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Combining weighting functions

With two possible preconditioning operators, ${\bf W}_{\rm m}$ and ${\bf W}_{\rm d}$, the question remains, what is the best strategy for combining them?

The first strategy that I propose is to calculate a model-space weighting function, ${\bf W}_{\rm m}$, and use it to create a new preconditioned system with the form of

\begin{displaymath}
{\bf d} = {\bf A} \; {\bf W}_{\rm m} \; {\bf x} = {\bf B} \; {\bf x}.\end{displaymath}

Now probe the composite operator, ${\bf B}$, for a data-space weighting function for the new system,  
 \begin{displaymath}
\tilde{\bf W}_{\rm d}^{2} = \frac{ <{\rm\bf diag} ({\bf d}_{...
 ...epsilon_{\rm d} {\bf I}} \approx 
\frac{1}{{\bf B}\, {\bf B}'}.\end{displaymath} (12)
The new data-space weighting function is dimensionless, and can be applied in consort with the model-space operator. This leads to a new system of equations,
\begin{eqnarray}
\tilde{\bf W}_{\rm d} \; {\bf d} &=& \tilde{\bf W}_{\rm d} \; {...
 ...} \hspace{0.15in}
{\bf m}& =& {\bf W}_{\rm m}\; {\bf x}, \nonumber\end{eqnarray} (13)
with appropriate model-space and data-space preconditioning operators. The adjoint solution to this system is given by
\begin{displaymath}
{\bf m} = {\bf W}_{\rm m}^2 \; {\bf A}' \; \tilde{\bf W}_{\rm d}^2 
\; {\bf d}.\end{displaymath} (14)

A second alternative strategy is the corollary of this: create a new system that is preconditioned by an appropriate data-space weighting function, and then calculate a model-space weighing function based on the new system.


next up previous print clean
Next: Numerical comparisons Up: Data-space weighting functions Previous: Data-space weighting functions
Stanford Exploration Project
4/29/2001