1. Forward Discrete Probability Propagation Rasit Onur Topaloglu Ph.D. candidate rot@ucsd.edu

2. SPICE Hierarchy and the Problem ex: g m =  2*k*I D Level1 Level4 Level0 TL eff W eff t ox Level2 Level3 00 NSUB  ms C ox FF k Q dep V th IDID gmgm r out Level5 SPICE formulas are hierarchical; hence can progressively relate physical parameters to device parameters in connectivity graphs Recent models attribute process variations to physical parameters physical electrical/ mathematical Probability density functions can be input at Level0 independently

3. Probability Propagation Estimation of device parameters at highest level needed to examine effects of process variations An analytic solution not possible since functions highly non- linear and Gaussian approximations not accurate in deep sub-micron GOALS Algebraic tractability : enable manual applicability by designers Speed : be comparable or outperform Monte Carlo in quick estimation A method to propagate pdf’s to highest level necessary Flexibility : be able to use non-standard densities to outperform parametric belief propagation

4. Shortcomings of Monte Carlo No algebraically tractability : No manual estimation by designers possible due to large number of iterations and random sampling Limited to standard distributions : Random number generators in CAD tools only provide certain distributions, hence a new module usually needs to be programmed Speed : No quick convergence to an estimate distribution due to random sampling unless a large number of costly iterations employed

5. A Reminder on Applying Monte Carlo for Probability Propagation gmgm Level1 Level4 Level0  n n V FB NSUB LW V th C ox t ox IDID k Level2 Level3 Pick independent samples from distributions of Level0 parameters Compute functions using these samples until highest level reached Construct a histogram to approximate the distribution Repeat while desired accuracy is not yet reached:

6. Parametric Belief Propagation Each node receives and sends messages to parents and children until equilibrium Parent to child (  ) : causal information Parent to parent ( ) : diagnostic information Calculations handled at each node:

7. Parametric Belief Propagation When arrows in the hierarchy tree indicate linear addition operations on Gaussians, analytic formulations possible Not straightforward for other distributions or non- standard distributions

8. Implementing FDPP F (Forward) : Given a function, estimates the distribution of next node in the formula hierarchy using samples Q (Quantize) : Discretizes a pdf to operate on its samples Analytic operation on continuous distributions difficult; instead work in discrete domain and convert back to continuous domain at the end: Q -1 (De-Quantize) : Converts a discrete pdf back to continuous domain : implemented as an interpolation function B (Band-pass) : Decrements number of samples using a threshold on sample probabilities R (Re-bin) : Decreases number of samples by combining close samples together

9. T NSUB PHIf Necessary Operators (Q, F, B, R, Q -1 ) on a Connectivity Graph QQ F B F, B and R repeated until we acquire the distribution of a high level parameter; Q and Q -1 used just once R Q -1 TL eff W eff t ox 00 NSUB  ms C ox FF k V th IDID gmgm r out Q dep

10. pdf(X) Q Operator Q N band-pass filters pdf(X) and divides into bins N in Q N indicates number of bins spdf(X)=  (X) X pdf(X) spdf(X) X Use N>(2/m), where m is maximum derivative of pdf(X), thereby obeying a bound similar to Nyquist If quantizer uniform and  small, quantization error random variable Q is uniformly distributed, then Variance of quantization error: Increase number of bins to reduce quantization error

11. F Operator F operator implements a function over spdf’s using deterministic sampling Corresponding function in connectivity graph applied to deterministic pair-wise combination of impulse values to get the value of the new sample Heights of impulses (probabilities) multiplied to get probability of new sample

12. Effect of Non-linear Functions Application of functions cause accumulation in certain ranges Band-pass and re-bin operations needed after F operation Impulses after F, before B and R De-quantization would not result in a pdf Increased number of samples would induce a computational burden

13. Band-pass, B e, Operator Eliminate samples having values out of range (6  ): might cut off tails of bi-modal or long-tailed distributions Margin-based Definition: Error-based Definition: Eliminate samples having probabilities least likely to occur : can also eliminate samples in useful range hence offers more computational efficiency Implementation : eliminate samples with probabilities less than 1/e times the sample with the largest probability e should be chosen such that it is smaller than the ratio of products of maximum and minimum probability samples for nodes to which F is applied

14. Re-bin, R N, Operator Impulses after F Resulting spdf(X) Unite into one  bin Samples falling into the same bin congregated in one Total distortion given by m i : center of i’th bin can be used to select bin locations, where

15. Experimental Results Impulse representation for threshold voltage and transconductance are obtained through FDPP on the graph  (X) for gm  (X) for V th Matlab R12 used to evaluate FDPP method

16. A close match is observed after interpolation Monte Carlo – FDPP Comparison solid : FDPP dotted : Monte Carlo Pdf* of V th Pdf* of I D Correlation error introduced by the independence assumption of F operator results in negligible error as R operator helps distribute this error over the pdf state space

17. Monte Carlo – FDPP Comparison with a Low Sample Number Monte Carlo inaccurate for moderate number of samples Indicates FDPP can converge to an acceptable estimate with far less number of samples solid : FDPP with 100 samples Pdf* of  F noisy : Monte Carlo with 1000 samples solid : FDPP with 100 samples noisy : Monte Carlo with 100000 samples

18. Edges define a linear sum, ex: n5=n2+n3 Monte Carlo – FDPP Comparison Pdf of n7Benchmark example solid : FDPP dotted : Monte Carlo triangles:belief propagation Monte Carlo result is separated as FDPP and belief propagation neglect correlation

19. When distributions at internal nodes n4, n5, n6 re-sampled using Monte Carlo, all methods converge Faulty Application of Monte Carlo Pdf of n7Benchmark example solid : FDPP dotted : Monte Carlo triangles:belief propagation

20. Conclusions Forward Discrete Probability Propagation is introduced as an alternative to Monte Carlo and parametric belief propagation methods for quick estimation : FDPP should be preferred to MC when a faster convergence to real distribution is necessary with limited number of samples FDPP provides an algebraic intuition due to deterministic sampling and manual applicability due to using less number of samples FDPP can account for non-standard pdf’s where parametric methods are limited to certain ones