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.

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

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

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

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

Outline Stochastic Modeling and Problem Formulation Algorithm Experimental Results Conclusions

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]

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

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

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.

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.

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

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

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

Outline Stochastic Modeling and Problem Formulation Algorithm Experimental Results Conclusions

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

Algorithm Outline

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

Impact of Incremental Step Size of White Space

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

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

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

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.

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,

Outline Stochastic Modeling and Problem Formulation Algorithm Experimental Results Conclusions

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

Impact of Leff Variation Node #3429 3.06X 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-7 332.8 1.81X 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 3429 3.06X 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 6924 3.10X 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 11224 2.93X 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

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

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).

Thank you!