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

$${\bf d} = {\bf A} \; {\bf W}_{\rm m} \; {\bf x} = {\bf B} \; {\bf x}.$$

Now probe the composite operator, ${\bf B}$, for a data-space weighting function for the new system,  

$$\tilde{\bf W}_{\rm d}^{2} = \frac{ <{\rm\bf diag} ({\bf d}_{... ...epsilon_{\rm d} {\bf I}} \approx \frac{1}{{\bf B}\, {\bf B}'}.$$ (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

$${\bf m} = {\bf W}_{\rm m}^2 \; {\bf A}' \; \tilde{\bf W}_{\rm d}^2 \; {\bf d}.$$ (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.