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Weak error analysis of numerical methods for stochastic models of population processes

Summary: Weak error analysis of numerical methods for stochastic models of
population processes
David F. Anderson1
and Masanori Koyama2
September 28, 2011
The simplest, and most common, stochastic model for population processes, including those
from biochemistry and cell biology, are continuous time Markov chains. Simulation of such
models is often relatively straightforward as there are easily implementable methods for the gen-
eration of exact sample paths. However, when using ensemble averages to approximate expected
values, the computational complexity can become prohibitive as the number of computations
per path scales linearly with the number of jumps, or reactions, of the process. When such
methods become computationally intractable, approximate methods, which introduce a bias,
can become advantageous. In this paper, we provide a general framework for understanding
the weak error, or bias, induced by different numerical approximation techniques. The analysis
takes into account both the natural scalings within a given system and the step-size of the nu-
merical method. Further, the weak trapezoidal method is introduced in the current setting, and
is proven to be second order accurate in a weak sense, making it the first higher order method
in this setting. Examples are provided demonstrating both the main analytical results, and the
reduction in computational complexity achieved with the approximate methods.


Source: Anderson, David F. - Department of Mathematics, University of Wisconsin at Madison


Collections: Mathematics