14,876 research outputs found
Constrained Approximation of Effective Generators for Multiscale Stochastic Reaction Networks and Application to Conditioned Path Sampling
Efficient analysis and simulation of multiscale stochastic systems of
chemical kinetics is an ongoing area for research, and is the source of many
theoretical and computational challenges. In this paper, we present a
significant improvement to the constrained approach, which is a method for
computing effective dynamics of slowly changing quantities in these systems,
but which does not rely on the quasi-steady-state assumption (QSSA). The QSSA
can cause errors in the estimation of effective dynamics for systems where the
difference in timescales between the "fast" and "slow" variables is not so
pronounced.
This new application of the constrained approach allows us to compute the
effective generator of the slow variables, without the need for expensive
stochastic simulations. This is achieved by finding the null space of the
generator of the constrained system. For complex systems where this is not
possible, or where the constrained subsystem is itself multiscale, the
constrained approach can then be applied iteratively. This results in breaking
the problem down into finding the solutions to many small eigenvalue problems,
which can be efficiently solved using standard methods.
Since this methodology does not rely on the quasi steady-state assumption,
the effective dynamics that are approximated are highly accurate, and in the
case of systems with only monomolecular reactions, are exact. We will
demonstrate this with some numerics, and also use the effective generators to
sample paths of the slow variables which are conditioned on their endpoints, a
task which would be computationally intractable for the generator of the full
system.Comment: 31 pages, 7 figure
A Multi-Step Richardson-Romberg Extrapolation Method For Stochastic Approximation
We obtain an expansion of the implicit weak discretization error for the
target of stochastic approximation algorithms introduced and studied in
[Frikha2013]. This allows us to extend and develop the Richardson-Romberg
extrapolation method for Monte Carlo linear estimator (introduced in [Talay &
Tubaro 1990] and deeply studied in [Pag{\`e}s 2007]) to the framework of
stochastic optimization by means of stochastic approximation algorithm. We
notably apply the method to the estimation of the quantile of diffusion
processes. Numerical results confirm the theoretical analysis and show a
significant reduction in the initial computational cost.Comment: 31 pages, 1 figur
- …