185 research outputs found
Large Deviations and Importance Sampling for Systems of Slow-Fast Motion
In this paper we develop the large deviations principle and a rigorous
mathematical framework for asymptotically efficient importance sampling schemes
for general, fully dependent systems of stochastic differential equations of
slow and fast motion with small noise in the slow component. We assume
periodicity with respect to the fast component. Depending on the interaction of
the fast scale with the smallness of the noise, we get different behavior. We
examine how one range of interaction differs from the other one both for the
large deviations and for the importance sampling. We use the large deviations
results to identify asymptotically optimal importance sampling schemes in each
case. Standard Monte Carlo schemes perform poorly in the small noise limit. In
the presence of multiscale aspects one faces additional difficulties and
straightforward adaptation of importance sampling schemes for standard small
noise diffusions will not produce efficient schemes. It turns out that one has
to consider the so called cell problem from the homogenization theory for
Hamilton-Jacobi-Bellman equations in order to guarantee asymptotic optimality.
We use stochastic control arguments.Comment: More detailed proofs. Differences from the published version are
editorial and typographica
Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach
The aim of this paper is to investigate from the numerical point of view the
possibility of coupling the Hamilton-Jacobi-Bellman (HJB) equation and
Pontryagin's Minimum Principle (PMP) to solve some control problems. A rough
approximation of the value function computed by the HJB method is used to
obtain an initial guess for the PMP method. The advantage of our approach over
other initialization techniques (such as continuation or direct methods) is to
provide an initial guess close to the global minimum. Numerical tests involving
multiple minima, discontinuous control, singular arcs and state constraints are
considered. The CPU time for the proposed method is less than four minutes up
to dimension four, without code parallelization
Multi-scale modeling of follicular ovulation as a reachability problem
During each ovarian cycle, only a definite number of follicles ovulate, while
the others undergo a degeneration process called atresia. We have designed a
multi-scale mathematical model where ovulation and atresia result from a
hormonal controlled selection process. A 2D-conservation law describes the age
and maturity structuration of the follicular cell population. In this paper, we
focus on the operating mode of the control, through the study of the
characteristics of the conservation law. We describe in particular the set of
microscopic initial conditions leading to the macroscopic phenomenon of either
ovulation or atresia, in the framework of backwards reachable sets theory
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