9,396 research outputs found
Algorithms (X,sigma,eta) : quasi-random mutations for Evolution Strategies
International audienceRandomization is an efficient tool for global optimization. We here define a method which keeps : - the order 0 of evolutionary algorithms (no gradient) ; - the stochastic aspect of evolutionary algorithms ; - the efficiency of so-called "low-dispersion" points ; and which ensures under mild assumptions global convergence with linear convergence rate. We use i) sampling on a ball instead of Gaussian sampling (in a way inspired by trust regions), ii) an original rule for step-size adaptation ; iii) quasi-monte-carlo sampling (low dispersion points) instead of Monte-Carlo sampling. We prove in this framework linear convergence rates i) for global optimization and not only local optimization ; ii) under very mild assumptions on the regularity of the function (existence of derivatives is not required). Though the main scope of this paper is theoretical, numerical experiments are made to backup the mathematical results. Algorithm XSE: quasi-random mutations for evolution strategies. A. Auger, M. Jebalia, O. Teytaud. Proceedings of EA'2005
Some numerical methods for solving stochastic impulse control in natural gas storage facilities
The valuation of gas storage facilities is characterized as a stochastic impulse control problem with finite horizon resulting in Hamilton-Jacobi-Bellman (HJB) equations for the value function. In this context the two catagories of solving schemes for optimal switching are discussed in a stochastic control framework. We reviewed some numerical methods which include approaches related to partial differential equations (PDEs), Markov chain approximation, nonparametric regression, quantization method and some practitioners’ methods. This paper considers optimal switching problem arising in valuation of gas storage contracts for leasing the storage facilities, and investigates the recent developments as well as their advantages and disadvantages of each scheme based on dynamic programming principle (DPP
A comparison of sample-based Stochastic Optimal Control methods
In this paper, we compare the performance of two scenario-based numerical
methods to solve stochastic optimal control problems: scenario trees and
particles. The problem consists in finding strategies to control a dynamical
system perturbed by exogenous noises so as to minimize some expected cost along
a discrete and finite time horizon. We introduce the Mean Squared Error (MSE)
which is the expected -distance between the strategy given by the
algorithm and the optimal strategy, as a performance indicator for the two
models. We study the behaviour of the MSE with respect to the number of
scenarios used for discretization. The first model, widely studied in the
Stochastic Programming community, consists in approximating the noise diffusion
using a scenario tree representation. On a numerical example, we observe that
the number of scenarios needed to obtain a given precision grows exponentially
with the time horizon. In that sense, our conclusion on scenario trees is
equivalent to the one in the work by Shapiro (2006) and has been widely noticed
by practitioners. However, in the second part, we show using the same example
that, by mixing Stochastic Programming and Dynamic Programming ideas, the
particle method described by Carpentier et al (2009) copes with this numerical
difficulty: the number of scenarios needed to obtain a given precision now does
not depend on the time horizon. Unfortunately, we also observe that serious
obstacles still arise from the system state space dimension
Metamodel-based importance sampling for the simulation of rare events
In the field of structural reliability, the Monte-Carlo estimator is
considered as the reference probability estimator. However, it is still
untractable for real engineering cases since it requires a high number of runs
of the model. In order to reduce the number of computer experiments, many other
approaches known as reliability methods have been proposed. A certain approach
consists in replacing the original experiment by a surrogate which is much
faster to evaluate. Nevertheless, it is often difficult (or even impossible) to
quantify the error made by this substitution. In this paper an alternative
approach is developed. It takes advantage of the kriging meta-modeling and
importance sampling techniques. The proposed alternative estimator is finally
applied to a finite element based structural reliability analysis.Comment: 8 pages, 3 figures, 1 table. Preprint submitted to ICASP11
Mini-symposia entitled "Meta-models/surrogate models for uncertainty
propagation, sensitivity and reliability analysis
Multilevel Sparse Grid Methods for Elliptic Partial Differential Equations with Random Coefficients
Stochastic sampling methods are arguably the most direct and least intrusive
means of incorporating parametric uncertainty into numerical simulations of
partial differential equations with random inputs. However, to achieve an
overall error that is within a desired tolerance, a large number of sample
simulations may be required (to control the sampling error), each of which may
need to be run at high levels of spatial fidelity (to control the spatial
error). Multilevel sampling methods aim to achieve the same accuracy as
traditional sampling methods, but at a reduced computational cost, through the
use of a hierarchy of spatial discretization models. Multilevel algorithms
coordinate the number of samples needed at each discretization level by
minimizing the computational cost, subject to a given error tolerance. They can
be applied to a variety of sampling schemes, exploit nesting when available,
can be implemented in parallel and can be used to inform adaptive spatial
refinement strategies. We extend the multilevel sampling algorithm to sparse
grid stochastic collocation methods, discuss its numerical implementation and
demonstrate its efficiency both theoretically and by means of numerical
examples
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