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Preconditioner with a starting guess

We often have a starting solution $ \bold m_0$ . You might worry that you could not find the starting preconditioned variable $ \bold p_0= \bold S^{-1}\bold m_0$ , because you did not know the inverse of $ \bold S$ .

We solve this problem using a shifted unknown $ \tilde {\bold m}$ .

$ \bold 0 $ $ \approx$ $ \bold F\bold m -\bold d $ typical regression
$ \bold 0 $ $ \approx$ $ \bold F(\tilde{\bold m}+\bold m_0) -\bold d $ Define $ \bold m =\tilde{\bold m}+\bold m_0$
$ \bold 0 $ $ \approx$ $ \bold F\tilde{\bold m}+\bold F \bold m_0 -\bold d $  
$ \bold 0 $ $ \approx$ $ \bold F\tilde{\bold m}-\tilde {\bold d} $ Defines $ \tilde{\bold d}$
      Implicitly define $ \bold p$ by $ \tilde{\bold m}=\bold S\bold p$ .
$ \bold 0 $ $ \approx$ $ \bold F\bold S \bold p-\tilde {\bold d} $ You iterate for $ \bold p$ .
$ \tilde {\bold m}$ = $ \bold S\bold p$ from your definition
$ \bold m$ = $ \tilde{\bold m} + \bold m_0 $ Got the answer.

which solves the problem never needing $ \bold S^{-1}$ . Unfortunately, as we see later, this conclusion is only valid while there is no regularization.


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Next: Guessing the preconditioner Up: PRECONDITIONED DATA FITTING Previous: PRECONDITIONED DATA FITTING

2015-05-07