitivities, IEEE Trans. Circ. Theory, vol. 16, no. 3, pp. 318–323, 1969.
DOI Google Scholar
[21]
Y. Cao, S. T. Li, L. Petzold, and R. Serban, Adjoint sensitivity analysis for differential-algebraic equations: The adjoint DAE system and its numerical solution, SIAM J. Sci. Comput., vol. 24, no. 3, pp. 1076–1089, 2003.
DOI Google Scholar
[22]
B. S. Kang, G. J. Park, and J. S. Arora, A review of optimization of structures subjected to transient loads, Struct. Multidisc. Optim., vol. 31, no. 2, pp. 81–95, 2006.
DOI Google Scholar
[23]
F. Liu and P. Feldmann, A time-unrolling method to compute sensitivity of dynamic systems, in Proc. 2014 51st ACM/EDAC/IEEE Design Automation Conf., San Francisco, CA, USA, 2014, pp. 1–6.
DOI Google Scholar
[24]
K. V. Aadithya, E. Keiter, and T. Mei, DAGSENS: Directed acyclic graph based direct and adjoint transient sensitivity analysis for event-driven objective functions, in Proc. 2017 IEEE/ACM Int. Conf. on Computer-Aided Design, Irvine, CA, USA, 2017, pp. 155–162.
DOI Google Scholar
[25]
A. Meir and J. Roychowdhury, BLAST: Efficient computation of nonlinear delay sensitivities in electronic and biological networks using barycentric Lagrange enabled transient adjoint analysis, in Proc. 49th Annu. Design Automation Conf., San Francisco, CA, USA, 2012, pp. 301–310.
DOI Google Scholar
[26]
Z. Ilievski, H. Xu, A. Verhoeven, E. J. W. ter Maten, W. H. A. Schilders, and R. M. M. Mattheij, Adjoint transient sensitivity analysis in circuit simulation, in Scientific Computing in Electrical Engineering, G. Ciuprina and D. Ioan, eds. Berlin, Germany: Springer, 2007, pp. 183–189.
DOI
[27]
W. F. Hu, Z. C. Ye, and Y. Wang, Adjoint transient sensitivity analysis for objective functions associated to many time points, in Proc. 2020 57th ACM/IEEE Design Automation Conf., San Francisco, CA, USA, 2020, pp. 1–6.
DOI Google Scholar
[28]
C. W. Gear, The numerical integration of ordinary differential equations, Math. Comput., vol. 21, no. 98, pp. 146–156, 1967.
DOI Google Scholar
[29]
J. Roychowdhury, Numerical simulation and modelling of electronic and biochemical systems, Found. Trends Electron. Des. Autom., vol. 3, nos. 2&3, pp. 97–303, 2009.
DOI Google Scholar
[30]
F. Fernandez and D. A Tortorelli, Semi-analytical sensitivity analysis for nonlinear transient problems, Struct. Multidisc. Optim., vol. 58, no. 6, pp. 2387–2410, 2018.
DOI Google Scholar
[31]
B. E. Rapp, Microfluidics: Modeling, Mechanics and Mathematics. Cambridge, MA, USA: Elsevier, 2016.
DOI
[32]
Y. LeCun, D. Touretzky, G. Hinton, and T. Sejnowski, A theoretical framework for back-propagation, in Proc. 1988 Connectionist Models Summer School, Pittsburg, PA, USA, 1988, pp. 21–28.
Google Scholar
[33]
H. H. Tan and K. H. Lim, Review of second-order optimization techniques in artificial neural networks backpropagation, IOP Conf. Ser. Mater. Sci. Eng., vol. 495, p. 012003, 2019.
DOI Google Scholar
[34]
W. Kester, Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so you don’t get lost in the noise floor. https://www.analog.com/media/en/training-seminars/tutorials/MT-003.pdf, 2009.
[35]
K. S. Kundert, Accurate Fourier analysis for circuit simulators, in Proc. IEEE Custom Integrated Circuits Conf., San Diego, CA, USA, 1994, pp. 25–28.
Google Scholar
[36]
A. V. Oppenheim, R. W Schafer, and J. R. Buck, Discrete-Time Signal Processing. 2nd ed. Upper Saddle River, NJ, USA: Prentice-Hall, Inc., 1999.
[37]
G. A. Holton, Cubic spline interpolation, in value-at-risk: theory and practice, G. A. Holton, ed., 2nd ed., https://www.value-at-risk.net/cubic-spline-interpolation/, 2014.
[38]
C. A Hall and W. W. Meyer, Optimal error bounds for cubic spline interpolation, J. Approx. Theory, vol. 16, no. 2, pp. 105–122, 1976.
DOI Google Scholar
[39]
A. Hjorungnes and D. Gesbert, Complex-valued matrix differentiation: Techniques and key results, IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2740–2746, 2007.
DOI Google Scholar
[40]
E. R. Keiter, K. V. Aadithya, T. Mei, T. V. Russo, R. Schiek, P. E. Sholander, H. Thornquist, and J. Verley, Xyce parallel electronic simulator users’ guide version 6.7. https://xyce.sandia.gov/files/xyce/Users_Guide.pdf, 2017.
DOI
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Publication history

Received: 24 March 2022
Revised: 13 June 2022
Accepted: 01 July 2022
Published: 13 December 2022
Issue date: June 2023

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© The author(s) 2023.

Acknowledgements

This work was supported by the National Key R&D Program (No. 2018YFB2202701) from Ministry of Science and Technology, China.

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The articles published in this open access journal are distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).

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