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Accurate Clock Mesh Sizing via Sequential Quadratic ProgrammingPowerPoint Presentation

Accurate Clock Mesh Sizing via Sequential Quadratic Programming

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### Accurate Clock Mesh Sizing via Sequential Quadratic Programming

Venkata Rajesh Mekala, Yifang Liu, Xiaoji Ye, Jiang Hu, Peng Li

Department of ECE, Texas A&M University

From ISPD’10

Systematic way - Sequential Quadratic Programming (SQP) Programming

- One of the most popular and robust algorithms for nonlinear continuous optimization
- Mathematical theory based

Definitions about SQP Programming

- Original problem
- Lagrangian function
- Jacobian

Optimality condition in one dimensional problem Programming

- Optimal solution will exist in f’(x)=0 and f’’(x)>0

Optimality condition in SQP Programming

- Karush-Kuhn-Tucker (KKT) conditions
- Second order optimality condition
is positive definite

H means Hessian matrix

How to solve? Programming

- In one dimensional problem
- Newton’s method

- In SQP

Result in QP form Programming

Outline Programming

- Introduction
- Problem formulation
- SQP for clock network sizing
- Sensitivity analysis
- Algorithm overview
- Experimental results and Conclusions

Introduction Programming

- Why clock mesh?
- Uniform, low skew clock distribution
- Better tolerance to On-Chip Variation (OCV)

Introduction (cont.) Programming

- Disadvantages
- Larger area (metal resources)
- Higher power consumption
- Sophisticated delay model is hard to analyze highly coupled structure

Previous works Programming

- Using clock tree networks
- Moment-based sensitivity analysis
- restricted in clock tree

- SQP under a power budget
- Inaccurate

- Divide and Conquer using SLP
- applies only to clock tree

- Moment-based sensitivity analysis

Previous works (cont.) Programming

- Using non-clock tree networks
- Crosslinks
- difficult to extend to a mesh

- Clock mesh

- Crosslinks

Our Contributions Programming

- Adopt a current-source based gate modeling approach to speed up the accurate analysis
- Develop efficient adjoint sensitivity analysis to provide desirable info
- First clock mesh sizing using systematic solution search and accurate delay model

Problem formulation Programming

- Given a CDN consisting of a clock mesh driven by a clock tree
- Minimize power consumption while meeting skew constraints by sizing the mesh
- Power dissipation is approximated by mesh area
- Skew is presented in a delay variance form

Formulae and terms Programming

- I: set of interconnect in the mesh
- xi: size of element i
- wi: area of ith element
- S: set of sinks
- Dj: propaagation delay from clock tree root to sink j

π Programming model of a clock mesh

SQP for clock network sizing Programming

- Use QP solver to solve

Quasi-Newton approximation of Hessian Programming

- Using BFGS method
where

Sensitivity analysis Programming

Linearize the original circuit Programming

- Using linearized compact gate model
- Kirchhoff CL
and VL

Algorithm overview Programming

Experimental results Programming

- The benchmarks are taken from ISPD and ISCAS. The BPTM 65-nm technology transistor models have been used

Table of results Programming

Area-skew tradeoff by varying delta Programming

Runtime of CMSSQP Programming

Conclusions Programming

- Can easily extend for sizing buffers and mesh element simultaneously
- Achieve up to 33% area reduction
- Robust in dealing with any complex clock mesh network

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