10,321 research outputs found
Computing mean first exit times for stochastic processes using multi-level Monte Carlo
The multi-level approach developed by Giles (2008) can be used to estimate mean first exit times for stochastic differential equations, which are of interest in finance, physics and chemical kinetics. Multi-level improves the computational expense of standard Monte Carlo in this setting by an order of magnitude. More precisely, for a target accuracy of TOL, so that the root mean square error of the estimator is O(TOL), the O(TOL-4) cost of standard Monte Carlo can be reduced to O(TOL-3|log(TOL)|1/2) with a multi-level scheme. This result was established in Higham, Mao, Roj, Song, and Yin (2013), and illustrated on some scalar examples. Here, we briefly overview the algorithm and present some new computational results in higher dimensions
A New Approach To Estimate The Collision Probability For Automotive Applications
We revisit the computation of probability of collision in the context of
automotive collision avoidance (the estimation of a potential collision is also
referred to as conflict detection in other contexts). After reviewing existing
approaches to the definition and computation of a collision probability we
argue that the question "What is the probability of collision within the next
three seconds?" can be answered on the basis of a collision probability rate.
Using results on level crossings for vector stochastic processes we derive a
general expression for the upper bound of the distribution of the collision
probability rate. This expression is valid for arbitrary prediction models
including process noise. We demonstrate in several examples that distributions
obtained by large-scale Monte-Carlo simulations obey this bound and in many
cases approximately saturate the bound. We derive an approximation for the
distribution of the collision probability rate that can be computed on an
embedded platform. In order to efficiently sample this probability rate
distribution for determination of its characteristic shape an adaptive method
to obtain the sampling points is proposed. An upper bound of the probability of
collision is then obtained by one-dimensional numerical integration over the
time period of interest. A straightforward application of this method applies
to the collision of an extended object with a second point-like object. Using
an abstraction of the second object by salient points of its boundary we
propose an application of this method to two extended objects with arbitrary
orientation. Finally, the distribution of the collision probability rate is
identified as the distribution of the time-to-collision.Comment: Revised and restructured version, discussion of extended vehicles
expanded, section on TTC expanded, references added, other minor changes, 17
pages, 18 figure
A Compilation Target for Probabilistic Programming Languages
Forward inference techniques such as sequential Monte Carlo and particle
Markov chain Monte Carlo for probabilistic programming can be implemented in
any programming language by creative use of standardized operating system
functionality including processes, forking, mutexes, and shared memory.
Exploiting this we have defined, developed, and tested a probabilistic
programming language intermediate representation language we call probabilistic
C, which itself can be compiled to machine code by standard compilers and
linked to operating system libraries yielding an efficient, scalable, portable
probabilistic programming compilation target. This opens up a new hardware and
systems research path for optimizing probabilistic programming systems.Comment: In Proceedings of the 31st International Conference on Machine
Learning (ICML), 201
Estimating expected first passage times using multilevel Monte Carlo algorithm
In this paper we devise a method of numerically estimating the expected first passage times of stochastic processes. We use Monte Carlo path simulations with Milstein discretisation scheme to approximate the solutions of scalar stochastic differential equations. To further reduce the variance of the estimated expected stopping time and improve computational efficiency, we use the multi-level Monte Carlo algorithm, recently developed by Giles (2008a), and other variance-reduction techniques. Our numerical results show significant improvements over conventional Monte Carlo techniques
Multi-level Monte Carlo for continuous time Markov chains, with applications in biochemical kinetics
We show how to extend a recently proposed multi-level Monte Carlo approach to
the continuous time Markov chain setting, thereby greatly lowering the
computational complexity needed to compute expected values of functions of the
state of the system to a specified accuracy. The extension is non-trivial,
exploiting a coupling of the requisite processes that is easy to simulate while
providing a small variance for the estimator. Further, and in a stark departure
from other implementations of multi-level Monte Carlo, we show how to produce
an unbiased estimator that is significantly less computationally expensive than
the usual unbiased estimator arising from exact algorithms in conjunction with
crude Monte Carlo. We thereby dramatically improve, in a quantifiable manner,
the basic computational complexity of current approaches that have many names
and variants across the scientific literature, including the
Bortz-Kalos-Lebowitz algorithm, discrete event simulation, dynamic Monte Carlo,
kinetic Monte Carlo, the n-fold way, the next reaction method,the
residence-time algorithm, the stochastic simulation algorithm, Gillespie's
algorithm, and tau-leaping. The new algorithm applies generically, but we also
give an example where the coupling idea alone, even without a multi-level
discretization, can be used to improve efficiency by exploiting system
structure. Stochastically modeled chemical reaction networks provide a very
important application for this work. Hence, we use this context for our
notation, terminology, natural scalings, and computational examples.Comment: Improved description of the constants in statement of Theorem
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