Importance Sampling in Rare Event Simulation

Rare events are an important limitation of discrete-event simulation in some areas of applications. The existing rare-event simulation (RES) techniques, such as importance sampling (IS), splitting, and RESTART, have been used for RES so far. However, these techniques not only have some limitations, but also are not general or easy to use. In this paper, the partition of the region (PoR) method is used as a variance reduction technique in a new RES technique for usage with a wide range of modeling and simulation languages. Three variants of the proposed technique will be introduced: simple PoR, recursive PoR, and advanced PoR. Simple and recursive PoR methods are powerful techniques that have some important features such as bounded relative error; however, they are restricted to some specific models. Advanced PoR is a more general technique that integrates the PoR method and the IS technique. To evaluate the efficiency of the proposed technique and its variants, the comparative simulation results of some illustrative examples using stochastic activity networks will be presented.

We review two types of adaptive Monte Carlo methods for rare event simulations. These methods are based on importance sampling. The first approach selects importance sampling distributions by minimizing the variance of importance sampling estimator. The second approach selects importance sampling distributions by minimizing the cross entropy to the optimal importance sampling distribution. We also review the basic concepts of importance sampling in the rare event simulation context. To make the basic concepts concrete, we introduce these ideas via the study of rare events of M/M/1 queues.

Importance sampling (IS) is a change-of-measure technique for speeding up the simulation of rare events in stochastic systems. In this paper, we presented an approach uses Importance Sampling technique for efficient estimation of software reliability via Markov software usage models in statistical testing. By suitable changes of the probabilities of state transitions during test, an iterative method based on the Ali-Silvey distance is proposed for this choice, and an unbiased reliability estimator with zero variance is obtained. A learning algorithm for the computation of optimal transition probabilities of the Markov chain usage model is also presented and experimental results of this algorithm are reported.

Rare event simulation in the context of queueing networks has been an active area of research for more than two decades. A commonly used technique to increase the efficiency of Monte Carlo simulation is importance sampling. However, there are few rigorous results on the design of efficient or asymptotically optimal importance sampling schemes for queueing networks. Using a recently developed game/subsolution approach, we construct simple and efficient state-dependent importance sampling schemes for simulating buffer overflows in stable open Jackson networks. The sampling distributions do not depend on the particular event of interest, and hence overflow probabilities for different events can be estimated simultaneously. A by-product of the analysis is the identification of the minimizing trajectory for the calculus of variation problem that is associated with the sample-path large deviation rate function.

When simulating small probabilities, say of order 10 -6 or less, by importance sampling, an established principle is to choose the importance sampling distribution as close to the conditional distribution given the rare event as possible. Implementing this often leads into large deviations calculations and exponential change of measure. We survey some of the standard examples where this approach works and supplement existing counterexamples with new ones. Difficulties often arise as consequence of reflecting barriers and we present an algorithm which at least in simple cases is able to deal with this problem. Also the case of heavy-tailed distributions is considered

Importance sampling is a change-of-measure technique for speeding up the simulation of rare events in stochastic systems. In this paper the authors establish a number of properties characterizing optimal importance sampling measures for Markovian systems. The authors show how these properties may be used to compute the optimal measure and give specific results for a tandem queueing system. The authors' approach has no apparent computational advantage over other direct methods, but it suggests a number of heuristic approximations which may lead to computationally attractive methods.

The Flexible Rare Event Sampling Harness System (FRESHS)

An importance sampling technique in Monte Carlo methods for SDEs with a.s. stable and mean-square unstable equilibrium

This paper surveys efficient techniques for estimating, via simulation, the probabilities of certain rare events in queueing and reliability models. The rare events of interest are long waiting times or buffer overflows in queueing systems, and system failure events in reliability models of highly dependable computing systems. The general approach to speeding up such simulations is to accelerate the occurrence of the rare events by using importance sampling. In importance sampling, the system is simulated using a new set of input probability distributions, and unbiased estimates are recovered by multiplying the simulation output by a likelihood ratio. Our focus is on describing asymptotically optimal importance sampling techniques. Using asymptotically optimal importance sampling, the number of samples required to get accurate estimates grows slowly compared to the rate at which the probability of the rare event approaches zero. In practice, this means that run lengths can be reduced by many orders of magnitude, compared to standard simulation. In certain cases, asymptotically optimal importance sampling results in estimates having bounded relative error. With bounded relative error, only a fixed number of samples are required to get accurate estimates, no matter how rare the event of interest is. The queueing systems studied include simple queues (e.g., GI/GI/1), Jackson networks, discrete time queues with multiple autocorrelated arrival processes that arise in the analysis of Asynchronous Transfer Mode communications switches, and tree structured networks of such switches. Both Markovian and non-Markovian reliability models are treated.

<p>Rare events, such as heat waves, floods, or hurricanes, play a crucial role in climate dynamics mainly due to the large impact they have. Predicting the occurrence of such events is thus a major challenge.<span> </span></p><p>In this talk, we introduce the relevant mathematical object for predicting a future event: the committor function is the probability that an event will occur, conditioned on the current state of the system. Computing this quantity from observations is an extremely difficult task since rare events have a very low probability of occurring and may not even have been observed in measurements made to date. Similarly, direct simulation of such events with comprehensive climate models comes at a prohibitive computational cost. Hence, rare event algorithms have been devised to simulate rare events efficiently, avoiding the computation of long periods of typical fluctuations.</p><p>The effectiveness of these algorithms strongly relies on the knowledge of a measure of how close the event of interest is to occur, called the “score function”. The main difficulty is that the optimal score function is the committor function which is exactly the quantity to be computed. Therefore, it is very natural to consider an iterative procedure where the data produced by the algorithm is used to improve the score function, which in turn improves the algorithm, and so on.</p><p>In this presentation, we propose a data-driven approach for computing the committor function, based on a Markov chain approximation of the dynamics of the system (the analogue method). We first illustrate this approach for a paradigmatic toy model of multistability for atmospheric dynamics with six variables (the Charney-Devore model). Secondly, we apply this methodology to data generated from a climate model, in order to study and predict the occurrence of extreme heat waves. In both cases, we show that it is possible to obtain fairly precise estimates of the committor function, even when few observations are available.</p><p>In the second part of the talk, we show the advantage of coupling the analogue Markov chain with a rare event algorithm. Indeed, the committor learned with the analogue Markov chain can be used as a score function performing better than user-defined score functions, as we show for the Charney-Devore model.<span> </span></p><p>This new approach is promising for studying rare events in complex dynamics: the rare events can be simulated with a minimal prior knowledge and the results are much more precise than those obtained with a user-designed score function.</p>

Simulation is widely used to estimate losses due to default and other credit events in financial portfolios. The accurate measurement of credit risk can be modeled as a rare event simulation problem. While Monte Carlo simulation is time-consuming for rare events, importance sampling techniques can effectively reduce the simulation time, thus improving simulation efficiency. This chapter proposes a new importance sampling method to estimate rare event probability in simulation models. The optimal importance sampling distributions are derived in terms of expectation in the normal copula model developed in finance. In the normal copula model, dependency is introduced through a set of common factors of multiple obligors. The intriguing dependence between defaults of multiple obligors imposes hurdles in simulation. The simulated results demonstrate the effectiveness of the proposed approach to solving the portfolio credit risk problem.

We consider stable open Jackson networks and study the rare events of excessive backlogs. Although, these events occur rarely they can be critical, since they can impair the functioning of the network. We attempt to estimate the probability of these events by simulations. Since, the direct simulation of rare events takes a very long time, this procedure is very costly. Instead, we devise a method for changing the network to speed up the simulation of rare events. We try to pursue this idea with the help of Large Deviation theory. This approach, under certain assumptions, results in a system of differential equations which may be difficult to solve. To circumvent this, we develop a heuristic method which gives the rule for changing the network for the purpose of simulations. We illustrate, by examples, that our method of simulations can be several orders of magnitude faster than direct simulations.

Extreme climate events have major impacts on human societies and ecosystems. The most detrimental events are often extremely rare, with return time of centuries or even millennia. Studying these events in the context of climate change is crucial to help adaptation efforts globally. Yet the study of these extremely rare events is extremely challenging due to the lack of data. Indeed, such events have likely not been observed in the instrumental period. An alternative is to use a global climate model to simulate these extremely rare events. However, this comes at a huge computational cost : gathering good statistics on centennial events would require to run a few thousand years of simulation. Rare event algorithms have recently been introduced in the field of climate science to tackle this difficulty [1]. By concentrating the computational effort on the trajectories most susceptible to lead to the extreme event of interest, they allow for the sampling of extremely rare events at a much lower computational cost than standard simulations. In this study, we run a rare event algorithm to sample extreme heatwave seasons in a heatwave hotspot of South Asia, using the intermediate complexity model Plasim. We compare the outcome of the algorithm against an extremely long – 8000 years – control run. This comparison allows us to demonstrate that the algorithm not only estimates return times with high precision (as shown in previous work), but also exhibits high precision in the estimation of composite statistics: composite maps conditioned on centennial heatwave seasons estimated from the algorithm, are in very good agreement with the ones from the 8000-year long control simulation. Our results suggest that extreme heatwave seasons in the studied region are associated with a quasi-stationary atmospheric wave-pattern stretching from the North Atlantic towards South Asia. We also show that the algorithm correctly estimates the intensity-duration-frequency statistics of subseasonal heatwaves occuring within centennial heatwave seasons. Thus rare event algorithms could, for instance, be combined with seasonal forecasts to provide information regarding expected number of heatwave days and the distribution of the duration and intensity in an extreme heatwave season, which could be useful for adaptation planning.  References[1] F. Ragone, J. Wouters, and F. Bouchet, “Computation of extreme heat waves in climate models using a large deviation algorithm,” Proc Natl Acad Sci USA, vol. 115, pp. 24–29, Jan. 2018. 

In this paper, we present a variance minimization (VM) procedure for rare event simulation in tandem queueing networks. We prove that the VM method can produce a zero variance. The VM method is suitable to compute optimal importance sampling (IS) parameters for small scale tandem networks. For large scale tandem networks we propose a sub-optimal IS (SOIS) method, which projects the optimal biased transition probabilities of the corresponding small scale system into those of a large scale system. In other words, we establish an efficient IS method for a large scale system by zooming into a small scale system and then projecting our findings into the large scale system. The numerical results show that our SOIS method can produce accurate results with very short CPU time, while many other methods often require much longer.

Birth of a new distinct phase is a phenomenon encountered in a myriad of processes, and has wide ranging consequences in material processing, biological self-assembly, separations and several other processes. Several phase transitions are nucleation driven. The nucleation events occur over nanosecond timescales and involve hundreds to thousands of molecules. These length and timescales are difficult to access in experiments, thereby making experimental studies of nucleation challenging. On the other hand, molecular simulations sample the nanosecond and nanometer scales making them ideal to study nucleation. However, nucleation is a rare event, meaning that the waiting time to observe one nucleation event is significant. This makes simulation studies of rare events challenging. The project focused on a multi-pronged approach to address such challenges to develop the next generation rare event sampling methods for molecular simulations. The key outcomes of our work include developing more effective methods for sampling rare events, utilizing machine learning to better elucidate nucleation mechanisms, development of software for easy implementation of the methodologies, and applications of the methods to realistic systems to push the method applicability beyond model systems. Overall, this work has enabled pushing the frontiers of molecular simulations to study rare events with a focus on nucleation in aqueous solutions.