Landweber iteration
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, and it can be now viewed as a special case of many other more general methods.
The original Landweber algorithm attempts to recover a signal x from (noisy) measurements y. The linear version assumes that 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 . However, if A is ill-conditioned, the explicit solution is a poor choice since it is sensitive to any noise in the data y. If A is singular, this explicit solution doesn't even exist. The Landweber algorithm is an attempt to regularize the problem, and is one of the alternatives to Tikhonov regularization. We may view the Landweber algorithm as solving:
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