Matrix Extension with Symmetry and Its Application to Filter Banks. (arXiv:1001.1117v1 [cs.IT])

In this paper, we completely solve the matrix extension problem with symmetry
and provide a step-by-step algorithm to construct such a desired matrix
$mathsf{P}_e$ from a given matrix $mathsf{P}$. Furthermore, using a cascade
structure, we obtain a complete representation of any $r imes s$ paraunitary
matrix $mathsf{P}$ having compatible symmetry, which in turn leads to an
algorithm for deriving a desired matrix $mathsf{P}_e$ from a given matrix
$mathsf{P}$. Matrix extension plays an important role in many areas such as
electronic engineering, system sciences, applied mathematics, and pure
mathematics. As an application of our general results on matrix extension with
symmetry, we obtain a satisfactory algorithm for constructing symmetric
paraunitary filter banks and symmetric orthonormal multiwavelets by deriving
high-pass filters with symmetry from any given low-pass filters with symmetry.
Several examples are provided to illustrate the proposed algorithms and results
in this paper.

Published by

Original source : http://arxiv.org/abs/1001.1117...

Eulerian and Semi-Lagrangian Methods for Convection-Diffusion for Differential Forms. (arXiv:1001.1031v1 [math.NA])

We consider generalized linear transient convection-diffusion problems for
differential forms on bounded domains in $mathbb{R}^{n}$. These involve Lie
derivatives with respect to a prescribed smooth vector field. We construct both
new Eulerian and semi-Lagrangian approaches to the discretization of the Lie
derivatives in the context of a Galerkin approximation based on discrete
differential forms. Details of implementation are discussed as well as an
application to the discretization of eddy current equations in moving media.

Published by

Original source : http://arxiv.org/abs/1001.1031...

Perturbation expansions of signal subspaces for long signals. (arXiv:1001.1051v1 [math.NA])

Singular Spectrum Analysis and many other subspace-based methods of signal
processing are implicitly relying on the assumption of close proximity of
unperturbed and perturbed signal subspaces extracted by the Singular Value
Decomposition of special "signal" and "perturbed signal" matrices. In this
paper, the analysis of the main principal angle between these subspaces is
performed in terms of the perturbation expansions of the corresponding
orthogonal projectors. Applicable upper bounds are derived. The main attention
is paid to the asymptotical case when the length of the time series tends to
infinity. Results concerning conditions for convergence, rate of convergence,
and the main terms of proximity are presented.

Published by

Original source : http://arxiv.org/abs/1001.1051...

Numerical studies of the metamodel fitting and validation processes. (arXiv:1001.1049v1 [math.NA])

Complex computer codes, for instance simulating physical phenomena, are often
too time expensive to be directly used to perform uncertainty, sensitivity,
optimization and robustness analyses. A widely accepted method to circumvent
this problem consists in replacing cpu time expensive computer models by cpu
inexpensive mathematical functions, called metamodels. In this paper, we focus
on the Gaussian process metamodel and two essential steps of its definition
phase. First, the initial design of the computer code input variables (which
allows to fit the metamodel) has to honor adequate space filling properties. We
adopt a numerical approach to compare the performance of different types of
space filling designs, in the class of the optimal Latin hypercube samples, in
terms of the predictivity of the subsequent fitted metamodel. We conclude that
such samples with minimal wrap-around discrepancy are particularly well-suited
for the Gaussian process metamodel fitting. Second, the metamodel validation
process consists in evaluating the metamodel predictivity with respect to the
initial computer code. We propose and test an algorithm which optimizes the
distance between the validation points and the metamodel learning points in
order to estimate the true metamodel predictivity with a minimum number of
validation points. Comparisons with classical validation algorithms and
application to a nuclear safety computer code show the relevance of this new
sequential validation design.

Published by

Original source : http://arxiv.org/abs/1001.1049...