Statistical static timing analysis using Markov chain Monte Carlo

Approximate Bayesian computation (ABC) techniques permit inferences in complex demographic models, but are computationally inefficient. A Markov chain Monte Carlo (MCMC) approach has been proposed (Marjoram et al. 2003), but it suffers from computational problems and poor mixing. We propose several methodological developments to overcome the shortcomings of this MCMC approach and hence realize substantial computational advances over standard ABC. The principal idea is to relax the tolerance within MCMC to permit good mixing, but retain a good approximation to the posterior by a combination of subsampling the output and regression adjustment. We also propose to use a partial least-squares (PLS) transformation to choose informative statistics. The accuracy of our approach is examined in the case of the divergence of two populations with and without migration. In that case, our ABC-MCMC approach needs considerably lower computation time to reach the same accuracy than conventional ABC. We then apply our method to a more complex case with the estimation of divergence times and migration rates between three African populations.

In recent years, there has been a lot of research into statistical static timing analysis (SSTA) to compute the critical path delay of a circuit under timing variation. In order to compute the true critical path delay, however, false paths that cannot be sensitized by any input vector must be identified first. Since SSTA is unable to capture the dynamic timing behavior of a circuit, it is completely blind to false paths, and thus it overestimates the circuit timing. In this paper, we propose a new concept of timing analysis approach called statistical dynamic timing analysis (SDTA), which is able to precisely express the statistical behavior of dynamic transitions at the output of gate into a compact form and to directly evaluate and propagate the expressions throughout the circuit, by which the false paths can be cleaned effectively. In addition, to be practical, we propose a couple of techniques that enable a fast computation of the SDTA. We tested the proposed approach on ISCAS benchmarks and carry skip adders under timing variation to show its accuracy in computing the distribution of the true critical path delay of a circuit. In summary, compared to the previous approach of false path-aware statistical timing analysis, our timing analysis technique is able to reduce the accuracy error in the mean and standard deviation of true critical path delay distribution from 9.8% to 1.9% and from 29.4% to -3.4%, respectively.

Existing statistical static timing analysis (SSTA) techniques suffer from limited modeling capability by using a linear delay model with Gaussian distribution, or have scalability problems due to expensive operations involved to handle non-Gaussian variation sources or non-linear delays. To overcome these limitations, we propose a novel SSTA technique to handle both nonlinear delay dependency and non- Gaussian variation sources simultaneously. We develop efficient algorithms to perform all statistical atomic operations (such as max and add) efficiently via either closed- form formulas or one-dimensional lookup tables. The resulting timing quantity provably preserves the correlation with variation sources to the third-order. We prove that the complexity of our algorithm is linear in both variation sources and circuit sizes, hence our algorithm scales well for large designs. Compared to Monte Carlo simulation for non-Gaussian variation sources and nonlinear delay models, our approach predicts all timing characteristics of circuit delay with less than 2% error.

In this paper, we propose a calculation method of gate delay for SSTA (Statistical Static Timing Analysis) considering MIS (Multiple Input Switching). In SSTA, statistical maximum/minimum operation is necessary to calculate the latest/fastest arrival time of multiple input gate. Most SSTA approaches calculate the distribution in the latest/fastest arrival time under SIS (Single Input Switching assumption), resulting in ignoring the effect of MIS on the gate delay and the output transition time. MIS occurs when multiple inputs of a gate switch nearly simultaneously. Thus, ignoring MIS causes error in the statistical maximum/minimum operation in SSTA. We propose a statistical gate delay model considering MIS. We verify the proposed method by SPICE based Monte Carlo simulations. Experimental results show that the neglect of MIS effect leads to 80% error in worst case. The error of the proposed method is less than 20%.

Variational Bayes approximation of factor stochastic volatility models

Statistical static timing analysis (SSTA) methods, which model process variations statistically as probability distribution function rather than deterministically, have been thoroughly performed on traditional zero clock skew circuits. In the traditional zero clock skew circuits, the synchronizing clock signal is designed to arrive in phase with respect to each register. However, designers will often schedule the clock skew to different registers in order to decrease the minimum clock period of the entire circuit. Clock skew scheduling imparts very different timing constraints that are based, in part, on the topology of the circuit. In this paper, SSTA is applied to nonzero clock skew circuits in order to determine the accuracy improvement relative to their zero skew counterparts, and also to assess how the results of skew scheduling might be impacted with more accurate statistical modeling. For 99.7% timing yield (3σ variation), SSTA is observed to improve the accuracy, and therefore increase the timing margin, of nonzero clock skew circuits by up to 2.5×, and on average by 1.3×, the amount seen by zero skew circuits.

The recent proliferation of Markov Chain Monte Carlo (MCMC) approaches has led to the use of the Bayesian inference in a wide variety of fields, including reliability engineering. With the current (and future) proliferation of new products, old problems continue to hamper us, while new challenges keep appearing. In Bayesian reliability, these include but are not limited to: (1) achieving and making use of prior information; (2) applying small data sets or system operating/environmental (SOE) data with big and complex data; and (3) making posterior inferences from high-dimensional numerical integration. To deal with old problems while meeting new challenges, this paper proposes an improved procedure for Bayesian reliability inference with MCMC, discussing modern reliability data and noting some applications where the Bayesian reliability approach with MCMC can be used. It also explores opportunities to use Bayesian reliability models to create stronger statistical methods from prior to posterior. Finally, it outlines some practical concerns and remaining challenges for future research.

In genetic counseling for cancer risk, the probability of carrying a mutation of a cancer-causing gene plays an important role. Family history of various cancers is important in calculating this probability. BRCAPRO is a widely used software for calculating the probability of carrying mutations in BRCA1 and BRCA2 genes given the family history of breast and ovarian cancer in first- and second-degree relatives. BRCAPRO uses an analytical (exact) calculational procedure. Using Markov chain Monte Carlo (MCMC) methods, we extend BRCAPRO to handle, in principle, any type of cancer, family history, any number of genes and alleles that each gene may have. When the information used in this MCMC approach is the same as for BRCAPRO (two genes: BRCA1 and BRCA2; two cancers: breast and ovarian; first- and second-degree relatives only), the two approaches give essentially the same answer. Extending the model to include (1) prostate cancer, (2) two mutated alleles of BRCA2, namely, mutations in Ovarian Cancer Cluster Region (OCCR) and non-OCCR region, and (3) relatives of degree greater than second-degree, leads to different carrier probabilities. The MCMC approach is a useful tool in building a comprehensive model to give accurate estimates of carrier probabilities. Such an approach will be even more important as additional information about the genetics of various cancers becomes available.

Density-estimation likelihood-free inference (DELFI) has recently been\nproposed as an efficient method for simulation-based cosmological parameter\ninference. Compared to the standard likelihood-based Markov Chain Monte Carlo\n(MCMC) approach, DELFI has several advantages: it is highly parallelizable,\nthere is no need to assume a possibly incorrect functional form for the\nlikelihood and complicated effects (e.g the mask and detector systematics) are\neasier to handle with forward models. In light of this, we present two DELFI\npipelines to perform weak lensing parameter inference with lognormal\nrealizations of the tomographic shear field -- using the C_l summary statistic.\nThe first pipeline accounts for the non-Gaussianities of the shear field,\nintrinsic alignments and photometric-redshift error. We validate that it is\naccurate enough for Stage III experiments and estimate that O(1000) simulations\nare needed to perform inference on Stage IV data. By comparing the second DELFI\npipeline, which makes no assumption about the functional form of the\nlikelihood, with the standard MCMC approach, which assumes a Gaussian\nlikelihood, we test the impact of the Gaussian likelihood approximation in the\nMCMC analysis. We find it has a negligible impact on Stage IV parameter\nconstraints. Our pipeline is a step towards seamlessly propagating all\ndata-processing, instrumental, theoretical and astrophysical systematics\nthrough to the final parameter constraints.\n

We propose a new method in which a generative network (GN) set within a reduced-order model (ROM) framework is used to solve inverse problems for partial differential equations (PDE). The aim is to match available measurements and estimate the corresponding uncertainties associated with the states and parameters of a numerical physical simulation. We train the GN using only unconditional simulations of the discretized PDE model. A GN is used here to exploit the ability of these methods to generate realistic outputs from complex probability distributions, such as the ones that represent the possible states and parameters of a physical problem. We compare the proposed method with the gold standard Markov chain Monte Carlo (MCMC). Additionally, we suggest the use of a novel type of time-stepping regularization to improve the representativeness of the physical solution, and we present a new way of evaluating the GN training, taking advantage of the real/generated sample structure. We apply the proposed approaches to a spatio-temporal compartmental model in epidemiology. The results show that the proposed GN-based ROM can efficiently quantify uncertainty and accurately match the measurements and the gold standard. This is achieved using only a limited number of unconditional simulations from the full-order numerical PDE model (40 simulations). The GN-based ROM operates 60 times faster than the gold standard (MCMC), while producing uncertainties that closely match those generated by the MCMC approach. The proposed method is a general framework for quantifying uncertainties in numerical physical simulations and is not restricted to the specific physics of this application.

Medical imaging systems are commonly assessed and optimized by use of objective measures of image quality (IQ). The Ideal Observer (IO) performance has been advocated to provide a figure-of-merit for use in assessing and optimizing imaging systems because the IO sets an upper performance limit among all observers. When joint signal detection and localization tasks are considered, the IO that employs a modified generalized likelihood ratio test maximizes observer performance as characterized by the localization receiver operating characteristic (LROC) curve. Computations of likelihood ratios are analytically intractable in the majority of cases. Therefore, sampling-based methods that employ Markov-Chain Monte Carlo (MCMC) techniques have been developed to approximate the likelihood ratios. However, the applications of MCMC methods have been limited to relatively simple object models. Supervised learning-based methods that employ convolutional neural networks have been recently developed to approximate the IO for binary signal detection tasks. In this paper, the ability of supervised learning-based methods to approximate the IO for joint signal detection and localization tasks is explored. Both background-known-exactly and background-known-statistically signal detection and localization tasks are considered. The considered object models include a lumpy object model and a clustered lumpy model, and the considered measurement noise models include Laplacian noise, Gaussian noise, and mixed Poisson-Gaussian noise. The LROC curves produced by the supervised learning-based method are compared to those produced by the MCMC approach or analytical computation when feasible. The potential utility of the proposed method for computing objective measures of IQ for optimizing imaging system performance is explored.

In this paper, we explore the implementation of Monte Carlo based statistical static timing analysis (SSTA) on a Graphics Processing Unit (GPU). SSTA via Monte Carlo simulations is a computationally expensive, but important step required to achieve design timing closure. It provides an accurate estimate of delay variations and their impact on design yield. The large number of threads that can be computed in parallel on a GPU suggests a natural fit for the problem of Monte Carlo based SSTA to the GPU platform. Our implementation performs multiple delay simulations at a single gate in parallel. A parallel implementation of the Mersenne Twister pseudo-random number generator on the GPU, followed by Box-Muller transformations (also implemented on the GPU) is used for generating gate delay numbers from a normal distribution. The μ and σ of the pin-to-output delay distributions for all inputs and for every gate, are obtained using a memory lookup, which benefits from the large memory bandwidth of the GPU. Threads which execute in parallel have no data/control dependencies on each other. All threads compute identical instructions, but on different data, as required by the Single Instruction Multiple Data (SIMD) programming semantics of the GPU. Our approach is implemented on a NVIDIA GeForce GTX 8800 GPU card. Our results indicate that our approach can obtain an average speedup of about 260x as compared to a serial CPU implementation. With the recently announced quad 8800 GPU cards, we estimate that our approach would attain a speedup of over 785x. The correctness of the Monte Carlo based SSTA implemented on a GPU has been verified by comparing its results with a CPU based implementation.

Traditional multiple target tracking (MTT) algorithms such as JPDA, MHT, a basic assume that the targets are points source. This is unrealistic in many cases. We consider targets with certain geometrical shape and they may give multiple measurements using the Markov Chain Monte Carlo (MCMC) approach. We aim at estimating the states of targets, their shape parameters, and number of targets. The proposed approach is based on the clustering process of finite mixture models (FMM), where the parameters of the FMM are obtained by the MCMC sampler. The states of the targets are estimated by equivalent measurement (EQM). The final experiment of three target tracking verifies the proposed algorithm.

This paper introduces a novel compact implicit model for a probabilistic set of waveforms (PSoW) which arise as representations for uncertain signal waveforms in Statistical Static Timing Analysis (SSTA). In traditional SSTA tools, signals are just represented as (distributions of) arrival time and slew. In our approach, to increase accuracy, PSoW's are used instead. However, to represent PSoW's explicitly, a very large amount of data is necessary, which can be problematic. To solve this problem, a compact implicit model is introduced, which can be characterized with just a handful of parameters. The results obtained show that the implicit model can generate real-life PSoW's with high accuracy.

Bayesian identification of a cracked plate using a population-based Markov Chain Monte Carlo method