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High-Speed Circuit-Tuning Techniques Based on Lagrangian Relaxation
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  1. High-Speed Circuit-Tuning Techniques Based on Lagrangian Relaxation Charlie Chung-Ping Chen ICCAD 99’ Embedded Tutorial Session 12A chen@engr.wisc.edu (608)2651145

  2. People Involved • Joint work Charlie Chen, University of Wisconsin at Madison Chris Chu, Iowa State University D. F. Wong, University of Texas at Austin • Publication “Fast and Exact Simultaneous Gate and Wire Sizing by Lagrangian Relaxation”, IEEE Transactions on Computer-Aided Design, July 1999

  3. Acknowledgement • Strategic CAD Labs, Intel Corp. Steve Burns, Prashant Sawkar, N. Sherwani, and Noel Menezes • IBM T. J. Watson Center Chandu Visweswariah • C. Kime, L. He (UWisc-Madison)

  4. Outline • Motivation • Overview of Circuit Tuning Techniques • Lagrangian Relaxation Based Circuit Tuning

  5. Motivation • Double the work load and design complexity every 18 months

  6. Motivation • Trends • Increased custom design • Aggressive tuning for performance improvement • Shorter time to market • Interconnect effects severe • Signal integrity issues emerging • Circuit Tuning • Can significantly improve circuit performance and signal integrity without major modification

  7. Manual Sizing 1000+ iterations • Pros • Takes advantage of human experience • Reliable • Simultaneously combines with other optimization techniques directly • Cons • Slow, tedious, limited, and error-prone procedure • Rely too much on experience, requires solid training • Optimality not guaranteed (don’t know when to stop) Change Simulate Satisfy?

  8. Automatic Circuit Tuning • Pros • Fast • Achieves the best performance with interconnect considerations • Explores alternatives (power/delay/noise tradeoff) • Boosts productivity • Optimality guaranty (for convex problems) • Insures timing and reliability • Cons • Complicated tool development and support ($$) • Tool testing, integration, and training

  9. Good Tuning Algorithm • Fast • Optimality guaranteed (for convex problem) • Versatile • Easy to use • Solution quality index (error bound to the optimal solution) • Simple (Easy to develop and maintain)

  10. Static vs. Dynamic Sizing • Static Sizing • Stage Based • Nature circuit decomposition, large scale tuning capability • Very reasonable accuracy (when using good model) • No need for sensitization vectors • Solves for all critical paths in a polynomial formulation • False paths; Potentially inaccurate modeling of slopes of input excitation • Dynamic Sizing • Simulation based • More accurate • No false path problems • Need good input vectors; good for circuits for which critical paths are known and limited • Takes care of a few scenario only • Relatively slower

  11. A Simple Sizing Problem • Minimize the maximum delay Dmax by changing w1,…,wn w7 w9 w4 w1 D1<Dmax a w5 w10 D2<Dmax b w2 w6 w11 w3 w8

  12. Existing Sizing Works • Algorithm: fast, non-optimal for general problem formulation • TILOS (J. Fishburn, A. Dunlop, ICCAD 85’) • Weight Delay Optimization (J. Cong et al., ICCAD 95’) • Mathematical Programming: slower, optimal • Geometrical Programming (TILOS) • Augmented Lagrangian (D. P. Marple et al., 86’) • Sequential Linear Programming (S. Sapatnekar et al.) • Interior Point Method (S. Sapatnekar et al., TCAD 93’) • Sequential Quadratic Programming (N. Menezes et al., DAC 95’) • Augmented Lagrangian + Adjoin Sensitivity (C. Visweswariah et al., ICCAD 96’, ICCAD97’) • Is there any method that is fast and optimal?

  13. Converge? Augmented Lagrangian Weighted Delay SQP Fast Optimal TILOS SLP ? Mathematical Programming Algorithm

  14. Heuristic Approach • TILOS: (J. Fishburn etc ICCAD 85’) • Find all the sensitivities associated with each gate • Up-Size one gate only with the maximum sensitivity • To minimize the object function Minimize Dmax w2 w1 w3 a w4 D1<Dmax w5 w6 D2<Dmax b w11 w9 w7 w8 w10

  15. Weighted Delay Optimization • J. Cong ICCAD 95’ • Size one wire at a time in DFS order • To minimize the weighted delay • best weight? Minimize l1D1 +l2D2 Drivers Loads w1 w2 w3 l1D1 l2D2 w5 w4

  16. Mathematical Programming • Problem Formulation: • Lagrangian: • Optimality (Necessary) Condition: (Kuhn-Tucker Condition)

  17. PSLP v.s. SQP • Penalty Sequential Linear Programming • Sequential Quadratic Programming

  18. Lagrangian Methods • Augmented Lagrangian • Lagrangian Relaxation

  19. Lagrangian Relaxation Theory • LRS (Lagrangian Relaxation Subproblem) • There exist Lagrangian multipliers will lead LRS to find the optimal solution for convex programming problem • The optimal solution for any LRS is a lower bound of the original problem for any type of problem

  20. Lagrangian Relaxation Lagrangian Relaxation Weighted Delay!

  21. Lagrangian Relaxation Augmented Lagrangian Weighted Delay SQP TILOS SLP Lagrangian Relaxation Sink Weights=Multipliers Mathematical Programming Algorithm

  22. Lagrangian Relaxation Framework Update Multipliers Weighted Delay Optimization Converge?

  23. Lagrangian Relaxation Framework Dmax l1 l2 D1 D2 Dmax l1 l2 D1 D2 More Critical -> More Resource -> More Weight D1 D2

  24. Weighted Minimization • Traverse the circuit in topological order • Resize each component to minimize Lagrangian during visit Minimize l1D1 +l2D2 w1 a D1 D2 b w2 w3

  25. Multipliers Adjustmenta subgradient approach • Subgradient: An extension definition of gradient for non-smooth function • Experience: Simple heuristic implementation can achieve very good convergence rate • Reference: Non-smooth function optimization: N. Z. Shor

  26. Path Delay Formulation d1 d2 Aa D1 Ab d3 D2 Ac • Exponential growing • More accurate • Can exclude false paths

  27. Stage Delay Formulation d1 d2 Ae Aa D1 Ab d3 D2 Ac • Polynomial size • Less accurate • Contains false paths

  28. Compatible? ? Path Based Stage Based

  29. Both Multipliers Satisfy KCL(Flow Conservation) l41 l51 l42 l52 Path Based Stage Based l43 1 l31 1 4 4 3 2 2 3 l32 5 5 l53 l3,in =l3,out l43 +l53=l31 +l32

  30. Mixed Delay Formulation Stage Based Stage Based Path Based

  31. Compatible? Lagrangian Relaxation Both Multipliers Satisfy KCL Stage Based Path Based

  32. Hierarchical Objective Function Decomposition • Divide the Lagrangian into who terms (containing or not containing variable wi ) • Hierarchically update the Lagrangian during resizing

  33. Intermediate Variables Cancellation Ae Aa D1 Ab D2 Ac lae lae +lbe =le1 +le2 lbe le1 + le2 lc2 lae (Aa+ d1 )+lbe (Ab+ d1)+le1 (d2 - D1 )+ le2 (d3 - D2 )

  34. Decomposition and Pruning • Flow Decomposition • Prune out all the gates with zero multipliers

  35. Complimentary Condition Implications • li (Di-Dmax )= 0 • Optimal Solution • Critical Path, weight l i >= 0.0, path delay=Dmax • Non-critical path, weight l i = 0.0, path delay < Dmax

  36. Convergence Sequence Optimal Solution Max Delay Any Feasible Maximum Delay= Upper Bound Lagrangian=Lower Bound Weighted Delay<=Maximum Delay # Iteration

  37. Transistor Sizing Extension

  38. Runtime and Storage Requirement

  39. Runtime versus Circuit Size

  40. Storage versus Circuit Size

  41. Convergence of Subgradient Optimization

  42. Area vs. Delay Tradeoff Curve

  43. Conclusion • Lagrangian Relaxation • General mathematical programming algorithm • Optimality guarantee for convex programming problem • Versatile • No extra complication (no quadratic penalty function) • Lagrangian multiplier provides connections between mathematical programming and algorithmic approaches • Multipliers satisfy KCL (flow conservation) • Hierarchical update objective function provides extreme efficiency • Solution quality guaranteed (by providing lower bound)