546 research outputs found
Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks
We consider a standard splitting algorithm for the rare-event simulation of
overflow probabilities in any subset of stations in a Jackson network at level
n, starting at a fixed initial position. It was shown in DeanDup09 that a
subsolution to the Isaacs equation guarantees that a subexponential number of
function evaluations (in n) suffice to estimate such overflow probabilities
within a given relative accuracy. Our analysis here shows that in fact
O(n^{2{\beta}+1}) function evaluations suffice to achieve a given relative
precision, where {\beta} is the number of bottleneck stations in the network.
This is the first rigorous analysis that allows to favorably compare splitting
against directly computing the overflow probability of interest, which can be
evaluated by solving a linear system of equations with O(n^{d}) variables.Comment: 23 page
Marginalization for rare event simulation in switching diffusions
In this paper we use splitting technique to estimate the probability of
hitting a rare but critical set by the continuous component of a switching
diffusion. Instead of following classical approach we use Wonham filter to
achieve multiple goals including reduction of asymptotic variance and exemption
from sampling the discrete components
Large deviations principle for the Adaptive Multilevel Splitting Algorithm in an idealized setting
The Adaptive Multilevel Splitting (AMS) algorithm is a powerful and versatile
method for the simulation of rare events. It is based on an interacting (via a
mutation-selection procedure) system of replicas, and depends on two integer
parameters: n N * the size of the system and the number k {1, . . .
, n -- 1} of the replicas that are eliminated and resampled at each iteration.
In an idealized setting, we analyze the performance of this algorithm in terms
of a Large Deviations Principle when n goes to infinity, for the estimation of
the (small) probability P(X \textgreater{} a) where a is a given threshold and
X is real-valued random variable. The proof uses the technique introduced in
[BLR15]: in order to study the log-Laplace transform, we rely on an auxiliary
functional equation. Such Large Deviations Principle results are potentially
useful to study the algorithm beyond the idealized setting, in particular to
compute rare transitions probabilities for complex high-dimensional stochastic
processes
A comparison of RESTART implementations
The RESTART method is a widely applicable simulation technique for the estimation of rare event probabilities. The method is based on the idea to restart the simulation in certain system states, in order to generate more occurrences of the rare event. One of the main questions for any RESTART implementation is how and when to restart the simulation, in order to achieve the most accurate results for a fixed simulation effort. We investigate and compare, both theoretically and empirically, different implementations of the RESTART method. We find that the original RESTART implementation, in which each path is split into a fixed number of copies, may not be the most efficient one. It is generally better to fix the total simulation effort for each stage of the simulation. Furthermore, given this effort, the best strategy is to restart an equal number of times from each state, rather than to restart each time from a randomly chosen stat
Analysis of Adaptive Multilevel Splitting algorithms in an idealized case
The Adaptive Multilevel Splitting algorithm is a very powerful and versatile
method to estimate rare events probabilities. It is an iterative procedure on
an interacting particle system, where at each step, the less well-adapted
particles among are killed while new better adapted particles are
resampled according to a conditional law. We analyze the algorithm in the
idealized setting of an exact resampling and prove that the estimator of the
rare event probability is unbiased whatever . We also obtain a precise
asymptotic expansion for the variance of the estimator and the cost of the
algorithm in the large limit, for a fixed
Probabilistic Reachability Analysis for Large Scale Stochastic Hybrid Systems
This paper studies probabilistic reachability analysis for large scale stochastic hybrid systems (SHS) as a problem of rare event estimation. In literature, advanced rare event estimation theory has recently been embedded within a stochastic analysis framework, and this has led to significant novel results in rare event estimation for a diffusion process using sequential MC simulation. This paper presents this rare event estimation theory directly in terms of probabilistic reachability analysis of an SHS, and develops novel theory which allows to extend the novel results for application to a large scale SHS where a very huge number of rare discrete modes may contribute significantly to the reach probability. Essentially, the approach taken is to introduce an aggregation of the discrete modes, and to develop importance sampling relative to the rare switching between the aggregation modes. The practical working of this approach is demonstrated for the safety verification of an advanced air traffic control example
Effective branching splitting method under cost contraint
This paper deals with the splitting method first introduced in rare event analysis. In this technique, the sample paths are split into R multiple copies at various stages during the simulation. Given the cost, the optimization of the algorithm suggests to sample a number of subtrials which may be non-integer and even unknown but estimated. In this paper, we present three different approaches to face this problem which provide precise estimates of the relative error between P(A) and its estimator
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