Preconditioning

Next: INTERVAL VELOCITY Up: OPPORTUNITIES FOR SMART DIRECTIONS Previous: The meaning of the

## 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 we must restart from 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.

 (28)

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

 (29)

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 preconditioning introduces the preconditioning variable . Our regression pair becomes:

 (30)

Here is the problem: We now require both and operators. In 2- and 3-dimensional spaces, we do not know very many operators with an easy inverse. That reason is why I found myself pushed to come up with the helix methodology of Chapter --because it provides invertible operators for smoothing and roughening.

 Preconditioning

Next: INTERVAL VELOCITY Up: OPPORTUNITIES FOR SMART DIRECTIONS Previous: The meaning of the

2015-05-07