Hybrid-norm and Fortran 2003: Separating the physics from the solver |

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IRLS methods tend to be difficult to use because the user must choose carefully the number of relinearizations and the numbers of steps between relinearizations carefully. (Claerbout, 2009) suggested an alternate approach that dynamically changes the weighting function every iteration and uses a Taylor expansion of the standard conjugate direction algorithm to update the solution. Further he suggests a different norm, the hybrid norm, that creates a smooth transition between the standard L2 problem and a L1 problem. Given an error function and a residual vector the hybrid norm is defined

where is a user supplied bad-data percentile.

Creating an inversion framework that supports a Taylor expansion approach
to conjugate directions requires adding two additional features to our
vector class description. First, we need to associate a norm to each vector.
Second we need to be able to multiply a vector by another vector, element
by element.
Adding support for the hybrid norm requires more changes. A vector
must now have a bad-data percentage associated with it, it must be
able to find its *ith* percentile value, and create a vector
with this value.

Hybrid-norm and Fortran 2003: Separating the physics from the solver |

2010-11-26