754 research outputs found
Efficient decomposition of cosmic microwave background polarization maps into pure E, pure B, and ambiguous components
Separation of the B component of a cosmic microwave background (CMB)
polarization map from the much larger E component is an essential step in CMB
polarimetry. For a map with incomplete sky coverage, this separation is
necessarily hampered by the presence of "ambiguous" modes which could be either
E or B modes. I present an efficient pixel-space algorithm for removing the
ambiguous modes and separating the map into "pure" E and B components. The
method, which works for arbitrary geometries, does not involve generating a
complete basis of such modes and scales the cube of the number of pixels on the
boundary of the map.Comment: Minor changes to previous version. Accepted for publication in Phys.
Rev.
Biharmonic Conformal Field Theories
The main result of this paper is the construction of a conformally covariant
operator in two dimensions acting on scalar fields and containing fourth order
derivatives. In this way it is possible to derive a class of Lagrangians
invariant under conformal transformations. They define conformal field theories
satisfying equations of the biharmonic type. Two kinds of these biharmonic
field theories are distinguished, characterized by the possibility or not of
the scalar fields to transform non-trivially under Weyl transformations. Both
cases are relevant for string theory and two dimensional gravity. The
biharmonic conformal field theories provide higher order corrections to the
equations of motion of the metric and give a possibility of adding new terms to
the Polyakov action.Comment: 11 pages, LaTeX, no figure
Automated code generation for discontinuous Galerkin methods
A compiler approach for generating low-level computer code from high-level
input for discontinuous Galerkin finite element forms is presented. The input
language mirrors conventional mathematical notation, and the compiler generates
efficient code in a standard programming language. This facilitates the rapid
generation of efficient code for general equations in varying spatial
dimensions. Key concepts underlying the compiler approach and the automated
generation of computer code are elaborated. The approach is demonstrated for a
range of common problems, including the Poisson, biharmonic,
advection--diffusion and Stokes equations
Optimising Spatial and Tonal Data for PDE-based Inpainting
Some recent methods for lossy signal and image compression store only a few
selected pixels and fill in the missing structures by inpainting with a partial
differential equation (PDE). Suitable operators include the Laplacian, the
biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The
quality of such approaches depends substantially on the selection of the data
that is kept. Optimising this data in the domain and codomain gives rise to
challenging mathematical problems that shall be addressed in our work.
In the 1D case, we prove results that provide insights into the difficulty of
this problem, and we give evidence that a splitting into spatial and tonal
(i.e. function value) optimisation does hardly deteriorate the results. In the
2D setting, we present generic algorithms that achieve a high reconstruction
quality even if the specified data is very sparse. To optimise the spatial
data, we use a probabilistic sparsification, followed by a nonlocal pixel
exchange that avoids getting trapped in bad local optima. After this spatial
optimisation we perform a tonal optimisation that modifies the function values
in order to reduce the global reconstruction error. For homogeneous diffusion
inpainting, this comes down to a least squares problem for which we prove that
it has a unique solution. We demonstrate that it can be found efficiently with
a gradient descent approach that is accelerated with fast explicit diffusion
(FED) cycles. Our framework allows to specify the desired density of the
inpainting mask a priori. Moreover, is more generic than other data
optimisation approaches for the sparse inpainting problem, since it can also be
extended to nonlinear inpainting operators such as EED. This is exploited to
achieve reconstructions with state-of-the-art quality.
We also give an extensive literature survey on PDE-based image compression
methods
Biharmonic fields and mesh completion
We discuss bi-harmonic fields which approximate signed distance fields. We conclude that the bi-harmonic field approximation can be a powerful tool for mesh completion in general and complex cases. We present an adaptive, multigrid algorithm to extrapolate signed distance fields. By defining a volume mask in a closed region bounding the area that must be repaired, the algorithm computes a signed distance field in well-defined regions and uses it as an over-determined boundary condition constraint for the biharmonic field computation in the remaining regions. The algorithm operates locally, within an expanded bounding box of each hole, and therefore scales well with the number of holes in a single, complex model. We discuss this approximation in practical examples in the case of triangular meshes resulting from laser scan acquisitions which require massive hole repair. We conclude that the proposed algorithm is robust and general, and is able to deal with complex topological casesPeer ReviewedPostprint (author's final draft
The thermodynamics and roughening of solid-solid interfaces
The dynamics of sharp interfaces separating two non-hydrostatically stressed
solids is analyzed using the idea that the rate of mass transport across the
interface is proportional to the thermodynamic potential difference across the
interface. The solids are allowed to exchange mass by transforming one solid
into the other, thermodynamic relations for the transformation of a mass
element are derived and a linear stability analysis of the interface is carried
out. The stability is shown to depend on the order of the phase transition
occurring at the interface. Numerical simulations are performed in the
non-linear regime to investigate the evolution and roughening of the interface.
It is shown that even small contrasts in the referential densities of the
solids may lead to the formation of finger like structures aligned with the
principal direction of the far field stress.Comment: (24 pages, 8 figures; V2: added figures, text revisions
- …