54,364 research outputs found
Diffusion, super-diffusion and coalescence from single step
From the exact single step evolution equation of the two-point correlation
function of a particle distribution subjected to a stochastic displacement
field \bu(\bx), we derive different dynamical regimes when \bu(\bx) is
iterated to build a velocity field. First we show that spatially uncorrelated
fields \bu(\bx) lead to both standard and anomalous diffusion equation. When
the field \bu(\bx) is spatially correlated each particle performs a simple
free Brownian motion, but the trajectories of different particles result to be
mutually correlated. The two-point statistical properties of the field
\bu(\bx) induce two-point spatial correlations in the particle distribution
satisfying a simple but non-trivial diffusion-like equation. These
displacement-displacement correlations lead the system to three possible
regimes: coalescence, simple clustering and a combination of the two. The
existence of these different regimes, in the one-dimensional system, is shown
through computer simulations and a simple theoretical argument.Comment: RevTeX (iopstyle) 19 pages, 5 eps-figure
First-passage and first-exit times of a Bessel-like stochastic process
We study a stochastic process related to the Bessel and the Rayleigh
processes, with various applications in physics, chemistry, biology, economics,
finance and other fields. The stochastic differential equation is , where is the Wiener process. Due to the
singularity of the drift term for , different natures of boundary at
the origin arise depending on the real parameter : entrance, exit, and
regular. For each of them we calculate analytically and numerically the
probability density functions of first-passage times or first-exit times.
Nontrivial behaviour is observed in the case of a regular boundary.Comment: 15 pages, 6 figures, submitted to Physical Review
Intrinsic Gaussian processes on complex constrained domains
We propose a class of intrinsic Gaussian processes (in-GPs) for
interpolation, regression and classification on manifolds with a primary focus
on complex constrained domains or irregular shaped spaces arising as subsets or
submanifolds of R, R2, R3 and beyond. For example, in-GPs can accommodate
spatial domains arising as complex subsets of Euclidean space. in-GPs respect
the potentially complex boundary or interior conditions as well as the
intrinsic geometry of the spaces. The key novelty of the proposed approach is
to utilise the relationship between heat kernels and the transition density of
Brownian motion on manifolds for constructing and approximating valid and
computationally feasible covariance kernels. This enables in-GPs to be
practically applied in great generality, while existing approaches for
smoothing on constrained domains are limited to simple special cases. The broad
utilities of the in-GP approach is illustrated through simulation studies and
data examples
ParMooN - a modernized program package based on mapped finite elements
{\sc ParMooN} is a program package for the numerical solution of elliptic and
parabolic partial differential equations. It inherits the distinct features of
its predecessor {\sc MooNMD} \cite{JM04}: strict decoupling of geometry and
finite element spaces, implementation of mapped finite elements as their
definition can be found in textbooks, and a geometric multigrid preconditioner
with the option to use different finite element spaces on different levels of
the multigrid hierarchy. After having presented some thoughts about in-house
research codes, this paper focuses on aspects of the parallelization for a
distributed memory environment, which is the main novelty of {\sc ParMooN}.
Numerical studies, performed on compute servers, assess the efficiency of the
parallelized geometric multigrid preconditioner in comparison with some
parallel solvers that are available in the library {\sc PETSc}. The results of
these studies give a first indication whether the cumbersome implementation of
the parallelized geometric multigrid method was worthwhile or not.Comment: partly supported by European Union (EU), Horizon 2020, Marie
Sk{\l}odowska-Curie Innovative Training Networks (ITN-EID), MIMESIS, grant
number 67571
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