1. Efficient Decoupling Capacitance Budgeting Considering Operation and Process Variations
Yiyu Shi*, Jinjun Xiong+, Chunchen Liu* and Lei He*
*Electrical Engineering Department, UCLA
+IBM T. J. Watson Research Center, Yorktown Heights, NY
This work is partially supported by NSF CAREER award and
a UC MICRO grant sponsored by Altera, RIO and Intel.

2. Motivation
The continuous semiconductor technology scaling leads to growing process variations, and statistical optimization has been actively researched to cope with process variations.
Stochastic gate sizing for power reduction [Bhardwaj:DAC’05, Mani:DAC’05]
Stochastic gate sizing for yield optimization [Davoodi:DAC’06, Sinha:ICCAD’05]
Stochastic buffer insertion to minimize delay [He:TCAD’07]
Adaptive body biasing with post-silicon tuning [Main:ICCAD’06]
However, all these work ignore operation variation such as
crosstalk difference over input vectors
power supply noise fluctuation over time
processor temperature variation over workload
A better design could be achieved by considering both operation and process variations
As a vehicle to demonstrate this point, we study the on-chip decoupling capacitance insertion and sizing (or decap budgeting) problem taking into account operation and process variations

3. Decap Budgeting Overview
Function
Load current causes the voltage droop/bounce
Suppress dynamic noise by supplying sudden current demands from local charge storage
Side effect of adding too much decap
Increased leakage
Increased die area
Risk of yield loss
Location matters
The closer to the turbulent point, the more noise reduction can be achieved
Need to add minimum amount of decap at proper location, yet sufficient for reducing noise
Load current
power supply
intrinsic
cap
decap
We define the noise as the integral over time of the area below Vn t0 t1

4. Decap Budgeting Problem Formulation
Objective
Find the distribution and location of the white space so the noise on power network is minimized
Constraints:
Circuit system constraints: KCL, KVL and circuit element equations
Decap constraints: amount of decap allowed at a location is limited
Limitation of existing work:
Most existing work in essence uses worst case load current in order to guarantee there is no noise violation, which is too pessimistic
It is not clear how to provide decap budgeting solution that is robust to current load under all kinds of operations for a circuit

5. Major Contribution of our work
In this paper, we develop a novel stochastic model for current loads, taking into account operation variation such as temporal and logic-induced correlations and process variations such as systematic and random Leff variation.
We propose a formal method to extract operation variation and formulate a new decap budgeting problem using the stochastic current model.
We develop an effective yet efficient iterative alternative programming algorithm and conduct experiments using industrial designs.
Experiments show that considering both operation and process variations can reduce over-design significantly. This demonstrates the importance of considering operation variation.
We convincingly demonstrate the significance of considering both operation and process variations and open a new research direction for optimizing signal, power and thermal integrity with consideration of operation variation

6. Outline Stochastic Modeling and Problem Formulation Algorithm
Experimental Results
Conclusions

7. Correlated Load Currents
Strong correlation between load currents due to
Operation variation
Currents at different ports have logic-induced correlation
Large number of ports with limited control bits
Currents at certain ports cannot reach maximum at the same time due to the inherent logic dependency for a given design
Currents at the same port have temporal correlation
System takes several clock cycles to execute one instruction
The currents cannot reach maximum at all the clock cycles
Process variation
Currents have intra-die variation due to process variation
The P/G network is robust to process variation, but the load currents have intra-die variation because the circuit suffers from process variation.
Leff variation is one of the primary variation sources and the variation is spatially correlated [Cao:DAC’05]

8. Current Sampling
Model the current in each clock cycle as a triangular waveform and assume constant rising\falling time
Other current waveforms can be used. It will not affect the algorithm
In our verification, we use the detailed non-simplified current waveform
Partition a circuit into blocks and assume no correlation between different blocks [Najm:ICCAD’05]
Extensive simulation for each block to get the peak current value in each clock cycle and at each port.
Assume there is only temporal correlation within certain number of clock cycles L
L can be the number of clock cycles to execute certain function

9. Stochastic Current Modeling
Divide peak current values into different sets according to the clock cycle and port number
The set contains peak current values at port k and in clock cycle j, j+L, j+2L,…
Example: Take L=2, and consider two ports in 8 consecutive clock cycles
Define to be the stochastic variable with the sample set
For example, has the samples 0.1, 0.3, 0.5, 0.7, and therefore has mean value 0,4
The correlation between and reflects the temporal correlation between clock cycle j1 and j2
The correlation between and reflects the logic induced correlation between port k1 and k2.
clock cycles j, temporal correlation
port k,
logic-induced correlation

10. Extraction of Correlations
The logic-induced correlation coefficient between port k1 and k2 at clock cycle j can be computed as
Temporal correlation coefficient between clock cycle j1 and j2 at port k can be computed as
To take process variation into consideration, sample each multiple times over different region, and the above two formulas can still be applied
We use the general definition of correlation coefficient. In our case, we have two.

11. Extraction of Correlations
As is not Gaussian, apply Independent Component Analysis [Hyvarinen’01] to remove the correlation between and get a new set of independent variables r1, r2, …
Each can be represented by the linear combination of r1, r2,…
Accordingly the waveform at each clock cycle can be reconstructed from those r1,r2,…, i.e.,
The new variables ri catch both the operation and process variations.
We use the general definition of correlation coefficient. In our case, we have two.

12. Example of Extracted Temporal Correlation
The correlation map for peak currents between different clock cycles of one port from an industry application.
The P/G network is modeled as RC mesh
The load currents are obtained by detailed simulation of the circuit
It can be seen that the correlation matrix can be clearly divided into four trunks, and L can be set as 10

13. Parameterized MNA Formulation
Original MNA formulation
With the design variables - decap area wi, the G, C matrices can be expressed as
Together with the stochastic current model, the MNA formulation becomes:
With parameters wi and ri
The objective now is to find the optimal solution for those parameters
More specifically, find the wi values that minimize the noise with the ri corresponding to the load currents which introduce the maximum noise

14. Stochastic Decap Formulation
Minimize the maximum noise sum over all ports
Subject to the stochastic current variable upper/lower bound
Subject to
Individual decap area constraint due to placement constraints
Total decap area constraint
Non-convex min/max optimization problem
Difficult to find exact optimal solution

15. Outline Stochastic Modeling and Problem Formulation Algorithm
Experimental Results
Conclusions

16. Iterative Programming Algorithm
Each iteration we increase the white space allowed until all the white space has been used up or it converges
Find the optimal decap budgeting for the giving max droop/bounce
update the max droop/bounce
update the decap budgeting
Find the input corresponding to the max. droop/bounce for the given decap budgeting
Cannot guarantee optimality, but can guarantee convergence and efficiency
Experimental results show our algorithm can achieve good optimization results

17. Algorithm Outline

18. Illustration of Iterative Programming
A3: (P3)
A1: (P3)
A0: Initial
A2: (P2)
A0: Initial noise curve at one randomly selected port
A1: The noise curve under the optimal decap budgeting for a giving droop/bounce
A2: The noise curve with the input corresponding to the max. droop/bounce for the decap budgeting in A1
A3: The noise curve under the optimal decap budgeting for the giving max droop/bounce in A2

19. Impact of Incremental Step Size of White Space

20. Sequential Linear Programming
We apply sequential linear programming (sLP) to solve each of the two sub-problems. For each sub-problem, we iteratively do the following two steps until the solution converges:
Compute the sensitivities of all the variables to the first order by moment matching.
Linearize the objective function with the sensitivities and the optimization problem becomes an LP
first order sensitivities

21. Sequential Quadratic Programming
To improve accuracy, sequential quadratic programming (sQP) can be applied instead of sLP. For each sub-problem, we iteratively do the following two steps until the solution converges:
Compute the sensitivities of all the variables to the first order by moment matching.
Linearize the objective function with the sensitivities and the optimization problem becomes an LP
second order sensitivities

22. Forward Euler Integration
The computation of all the sensitivities require to solve the equations of the type
in time domain, i.e.,
The method used to solve those differential equations are called integration method.
We first discretize time t as t1, t2, …, with step size h, then by Taylor expansion of x(t), we have

23. Forward Euler Integration (cont’d)
By using the first order approximation, we have
Together with the original discretized differential equation
Then from we can easily get
However, those kind of explicit integration method have the common problem of numerical instability.

24. Backward Euler Integration
Instead of the explicit method, we use one more Taylor expansion
Insert into the expansion of x0(t), we have
Together with the original discretized equation,

25. Outline Stochastic Modeling and Problem Formulation Algorithm
Experimental Results
Conclusions

26. Impact of Current Correlations
Model 1
Maximum current at all ports
Model 2
Stochastic model with logic-induced correlation
Model 3
Model 2 + temporal correlation
Node #
Noise (V*s)
Runtime (s)
Model 1
Model 2
Model 3
1284
6.33e-7
1.28e-7
4.10e-8
104.2
161.2
282.3
10490
5.21e-5
1.09e-5
4.80e-6
973.2
1430
2199
42280
7.92e-4
5.38e-4
9.13e-5
2732
3823
5238
166380
1.34e-2
5.37e-3
2.28e-3
3625
5798
7821
avg 1
1/2.68X
1/9.10X
1.50X
2.26X
Compared with the model assuming maximum currents at all ports, under the same decap area,
Stochastic model with spatial correlation only reduce the noise by up to 3X
Stochastic model with both spatial and temporal correlation reduce the noise by up to 9X

27. Impact of Leff Variation
Node # X
V.R.
sLP
sLP + Leff
mean (V*s)
std (V*s)
runtime (s)
1284
10%
9.28e-7
3.97e-7
184.2
6.14e-7
1.38e-7 X
20%
9.43e-7
4.55e-7
6.38e-7
1.86e-7
10490
1.03e-4
4.79e-5
1121
7.22e-5
1.23e-5 X
1.22e-4
4.38e-5
7.94e-5
2.06e-5
42280
2.29e-3
9.72e-4
2236
8.23e-4
1.01e-4 X
4.43e-3
1.01e-3
8.28e-4
1.92e-4
166380
2.06e-2
9.91e-3
3824
5.31e-3
8.92e-4 X
2.31e-2
1.03e-2
5.92e-3
9.33e-4
avg 1
1/2.02X
1/5.05X
2.73X
1/1.95X
1/4.05X
Compared with the stochastic model without considering Leff variation, the stochastic model with it reduce the average noise by up to 4X and the 3-sigma noise by up to 13X

28. Comparison between sLP and sQP
In terms of noise, sQP is much better than sLP for large test cases and slightly worse for the small test case.
In terms of runtime, sLP is on average 3.25X faster than sQP

29. Conclusions
In this paper, we develop a novel stochastic model for current loads, taking into account operation variation such as temporal and logic-induced correlations and process variations such as systematic and random Leff variation.
We propose a formal method to extract operation variation and formulate a new decap budgeting problem using the stochastic current model.
We develop an effective yet efficient iterative alternative programming algorithm and conduct experiments using industrial designs. Experimental results show that the noise can be reduced by up to 9X.
We also apply similar idea to temperature-aware clock routing [Hao:ispd’07] and microprocessor floorplanning (Section 8C.2).

30. Thank you!