Biology Reference
In-Depth Information
t
) is a vector of i.i.d. Gaussian random numbers
with each component having a mean of zero and a variance of 2
D
where
N
n
p
(0, 2
D
δ
δ
t
;
t
is the simulation time step size. Moving the particles according
to Eq. (18) creates a concentration field that, for
N
δ
→∞
, converges
to the exact solution of the diffusion equation as given in Eq. (16).
Homogeneous Neumann boundary conditions can be satisfied
by reflecting the particles at the boundary. Drawing the step
displacements in Eq. (18) from a multivariate Gaussian distribu-
tion readily extends the RW method to anisotropic diffusion
processes.
RW is a stochastic simulation method. This Monte Carlo
66,67
char-
acter limits its convergence capabilities (cf. Sec. 4.1), since the variance
of the mean of
N
i.i.d. random variables is given by 1/
N
times the
individual variance of a single random variable
68
(cf. Sec. 7.3).
Moreover, the solution deteriorates with increasing diffusion constant
D
as the variance of the random variables becomes larger. In the case of
small
D
(<<
√
t
), the motion of the particles can be masked by the sam-
pling noise. RW thus works best for an intermediate range of diffusion
constants.
δ
7.2. Particle Strength Exchange (PSE)
The Particle Strength Exchange (PSE)
103,104
method is a determinis-
tic pure particle method to simulate continuous diffusion processes
in space. It is based on approximating the diffusion operator by
a mass-conserving integral operator that can be consistently evaluated
on the particle locations (cf. Sec. 5.2.1). The PSE scheme has
been devised by Degond and Mas-Gallic for both isotropic
103
and anisotropic
104
diffusion. We illustrate the concept in the
isotropic case. Anisotropic PSE is analogous and follows a similar
derivation.
104
7.2.1. PSE for isotropic diffusion
In free space, i.e.
Ω=IR
d
, the isotropic PSE method
103
obtains an integral
approximation to the Laplace operator in Eq. (15) by considering the
