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Fitting success and solver success

Every time we run a data modeling program, we have access to two publishable numbers $1-\vert\bold r\vert/\vert\bold d\vert$ and $1-\vert\bold F\T\,\bold r\vert/\vert\bold F\T\,\bold d\vert$. The first says how well the model fits the data. The second says how well we did the job of finding out.

Define the residual $\bold r = \bold F \bold m - \bold d $ and the ``size'' of any vector, such as the data vector, as $\vert\bold d\vert=\sqrt{\bold d \cdot \bold d}$. The number $1-\vert\bold r\vert/\vert\bold d\vert$ is called the ``success at fitting data.'' (Any data-space weighting function should have been incorporated in both $\bold F$ and $\bold d$.)

While the data fitting success is of interest to everyone, the second number $1-\vert\bold F\T\,\bold r\vert/\vert\bold F\T\,\bold d\vert$ is of interest in QA (quality analysis). In giant problems, especially those arising in seismology, running iterations to completion is impractical. A question always of interest is whether or not enough iterations have been run. This number gives us guidance to where more effort could be worthwhile.

$ 0 \quad \le \quad {\rm Success} \quad \le \quad 1$

Fitting success:      $1-\vert\bold r\vert/\vert\bold d\vert$

Numerical success:      $1-\vert\bold F\T\,\bold r\vert/\vert\bold F\T\,\bold d\vert$


next up previous [pdf]

Next: Roundoff Up: KRYLOV SUBSPACE ITERATIVE METHODS Previous: A basic solver program

2014-12-01