556 research outputs found
Monte Carlo approximations of the Neumann problem
We introduce Monte Carlo methods to compute the solution of elliptic
equations with pure Neumann boundary conditions. We first prove that the
solution obtained by the stochastic representation has a zero mean value with
respect to the invariant measure of the stochastic process associated to the
equation. Pointwise approximations are computed by means of standard and new
simulation schemes especially devised for local time approximation on the
boundary of the domain. Global approximations are computed thanks to a
stochastic spectral formulation taking into account the property of zero mean
value of the solution. This stochastic formulation is asymptotically perfect in
terms of conditioning. Numerical examples are given on the Laplace operator on
a square domain with both pure Neumann and mixed Dirichlet-Neumann boundary
conditions. A more general convection-diffusion equation is also numerically
studied
Long time behavior of a mean-field model of interacting neurons
We study the long time behavior of the solution to some McKean-Vlasov
stochastic differential equation (SDE) driven by a Poisson process. In
neuroscience, this SDE models the asymptotic dynamic of the membrane potential
of a spiking neuron in a large network. We prove that for a small enough
interaction parameter, any solution converges to the unique (in this case)
invariant measure. To this aim, we first obtain global bounds on the jump rate
and derive a Volterra type integral equation satisfied by this rate. We then
replace temporary the interaction part of the equation by a deterministic
external quantity (we call it the external current). For constant current, we
obtain the convergence to the invariant measure. Using a perturbation method,
we extend this result to more general external currents. Finally, we prove the
result for the non-linear McKean-Vlasov equation
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