Preconditioning

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## Need for an invertible preconditioner

It is important to use regularization to solve many examples. It is important to precondition because in practice computer power is often a limiting factor. It is important to be able to begin from a nonzero starting solution because in nonlinear problems then we must restart from the result of an earlier solution. Putting all three requirements together leads to a little problem. It turns out the three together lead us to needing a preconditioning transformation that is invertible. Let us see why this is so.

 (15)

First we change variables from to . Clearly starts from , and . Then our regression pair becomes

 (16)

This result differs from the original regression in only two minor ways, (1) revised data, and (2) a little more general form of the regularization, the extra term .

Now let us introduce preconditioning. From the regularization we see this introduces the preconditioning variable . Our regression pair becomes:

 (17)

Here is the problem: Now we require both and operators. In 2- and 3-dimensional spaces we don't know very many operators with an easy inverse. Indeed, that is why I found myself pushed to come up with the helix methodology of the previous chapter - because it provides invertible operators for smoothing and roughening.

 Preconditioning

Next: OPPORTUNITIES FOR SMART DIRECTIONS Up: PRECONDITIONING THE REGULARIZATION Previous: The preconditioned solver

2011-08-20