4,617 research outputs found
Bayesian data assimilation in shape registration
In this paper we apply a Bayesian framework to the problem of geodesic curve matching. Given a template curve, the geodesic equations provide a mapping from initial conditions\ud
for the conjugate momentum onto topologically equivalent shapes. Here, we aim to recover the well defined posterior distribution on the initial momentum which gives rise to observed points on the target curve; this is achieved by explicitly including a reparameterisation in the formulation. Appropriate priors are chosen for the functions which together determine this field and the positions of the observation points, the initial momentum p0 and the reparameterisation vector field v, informed by regularity results about the forward model. Having done this, we illustrate how Maximum Likelihood Estimators (MLEs) can be used to find regions of high posterior density, but also how we can apply recently developed MCMC methods on function spaces to characterise the whole of the posterior density. These illustrative examples also include scenarios where the posterior distribution is multimodal and irregular, leading us to the conclusion that knowledge of a state of global maximal posterior density does not always give us the whole picture, and full posterior sampling can give better quantification of likely states and the overall uncertainty inherent in the problem
A water wave model with horizontal circulation and accurate dispersion
We describe a new water wave model which is variational, and combines a depth-averaged vertical (component of) vorticity with depth-dependent potential flow. The model facilitates the further restriction of the vertical profile of the velocity potential to n-th order polynomials or a finite element profile with a small number of elements (say), leading to a framework for efficient modelling of the interaction of steepening and breaking waves near the shore with a large-scale horizontal flow. The equations are derived from a constrained variational formulation which leads to conservation laws for energy, mass, momentum and vertical vorticity (or circulation). We show that the potential flow water wave equations and the shallow-water equations are recovered in the relevant limits, and provide approximate shock relations for the model which can be used in numerical schemes to model breaking waves
Embedded discontinuous Galerkin transport schemes with localised limiters
Motivated by finite element spaces used for representation of temperature in
the compatible finite element approach for numerical weather prediction, we
introduce locally bounded transport schemes for (partially-)continuous finite
element spaces. The underlying high-order transport scheme is constructed by
injecting the partially-continuous field into an embedding discontinuous finite
element space, applying a stable upwind discontinuous Galerkin (DG) scheme, and
projecting back into the partially-continuous space; we call this an embedded
DG scheme. We prove that this scheme is stable in L2 provided that the
underlying upwind DG scheme is. We then provide a framework for applying
limiters for embedded DG transport schemes. Standard DG limiters are applied
during the underlying DG scheme. We introduce a new localised form of
element-based flux-correction which we apply to limiting the projection back
into the partially-continuous space, so that the whole transport scheme is
bounded. We provide details in the specific case of tensor-product finite
element spaces on wedge elements that are discontinuous P1/Q1 in the horizontal
and continuous P2 in the vertical. The framework is illustrated with numerical
tests
Solving the Poisson equation on small aspect ratio domains using unstructured meshes
We discuss the ill conditioning of the matrix for the discretised Poisson
equation in the small aspect ratio limit, and motivate this problem in the
context of nonhydrostatic ocean modelling. Efficient iterative solvers for the
Poisson equation in small aspect ratio domains are crucial for the successful
development of nonhydrostatic ocean models on unstructured meshes. We introduce
a new multigrid preconditioner for the Poisson problem which can be used with
finite element discretisations on general unstructured meshes; this
preconditioner is motivated by the fact that the Poisson problem has a
condition number which is independent of aspect ratio when Dirichlet boundary
conditions are imposed on the top surface of the domain. This leads to the
first level in an algebraic multigrid solver (which can be extended by further
conventional algebraic multigrid stages), and an additive smoother. We
illustrate the method with numerical tests on unstructured meshes, which show
that the preconditioner makes a dramatic improvement on a more standard
multigrid preconditioner approach, and also show that the additive smoother
produces better results than standard SOR smoothing. This new solver method
makes it feasible to run nonhydrostatic unstructured mesh ocean models in small
aspect ratio domains.Comment: submitted to Ocean Modellin
Geodesic boundary value problems with symmetry
This paper shows how left and right actions of Lie groups on a manifold may
be used to complement one another in a variational reformulation of optimal
control problems equivalently as geodesic boundary value problems with
symmetry. We prove an equivalence theorem to this effect and illustrate it with
several examples. In finite-dimensions, we discuss geodesic flows on the Lie
groups SO(3) and SE(3) under the left and right actions of their respective Lie
algebras. In an infinite-dimensional example, we discuss optimal
large-deformation matching of one closed curve to another embedded in the same
plane. In the curve-matching example, the manifold \Emb(S^1, \mathbb{R}^2)
comprises the space of closed curves embedded in the plane
. The diffeomorphic left action \Diff(\mathbb{R}^2) deforms the
curve by a smooth invertible time-dependent transformation of the coordinate
system in which it is embedded, while leaving the parameterisation of the curve
invariant. The diffeomorphic right action \Diff(S^1) corresponds to a smooth
invertible reparameterisation of the domain coordinates of the curve. As
we show, this right action unlocks an important degree of freedom for
geodesically matching the curve shapes using an equivalent fixed boundary value
problem, without being constrained to match corresponding points along the
template and target curves at the endpoint in time.Comment: First version -- comments welcome
Estimating eddy diffusivities from noisy Lagrangian observations
The problem of estimating the eddy diffusivity from Lagrangian observations
in the presence of measurement error is studied in this paper. We consider a
class of incompressible velocity fields for which is can be rigorously proved
that the small scale dynamics can be parameterised in terms of an eddy
diffusivity tensor. We show, by means of analysis and numerical experiments,
that subsampling of the data is necessary for the accurate estimation of the
eddy diffusivity. The optimal sampling rate depends on the detailed properties
of the velocity field. Furthermore, we show that averaging over the data only
marginally reduces the bias of the estimator due to the multiscale structure of
the problem, but that it does significantly reduce the effect of observation
error
Vigorous star formation in a bulge-dominated ERO at z = 1.34
We present near-IR spectroscopy of three Extremely Red Objects (EROs) using
the OHS/CISCO spectrograph at Subaru telescope. One target exhibits a strong
emission line which we identify as H-alpha at z = 1.34. Using new and existing
ground-based optical and near-IR imaging, and archival HST imaging, we argue
that this target is essentially an elliptical galaxy, with an old stellar
population of around 4E11 solar masses, but having a dust-enshrouded
star-forming component with a SFR of some 50--100 solar masses per yr. There is
no evidence that the galaxy contains an AGN. Analysis of a further two targets,
which do not exhibit any features in our near-IR spectra, suggests that one is
a quiescent galaxy in the redshift range 1.2 < z < 1.6, but that the other
cannot be conclusively categorised as either star-forming or quiescent.
Even though our first target has many of the properties of an old elliptical,
the ongoing star formation means that it cannot have formed_all_ of its stellar
population at high redshift. While we cannot infer any robust values for the
star formation rate in ellipticals at z > 1 from this one object, we argue that
the presence of an object with such a high SFR in such a small sample suggests
that a non-negligible fraction of the elliptical galaxy population may have
formed a component of their stellar population at redshifts z ~ 1--2. We
suggest that this is evidence for ongoing star formation in the history of
elliptical galaxies.Comment: Accepted by MNRAS 1 April 2005. Ten pages, six figure
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