6,183 research outputs found
Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method
BackgroundBiochemical systems with relatively low numbers of components must be simulated stochastically in order to capture their inherent noise. Although there has recently been considerable work on discrete stochastic solvers, there is still a need for numerical methods that are both fast and accurate. The Bulirsch-Stoer method is an established method for solving ordinary differential equations that possesses both of these qualities.ResultsIn this paper, we present the Stochastic Bulirsch-Stoer method, a new numerical method for simulating discrete chemical reaction systems, inspired by its deterministic counterpart. It is able to achieve an excellent efficiency due to the fact that it is based on an approach with high deterministic order, allowing for larger stepsizes and leading to fast simulations. We compare it to the Euler ?-leap, as well as two more recent ?-leap methods, on a number of example problems, and find that as well as being very accurate, our method is the most robust, in terms of efficiency, of all the methods considered in this paper. The problems it is most suited for are those with increased populations that would be too slow to simulate using Gillespie’s stochastic simulation algorithm. For such problems, it is likely to achieve higher weak order in the moments.ConclusionsThe Stochastic Bulirsch-Stoer method is a novel stochastic solver that can be used for fast and accurate simulations. Crucially, compared to other similar methods, it better retains its high accuracy when the timesteps are increased. Thus the Stochastic Bulirsch-Stoer method is both computationally efficient and robust. These are key properties for any stochastic numerical method, as they must typically run many thousands of simulations
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
Ground state energy of the -Bose and Fermi gas at weak coupling from double extrapolation
We consider the ground state energy of the Lieb-Liniger gas with
interaction in the weak coupling regime . For bosons with repulsive
interaction, previous studies gave the expansion
.
Using a numerical solution of the Lieb-Liniger integral equation discretized
with points and finite strength of the interaction, we obtain very
accurate numerics for the next orders after extrapolation on and .
The coefficient of in the expansion is found approximately equal
to , accurate within all digits shown. This value
is supported by a numerical solution of the Bethe equations with particles
followed by extrapolation on and . It was identified as
by G. Lang. The next two coefficients are also
guessed from numerics. For balanced spin fermions with attractive
interaction, the best result so far for the ground state energy was
. An analogue
double extrapolation scheme leads to the value for the
coefficient of .Comment: 11 pages, 2 figures, 3 table
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