Krylov space solver in Fortran 2009: Beta version

Operator-based object-oriented solvers

SEP (Claerbout, 1999) has traditionally taken an approach which is described as either classical, traditional, or deterministic to iterative inversion. The classical approach attempts to find the model $\bf m$ that minimizes the data misfit. Given a recorded dataset $\bf d$ , and a linear operator $\bf L$ , we attempt to minimize the residual vector $\bf r$ which is defined as

$$\bf0 \approx \bf r = \bf d - \bf L \bf m.$$ (1)

In the simplest case where we are using steepest descent to solve the linear least squares inversion, we estimate $\bf m$ by mapping the initial residual (in this simple case $-\bf d$ ) back into the same space as the model to form a gradient vector $\bf g$ by applying the adjoint of $\bf L$ . We then map the gradient vector back into data-space by applying $\bf L$ to form $\bf\Delta \bf r$ . Finally, we find the scaling factor of $\bf\Delta \bf r$ that will make $\bf r + \bf\Delta \bf r$ as small as possible. We then repeat this procedure until $\bf r$ is suitably small. More complex inversion approaches are normally built on this basic concept.

Krylov space solver in Fortran 2009: Beta version