Han Zhao Advisor: Prof. Lei He TA: Fang Gong Final Project Quadratic Response Surface + Importance Sampling Han Zhao Advisor: Prof. Lei He TA: Fang Gong

Outline Quadratic Response Surface Modeling Importance Sampling Problem statement Reading & writing failure model Power & area model Results Importance Sampling Method Conclusions Future work

Outline Quadratic Response Surface Modeling Importance Sampling Problem statement Reading & writing failure model Power & area model Results Importance Sampling Method Conclusions Future work

Quadratic Response Surface Optimization problem x1 x2 x3 x4 Leff 1 Leff 2 Vth 1 Vth 2 Variable y Reading voltage Writing time Reading power Writing power Object function

Quadratic Response Surface Central composite designs Circumscribed Inscribed Faced Center point Nominal parameters Star points Variation Generate 15+ samples Hspice simulation Linear regression to find coefficients

Outline Quadratic Response Surface Modeling Importance Sampling Problem statement Reading & writing failure model Power & area model Results Importance Sampling Method Conclusions Future work

Reading Failure Model Polynomial function Optimal solutions Leff 1 (x1) (μm) Leff 2 (x2) Vth 1 (x3) (V) Vth 2 (x4) Voltage (y) 0.095 0.38 0.5 0.1632 0.105

Reading Failure Model compromise Lower part is optimal Upper part is optimal

Writing Failure Model Polynomial function Optimal solutions Leff 1 (x1) (μm) Leff 2 (x2) Vth 1 (x3) (V) Vth 2 (x4) Time (y) (ps) 0.095 0.2 6.0412

Writing Failure Model consistent Right part is optimal

Outline Quadratic Response Surface Modeling Importance Sampling Problem statement Reading & writing failure model Power & area model Results Importance Sampling Method Conclusions Future work

Optimal solutions for reading Optimal solutions for writing Power Model Reading Writing Optimal solutions for reading Optimal solutions for writing Leff 1 (x1) (μm) Leff 2 (x2) Vth 1 (x3) (V) Vth 2 (x4) Power (y) (mW) 0.105 0.095 0.31 0.5 0.0265 Leff 1 (x1) (μm) Leff 2 (x2) Vth 1 (x3) (V) Vth 2 (x4) Power (y) (mW) 0.105 0.5 0.0522

Power Model

Area Model Total area Only depend on effective length The optimal solution

Outline Quadratic Response Surface Modeling Importance Sampling Problem statement Reading & writing failure model Power & area model Results Importance Sampling Method Conclusions Future work

Results Leff 1 (x1) (μm) Leff 2 (x2) Vth 1 (x3) (V) Vth 2 (x4) Reading voltage 0.095 0.38 0.5 0.105 Writing time 0.2 power 0.31 Writing power Optimal Consistent with results from QMC exhaustive search

Outline Quadratic Response Surface Modeling Importance Sampling Problem statement Reading & writing failure model Power & area model Results Importance Sampling Method Conclusions Future work

Importance Sampling Motivation Method Optimal design makes the yield rate approach infinitely to 1 To find the yield rate (approximately 1-1e-6), we should run Hspice about 1.4e9 times! The CPU runtime is too long to count Method Find failure regions with uniform distribution (5*sigma range) Sample quite a few points in that failing areas Calculate the equivalent yield rate with formulas

Outline Quadratic Response Surface Modeling Importance Sampling Problem statement Reading & writing failure model Power & area model Results Importance Sampling Method Conclusions Future work

Results Failing regions Leff 1 (x1) (μm) Leff 2 (x2) Vth 1 (x3) (V) Optimal 0.095 0.5 0.2 Reading failure 0.08 0.2617 0.198 Writing failure 0.096 0.106 0.525 0.273 Failing regions

Results Results Yield rate Samples CPU runtime Optimal 1 – 1e-6 1.4e9 Too long! Reading 1 – 5.78e-7 500 1 min Writing 1 – 2.83e-4

Outline Quadratic Response Surface Modeling Importance Sampling Problem statement Reading & writing failure model Power & area model Results Importance Sampling Method Conclusions Future work

Conclusions Combine two methods to solve problem Future work Quadratic response surface is used to find optimal designs Importance sampling is used to find the yield rates under the optimal designs Results are consistent with exhaustive search and meet all requirements (e.g., CPU runtime) Future work Improve response surface model to find more accurate results Propose more general methods to find coefficients in formulas of importance sampling

Acknowledgement Thanks for discussions, suggestions and helps! TA: Fang Gong, Yiyu Shi Prof. Lei He Thanks for your attention!