A nonlinear PDE-based method for sparse deconvolution. (arXiv:1104.0240v1 [math.OC])

A nonlinear PDE-based method for sparse deconvolution. (arXiv:1104.0240v1 [math.OC]): "

In this paper, we introduce a new nonlinear evolution partial differential
equation for sparse deconvolution problems. The proposed PDE has the form of
continuity equation that arises in various research areas, e.g. fluid dynamics
and optimal transportation, and thus has some interesting physical and
geometric interpretations. The underlying optimization model that we consider
is the standard $\ell_1$ minimization with linear equality constraints, i.e.
$\min_u\{\|u\|_1 : Au=f\}$ with $A$ being an under-sampled convolution
operator. We show that our PDE preserves the $\ell_1$ norm while lowering the
residual $\|Au-f\|_2$. More importantly the solution of the PDE becomes sparser
asymptotically, which is illustrated numerically. Therefore, it can be treated
as a natural and helpful plug-in to some algorithms for $\ell_1$ minimization
problems, e.g. Bregman iterative methods introduced for sparse reconstruction
problems in [W. Yin, S. Osher, D. Goldfarb, and J. Darbon,SIAM J. Imaging Sci.,
1 (2008), pp. 143-168]. Numerical experiments show great improvements in terms
of both convergence speed and reconstruction quality.

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Optimisations for quadrature representations of finite element tensors through automated code generation. (arXiv:1104.0199v1 [cs.MS])

Optimisations for quadrature representations of finite element tensors through automated code generation. (arXiv:1104.0199v1 [cs.MS]): "

We examine aspects of the computation of finite element matrices and vectors
which are made possible by automated code generation. Given a variational form
in a syntax which resembles standard mathematical notation, the low-level
computer code for building finite element tensors, typically matrices, vectors
and scalars, can be generated automatically via a form compiler. In particular,
the generation of code for computing finite element matrices using a quadrature
approach is addressed. For quadrature representations, a number of optimisation
strategies which are made possible by automated code generation are presented.
The relative performance of two different automatically generated
representations of finite element matrices is examined, with a particular
emphasis on complicated variational forms. It is shown that approaches which
perform best for simple forms are not tractable for more complicated problems
in terms of run time performance, the time required to generate the code or the
size of the generated code. The approach and optimisations elaborated here are
effective for a range of variational forms.

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