A Coupling Approach to Rare Event Simulation Via Dynamic Importance Sampling
A Coupling Approach to Rare Event Simulation Via Dynamic Importance Sampling

Author: Benjamin Jiahong Zhang

Publisher:

Published: 2017

Total Pages: 109

ISBN-13:

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Rare event simulation involves using Monte Carlo methods to estimate probabilities of unlikely events and to understand the dynamics of a system conditioned on a rare event. An established class of algorithms based on large deviations theory and control theory constructs provably asymptotically efficient importance sampling estimators. Dynamic importance sampling is one these algorithms in which the choice of biasing distribution adapts in the course of a simulation according to the solution of an Isaacs partial differential equation or by solving a sequence of variational problems. However, obtaining the solution of either problem may be expensive, where the cost of solving these problems may be even more expensive than performing simple Monte Carlo exhaustively. Deterministic couplings induced by transport maps allows one to relate a complex probability distribution of interest to a simple reference distribution (e.g. a standard Gaussian) through a monotone, invertible function. This diverts the complexity of the distribution of interest into a transport map. We extend the notion of transport maps between probability distributions on Euclidean space to probability distributions on path space following a similar procedure to Itô’s coupling. The contraction principle is a key concept from large deviations theory that allows one to relate large deviations principles of different systems through deterministic couplings. We convey that with the ability to computationally construct transport maps, we can leverage the contraction principle to reformulate the sequence of variational problems required to implement dynamic importance sampling and make computation more amenable. We apply this approach to simple rotorcraft models. We conclude by outlining future directions of research such as using the coupling interpretation to accelerate rare event simulation via particle splitting, using transport maps to learn large deviations principles, and accelerating inference of rare events.

Introduction to Rare Event Simulation
Introduction to Rare Event Simulation

Author: James Bucklew

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 262

ISBN-13: 1475740786

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This book presents a unified theory of rare event simulation and the variance reduction technique known as importance sampling from the point of view of the probabilistic theory of large deviations. It allows us to view a vast assortment of simulation problems from a unified single perspective.

Importance Sampling
Importance Sampling

Author: Rajan Srinivasan

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 252

ISBN-13: 3662050528

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This research monograph deals with fast stochastic simulation based on im portance sampling (IS) principles and some of its applications. It is in large part devoted to an adaptive form of IS that has proved to be effective in appli cations that involve the estimation of probabilities of rare events. Rare events are often encountered in scientific and engineering processes. Their charac terization is especially important as their occurrence can have catastrophic consequences of varying proportions. Examples range from fracture due to material fatigue in engineering structures to exceedance of dangerous levels during river water floods to false target declarations in radar systems. Fast simulation using IS is essentially a forced Monte Carlo procedure designed to hasten the occurrence of rare events. Development of this simu lation method of analysis of scientific phenomena is usually attributed to the mathematician von Neumann, and others. Since its inception, MC simula tion has found a wide range of employment, from statistical thermodynamics in disordered systems to the analysis and design of engineering structures characterized by high complexity. Indeed, whenever an engineering problem is analytically intractable (which is often the case) and a solution by nu merical techniques prohibitively expensive computationally, a last resort to determine the input-output characteristics of, or states within, a system is to carry out a simulation.

Sequential Methods for Rare Event Simulations
Sequential Methods for Rare Event Simulations

Author: Shaojie Deng

Publisher:

Published: 2010

Total Pages:

ISBN-13:

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We consider rare events modeled as a Markov Chain hitting a certain rare set. A sequential importance sampling with resampling (SISR) method is introduced to provide a versatile approach for computing such probabilities of rare events. The method uses resampling to track the zero-variance importance measure associated with the event of interest. A general methodology for choosing the importance measure and resampling scheme to come up with an efficient estimator of the probability of occurrence of the rare event is developed and the distinction between light-tailed and heavy-tailed problems is highlighted. Applications include classic tail probabilities for sums of independent light-tailed or heavy-tailed random variables. Markovian extensions and simultaneous simulation are also given. The heuristics and the methodology can also be applied to more complex Monte Carlo problems that arise in recent works on the dynamic portfolio credit risk model.

Rare Event Simulation using Monte Carlo Methods
Rare Event Simulation using Monte Carlo Methods

Author: Gerardo Rubino

Publisher: John Wiley & Sons

Published: 2009-03-18

Total Pages: 278

ISBN-13: 9780470745410

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In a probabilistic model, a rare event is an event with a very small probability of occurrence. The forecasting of rare events is a formidable task but is important in many areas. For instance a catastrophic failure in a transport system or in a nuclear power plant, the failure of an information processing system in a bank, or in the communication network of a group of banks, leading to financial losses. Being able to evaluate the probability of rare events is therefore a critical issue. Monte Carlo Methods, the simulation of corresponding models, are used to analyze rare events. This book sets out to present the mathematical tools available for the efficient simulation of rare events. Importance sampling and splitting are presented along with an exposition of how to apply these tools to a variety of fields ranging from performance and dependability evaluation of complex systems, typically in computer science or in telecommunications, to chemical reaction analysis in biology or particle transport in physics. Graduate students, researchers and practitioners who wish to learn and apply rare event simulation techniques will find this book beneficial.

Efficient Analysis of Rare Events Associated with Individual Buffers in a Tandem Jackson Network
Efficient Analysis of Rare Events Associated with Individual Buffers in a Tandem Jackson Network

Author:

Publisher:

Published: 2004

Total Pages:

ISBN-13:

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For more than a decade, importance sampling has been a popular technique for the efficient estimation of rare event probabilities. This thesis presents an approach for applying balanced likelihood ratio importance sampling to estimate rare event probabilities in tandem Jackson networks. The rare event of interest is the probability that the content of the second buffer in a two node tandem Jackson network reaches some high level before it empties. Heuristic importance sampling distributions are derived that can be used to estimate this overflow probability in cases where the first buffer capacity is finite and infinite. In the proposed methods, the transition probabilities of the embedded discrete-time Markov chain are modified dynamically to bound the overall likelihood ratio of each cycle. The proposed importance sampling distributions differ from previous balanced likelihood ratio methods in that they are specified as functions of the contents of the buffers. When the first buffer capacity is infinite, the proposed importance sampling estimator yields bounded relative error except when the first server is the bottleneck. In the latter case, numerical results suggest that the relative error is linearly bounded in the buffer size. When the first buffer capacity is finite, empirical results indicate that the relative errors of these importance sampling estimators are bounded independent of the buffer size when the second server is the bottleneck and bounded linearly in the buffer size otherwise.

Simulation and the Monte Carlo Method
Simulation and the Monte Carlo Method

Author: Reuven Y. Rubinstein

Publisher: John Wiley & Sons

Published: 2016-10-21

Total Pages: 470

ISBN-13: 1118632389

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This accessible new edition explores the major topics in Monte Carlo simulation that have arisen over the past 30 years and presents a sound foundation for problem solving Simulation and the Monte Carlo Method, Third Edition reflects the latest developments in the field and presents a fully updated and comprehensive account of the state-of-the-art theory, methods and applications that have emerged in Monte Carlo simulation since the publication of the classic First Edition over more than a quarter of a century ago. While maintaining its accessible and intuitive approach, this revised edition features a wealth of up-to-date information that facilitates a deeper understanding of problem solving across a wide array of subject areas, such as engineering, statistics, computer science, mathematics, and the physical and life sciences. The book begins with a modernized introduction that addresses the basic concepts of probability, Markov processes, and convex optimization. Subsequent chapters discuss the dramatic changes that have occurred in the field of the Monte Carlo method, with coverage of many modern topics including: Markov Chain Monte Carlo, variance reduction techniques such as importance (re-)sampling, and the transform likelihood ratio method, the score function method for sensitivity analysis, the stochastic approximation method and the stochastic counter-part method for Monte Carlo optimization, the cross-entropy method for rare events estimation and combinatorial optimization, and application of Monte Carlo techniques for counting problems. An extensive range of exercises is provided at the end of each chapter, as well as a generous sampling of applied examples. The Third Edition features a new chapter on the highly versatile splitting method, with applications to rare-event estimation, counting, sampling, and optimization. A second new chapter introduces the stochastic enumeration method, which is a new fast sequential Monte Carlo method for tree search. In addition, the Third Edition features new material on: • Random number generation, including multiple-recursive generators and the Mersenne Twister • Simulation of Gaussian processes, Brownian motion, and diffusion processes • Multilevel Monte Carlo method • New enhancements of the cross-entropy (CE) method, including the “improved” CE method, which uses sampling from the zero-variance distribution to find the optimal importance sampling parameters • Over 100 algorithms in modern pseudo code with flow control • Over 25 new exercises Simulation and the Monte Carlo Method, Third Edition is an excellent text for upper-undergraduate and beginning graduate courses in stochastic simulation and Monte Carlo techniques. The book also serves as a valuable reference for professionals who would like to achieve a more formal understanding of the Monte Carlo method. Reuven Y. Rubinstein, DSc, was Professor Emeritus in the Faculty of Industrial Engineering and Management at Technion-Israel Institute of Technology. He served as a consultant at numerous large-scale organizations, such as IBM, Motorola, and NEC. The author of over 100 articles and six books, Dr. Rubinstein was also the inventor of the popular score-function method in simulation analysis and generic cross-entropy methods for combinatorial optimization and counting. Dirk P. Kroese, PhD, is a Professor of Mathematics and Statistics in the School of Mathematics and Physics of The University of Queensland, Australia. He has published over 100 articles and four books in a wide range of areas in applied probability and statistics, including Monte Carlo methods, cross-entropy, randomized algorithms, tele-traffic c theory, reliability, computational statistics, applied probability, and stochastic modeling.

Simulation Methods for Rare Events in Nonlinear Lightwave Systems
Simulation Methods for Rare Events in Nonlinear Lightwave Systems

Author:

Publisher:

Published: 2007

Total Pages: 6

ISBN-13:

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The objectives of this project were to develop new hybrid analytical/computational methods that are capable of simulating the rare events that are the determining factors of the performance of lightwave systems and devices. These methods use the following: (1) analytical techniques, such as perturbation and asymptotic methods, to guide numerical simulations using importance sampling; and (2) adaptive numerical methods, such as the multicanonical Monte Carlo and cross-entropy methods, to perform the simulation of rare events when guiding analytical models are not available. The above methods can be used to evaluate the performance of specific optical systems and devices, including ultra-high-precision optical clocks based upon mode-locked fiber lasers, and optical clocks and other devices based upon hybrid opto-electronic oscillators. In each case, the goal is to use the methods to develop models that can accurately predict the performance of these devices, as well as determine the failure modes that are the limiting factors in their performance. The author has developed methods based upon soliton perturbation theory and importance sampling to simulate rare events in lightwave systems, including mode-locked laser systems. A key step to using the methods based upon soliton perturbation theory is to use an approximate version of the system dynamics to determine the locations in the large-dimensional state space that most contribute to the desired rare events (e.g., errors). In this method, calculus of variations applied to the approximate system allows the most significant rare events to be located, and then fully detailed importance-sampled Monte-Carlo simulations in the vicinity of these locations properly determines the probabilities of these rare events and corrects for any errors made by the approximations in determining the system dynamics.

Splitting for Rare Event Simulation: A Large Deviations Approach to Design and Analysis
Splitting for Rare Event Simulation: A Large Deviations Approach to Design and Analysis

Author:

Publisher:

Published: 2007

Total Pages: 35

ISBN-13:

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Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set B before another set A, and it is assumed that this probability satisfies a large deviation scaling. A notion of subsolution is defined for the related calculus of variations problem, and two main results are proved under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized "asymptotically" in terms of the subsolution. The design of asymptotically optimal schemes is discussed, and numerical examples are presented.