15,979 research outputs found
Influence of stochastic domain growth on pattern nucleation for diffusive systems with internal noise
Numerous mathematical models exploring the emergence of complexity within developmental biology incorporate diffusion as the dominant mechanism of transport. However, self-organizing paradigms can exhibit the biologically undesirable property of extensive sensitivity, as illustrated by the behavior of the French-flag model in response to intrinsic noise and Turing’s model when subjected to fluctuations in initial conditions. Domain growth is known to be a stabilizing factor for the latter, though the interaction of intrinsic noise and domain growth is underexplored, even in the simplest of biophysical settings. Previously, we developed analytical Fourier methods and a description of domain growth that allowed us to characterize the effects of deterministic domain growth on stochastically diffusing systems. In this paper we extend our analysis to encompass stochastically growing domains. This form of growth can be used only to link the meso- and macroscopic domains as the “box-splitting” form of growth on the microscopic scale has an ill-defined thermodynamic limit. The extension is achieved by allowing the simulated particles to undergo random walks on a discretized domain, while stochastically controlling the length of each discretized compartment. Due to the dependence of diffusion on the domain discretization, we find that the description of diffusion cannot be uniquely derived. We apply these analytical methods to two justified descriptions, where it is shown that, under certain conditions, diffusion is able to support a consistent inhomogeneous state that is far removed from the deterministic equilibrium, without additional kinetics. Finally, a logistically growing domain is considered. Not only does this show that we can deal with nonmonotonic descriptions of stochastic growth, but it is also seen that diffusion on a stationary domain produces different effects to diffusion on a domain that is stationary “on average.
Stochastic determination of matrix determinants
Matrix determinants play an important role in data analysis, in particular
when Gaussian processes are involved. Due to currently exploding data volumes,
linear operations - matrices - acting on the data are often not accessible
directly but are only represented indirectly in form of a computer routine.
Such a routine implements the transformation a data vector undergoes under
matrix multiplication. While efficient probing routines to estimate a matrix's
diagonal or trace, based solely on such computationally affordable
matrix-vector multiplications, are well known and frequently used in signal
inference, there is no stochastic estimate for its determinant. We introduce a
probing method for the logarithm of a determinant of a linear operator. Our
method rests upon a reformulation of the log-determinant by an integral
representation and the transformation of the involved terms into stochastic
expressions. This stochastic determinant determination enables large-size
applications in Bayesian inference, in particular evidence calculations, model
comparison, and posterior determination.Comment: 8 pages, 5 figure
Searching for optimal variables in real multivariate stochastic data
By implementing a recent technique for the determination of stochastic
eigendirections of two coupled stochastic variables, we investigate the
evolution of fluctuations of NO2 concentrations at two monitoring stations in
the city of Lisbon, Portugal. We analyze the stochastic part of the
measurements recorded at the monitoring stations by means of a method where the
two concentrations are considered as stochastic variables evolving according to
a system of coupled stochastic differential equations. Analysis of their
structure allows for transforming the set of measured variables to a set of
derived variables, one of them with reduced stochasticity. For the specific
case of NO2 concentration measures, the set of derived variables are well
approximated by a global rotation of the original set of measured variables. We
conclude that the stochastic sources at each station are independent from each
other and typically have amplitudes of the order of the deterministic
contributions. Such findings show significant limitations when predicting such
quantities. Still, we briefly discuss how predictive power can be increased in
general in the light of our methods
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