159 research outputs found
A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise
We consider the numerical approximation of a general second order
semi--linear parabolic stochastic partial differential equation (SPDE) driven
by additive space-time noise. We introduce a new modified scheme using a linear
functional of the noise with a semi--implicit Euler--Maruyama method in time
and in space we analyse a finite element method (although extension to finite
differences or finite volumes would be possible). We prove convergence in the
root mean square norm for a diffusion reaction equation and diffusion
advection reaction equation. We present numerical results for a linear reaction
diffusion equation in two dimensions as well as a nonlinear example of
two-dimensional stochastic advection diffusion reaction equation. We see from
both the analysis and numerics that the proposed scheme has better convergence
properties than the standard semi--implicit Euler--Maruyama method
Numerical variational methods applied to cylinder buckling
We review and compare different computational variational methods applied to
a system of fourth order equations that arises as a model of cylinder buckling.
We describe both the discretization and implementation, in particular how to
deal with a 1 dimensional null space. We show that we can construct many
different solutions from a complex energy surface. We examine numerically
convergence in the spatial discretization and in the domain size. Finally we
give a physical interpretation of some of the solutions found.Comment: 23 pages, 12 figures, 6 table
Basis properties of the p, q-sine functions
We improve the currently known thresholds for basisness of the family of
periodically dilated p,q-sine functions. Our findings rely on a Beurling
decomposition of the corresponding change of coordinates in terms of shift
operators of infinite multiplicity. We also determine refined bounds on the
Riesz constant associated to this family. These results seal mathematical gaps
in the existing literature on the subject.Comment: 28 pages, 5 figures, computer codes included in appendi
Weak Convergence Of Tamed Exponential Integrators for Stochastic Differential Equations
We prove weak convergence of order one for a class of exponential based
integrators for SDEs with non-globally Lipschtiz drift. Our analysis covers
tamed versions of Geometric Brownian Motion (GBM) based methods as well as the
standard exponential schemes. The numerical performance of both the GBM and
exponential tamed methods through four different multi-level Monte Carlo
techniques are compared. We observe that for linear noise the standard
exponential tamed method requires severe restrictions on the stepsize unlike
the GBM tamed method.Comment: 24 pages, 3 figure
Strong Convergence of a GBM Based Tamed Integrator for SDEs and an Adaptive Implementation
We introduce a tamed exponential time integrator which exploits linear terms
in both the drift and diffusion for Stochastic Differential Equations (SDEs)
with a one sided globally Lipschitz drift term. Strong convergence of the
proposed scheme is proved, exploiting the boundedness of the geometric Brownian
motion (GBM) and we establish order 1 convergence for linear diffusion terms.
In our implementation we illustrate the efficiency of the proposed scheme
compared to existing fixed step methods and utilize it in an adaptive time
stepping scheme. Furthermore we extend the method to nonlinear diffusion terms
and show it remains competitive. The efficiency of these GBM based approaches
are illustrated by considering some well-known SDE models
Secretory vesicles are preferentially targeted to areas of low molecular SNARE density
Intercellular communication is commonly mediated by the regulated fusion, or exocytosis, of vesicles with the cell surface. SNARE (soluble N-ethymaleimide sensitive factor attachment protein receptor) proteins are the catalytic core of the secretory machinery, driving vesicle and plasma membrane merger. Plasma membrane SNAREs (tSNAREs) are proposed to reside in dense clusters containing many molecules, thus providing a concentrated reservoir to promote membrane fusion. However, biophysical experiments suggest that a small number of SNAREs are sufficient to drive a single fusion event. Here we show, using molecular imaging, that the majority of tSNARE molecules are spatially separated from secretory vesicles. Furthermore, the motilities of the individual tSNAREs are constrained in membrane micro-domains, maintaining a non-random molecular distribution and limiting the maximum number of molecules encountered by secretory vesicles. Together our results provide a new model for the molecular mechanism of regulated exocytosis and demonstrate the exquisite organization of the plasma membrane at the level of individual molecular machines
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