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The '''Landweber iteration''' or '''Landweber algorithm''' is an algorithm to solve [[ill-posed]] linear [[inverse problems]], and it has been extended to solve non-linear problems that involve constraints. The method was first proposed in the 1950s,<ref name="Landweber"/> and it can be now viewed as a special case of many other more general methods.<ref name="Combettes"/>
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== Basic algorithm ==
The original Landweber algorithm <ref name="Landweber"/> attempts to recover a signal ''x'' from measurements ''y''. The linear version assumes that <math>y=Ax</math> for a [[linear operator]] ''A''. When the problem is in finite [[dimensions]], ''A'' is just a matrix.
 
When ''A'' is [[nonsingular]], then an explicit solution is <math> x = A^{-1} y</math>. However, if ''A'' is [[ill-conditioned]], the explicit solution is a poor choice since it is sensitive to any errors made on ''y''. If ''A'' is [[Mathematical singularity|singular]], this explicit solution doesn't even exist. The Landweber algorithm is an attempt to [[Regularization (mathematics)|regularize]] the problem, and is one of the alternatives to [[Tikhonov regularization]].  We may view the Landweber algorithm as solving:
 
: <math> \min_x 0.5 \|Ax-y\|_2^2 </math>
 
using an iterative method. For [[ill-posed]] problems, the iterative method may be purposefully stopped before convergence.
 
The algorithm is given by the update
 
: <math> x_{k+1} = x_{k} - A^*(Ax_k - y). </math>
 
If we write <math> f(x) = 0.5 \|Ax-y\|_2^2 </math>, then the update can be written in terms of the [[gradient]]
 
: <math> x_{k+1} = x_k - \nabla f(x_k) </math>
 
and hence the algorithm is a special case of [[gradient descent]].
 
Discussion of the Landweber iteration as a [[regularization (mathematics)|regularization]] algorithm can be found in.<ref>Louis, A.K. (1989): Inverse und schlecht gestellte Probleme. Stuttgart, Teubner</ref><ref>Vainikko, G.M., Veretennikov, A.Y. (1986): Iteration Procedures in Ill-Posed Problems. Moscow, Nauka (in Russian)</ref>
 
== Nonlinear extension ==
In general, the updates generated by
<math> x_{k+1} = x_{k} - \tau \nabla f(x_k) </math>
will generate a sequence <math>f(x_k)</math> that [[convergence (mathematics)|converges]] to a minimizer of ''f'' whenever ''f'' is [[convex function|convex]]
and the stepsize <math>\tau</math> is chosen such that <math> 0 < \tau < 2/( \|A\|^2 ) </math> where <math> \|\cdot \| </math> is the [[spectral norm]].
 
Since this is special type of gradient descent, there currently is not much benefit to analyzing it on its own as the nonlinear Landweber, but such analysis was performed historically by many communities not aware of unifying frameworks.
 
The nonlinear Landweber problem has been studied in many papers in many communities; see, for example,.<ref>A convergence analysis of the Landweber iteration for nonlinear ill-posed problems
Martin Hanke, Andreas Neubauer and Otmar Scherzer. NUMERISCHE MATHEMATIK
Volume 72, Number 1 (1995), 21-37, DOI: 10.1007/s002110050158</ref>
 
== Extension to constrained problems ==
If ''f'' is a [[convex function]] and ''C'' is a [[convex set]], then the problem
 
: <math> \min_{x \in C} f(x) </math>
 
can be solved by the constrained, nonlinear Landweber iteration, given by:
 
: <math> x_{k+1} = \mathcal{P}_C( x_{k} - \tau \nabla f(x_k) )</math>
 
where <math>\mathcal{P}</math> is the [[projection (mathematics)|projection]] onto the set ''C''. Convergence is guaranteed when <math> 0 < \tau < 2/( \|A\|^2 ) </math>.<ref>Eicke, B.: Iteration methods for convexly constrained ill-posed problems in Hilbert space. Numer. Funct. Anal. Optim. 13, 413–429 (1992)</ref> This is again a special case of [[projected gradient descent]] (which is a special case of the [[Forward–backward algorithm (operator splitting)|forward–backward algorithm]]) as discussed in.<ref name="Combettes"/>
 
== Applications ==
Since the method has been around since the 1950s, it has been adopted by many scientific communities, especially those studying ill-posed problems. In particular, the [[computer vision]] community <ref>
Johansson, B., Elfving, T., Kozlovc, V., Censor, Y., Forssen, P.E., Granlund, G.; "The application of an oblique-projected Landweber method to a model of supervised learning", Math. Comput. Modelling, vol 43, pp 892–909 (2006)</ref> and the signal restoration community.<ref>Trussell, H.J., Civanlar, M.R.: The Landweber iteration and projection onto convex sets. IEEE Trans. Acoust., Speech, Signal Process. 33, 1632–1634 (1985)</ref> It is also used in [[image processing]], since many image problems, such as [[deblurring]], are ill-posed. Variants of this method have been used also in sparse approximation problems and compressed sensing settings.<ref>{{cite web
|url        = http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6136024&tag=1
|title      = Recipes for hard thresholding methods
|author      = Anastasios Kyrillidis and Volkan Cevher
}}</ref>
 
== References ==
<references>
<ref name="Landweber">Landweber, L. (1951): An iteration formula for Fredholm integral equations of the first kind.
Amer. J. Math. 73, 615–624</ref>
<ref name="Combettes">P. L. Combettes and J.-C. Pesquet, "Proximal splitting methods in signal processing," in: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, (H. H. Bauschke, R. S. Burachik, P. L. Combettes, V. Elser, D. R. Luke, and H. Wolkowicz, Editors), pp. 185–212. Springer, New York, 2011. [http://www.ann.jussieu.fr/~plc/prox.pdf PDF]</ref>
</references>
 
[[Category:Image processing]]
[[Category:Inverse problems]]
[[Category:Gradient methods]]

Latest revision as of 23:31, 17 July 2014

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You cɑn easily build muscle witҺ absolutely no efforts if you just manage to know how to do the same. You cоuld easily learn how to do thіs which is verү little studying on your part. You should learn how every musϲle can work and the different kinds of exercises that suit the muscle so that they can grow. One of the Best Nitric Oxide Supplements ways of building muscle it is important to make sure you know your diet. You should know that in order to build muscle you ѕhould not skimp on сalories, however, rememЬer yօu should make good choices with regard to food so that tɦe muscle that ƴou gain is not going to be covеred witҺ fat.

You coulԁ wonder what it is about the muscle that is amazing to look at. It іs defіnitely the low amount of fat percentage that is present in yoսr body that gives it such a ƅeautiful ѕіte for people to behold. Yes, that is true, the lower the amount of fat percentage present in your body it is easier for the person to be a body buіlder, but howeνer this does not constitute to be the main reason. Many individuals who have ɑ very low percentage of body fat, however, along with low fat percentage present in the body it is also a lot of solid and lean muscle, whіch makes muscle builԁing possible.

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