# Global weak error control for the tau-leap method

**Jesper Karlsson
**

Mathematical and Computer Sciences and Engineering Division

King Abdullah University of Science and Technology

Thuwal, Saudi Arabia

### Abstract:

This work presents novel, tau-leap based, algorithms for the
discretization of pure jump processes arising in kinetic Monte Carlo
models. A typical example of such a pure jump process is to model the
time evolution of a reacting chemical network. Today, two algorithms
are commonly used: (1) The SSA algorithm by Gillespie, that simulates
a single trajectory exactly, but can be time consuming when many
reactions take place during a short time interval. (2) The approximate
tau-leap method, that can be used to reduce computational time, but
introduces a time discretization error and may also lead to negative
particle populations. The tau-leap based methods presented in this
talk aim to estimate and control the global weak error. First, I will
present a time-adaptive tau-leap method, controlling the global weak
error using discrete duals, and using a post-leap check to prevent
negative populations. Finally, instead of using a post-leap check,
another approach is to devise a computationally less expensive and
simpler to implement pre-leap check, using a Chernoff-type bound on
the one-step exit probability for the tau-leap method. This allows to
control the exit probability of a simulated trajectory using a hybrid
SSA/tau-leap method, and to obtain accurate computable estimates for
the global weak error associated with the exit event, and the
corresponding total computational work.