1,769 research outputs found
New Douglas-Rachford algorithmic structures and their convergence analyses
In this paper we study new algorithmic structures with Douglas- Rachford (DR)
operators to solve convex feasibility problems. We propose to embed the basic
two-set-DR algorithmic operator into the String-Averaging Projections (SAP) and
into the Block-Iterative Pro- jection (BIP) algorithmic structures, thereby
creating new DR algo- rithmic schemes that include the recently proposed cyclic
Douglas- Rachford algorithm and the averaged DR algorithm as special cases. We
further propose and investigate a new multiple-set-DR algorithmic operator.
Convergence of all these algorithmic schemes is studied by using properties of
strongly quasi-nonexpansive operators and firmly nonexpansive operators.Comment: SIAM Journal on Optimization, accepted for publicatio
Bounded perturbation resilience of projected scaled gradient methods
We investigate projected scaled gradient (PSG) methods for convex
minimization problems. These methods perform a descent step along a diagonally
scaled gradient direction followed by a feasibility regaining step via
orthogonal projection onto the constraint set. This constitutes a generalized
algorithmic structure that encompasses as special cases the gradient projection
method, the projected Newton method, the projected Landweber-type methods and
the generalized Expectation-Maximization (EM)-type methods. We prove the
convergence of the PSG methods in the presence of bounded perturbations. This
resilience to bounded perturbations is relevant to the ability to apply the
recently developed superiorization methodology to PSG methods, in particular to
the EM algorithm.Comment: Computational Optimization and Applications, accepted for publicatio
The Convergence of Two Algorithms for Compressed Sensing Based Tomography
The constrained total variation minimization has been developed successfully for image reconstruction in computed tomography. In this paper, the block component averaging and diagonally-relaxed orthogonal projection methods are proposed to incorporate with the total variation minimization in the compressed sensing framework. The convergence of the algorithms under a certain condition is derived. Examples are given to illustrate their convergence behavior and noise performance
Triangular Gatzouras-Lalley-type planar carpets with overlaps
We construct a family of planar self-affine carpets with overlaps using lower
triangular matrices in a way that generalizes the original Gatzouras--Lalley
carpets defined by diagonal matrices. Assuming the rectangular open set
condition, Bara\'nski proved for this construction that for typical parameters,
which can be explicitly checked, the inequalities between the Hausdorff, box
and affinity dimension of the attractor are strict. We generalize this result
to overlapping constructions, where we allow complete columns to be shifted
along the horizontal axis or allow parallelograms to overlap within a column in
a transversal way. Our main result is to show sufficient conditions under which
these overlaps do not cause the drop of the dimension of the attractor. Several
examples are provided to illustrate the results, including a self-affine
smiley, a family of self-affine continuous curves, examples with overlaps and
an application of our results to some three-dimensional systems.Comment: 12 figures; v2: improved presentation, updated references, added a
three-dimensional example and an Appendix. Results unchange
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