Combining model-space and data-space 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?

Chemingui (1999) calculated both ${\bf W}_{\rm m}$ and ${\bf W}_{\rm d}$ from operator fold and data fold respectively, but observed that because they both contain the inverse units to the operator, ${\bf A}$, he should not apply them both at once. Instead he combined the two preconditioning operators, and solved the system

$$\begin{eqnarray} {\bf W}_{\rm d}^{n} \;{\bf d} &= &{\bf W}_{\rm d}^{n} \; {\bf A... ...m}^{n-1}\; {\bf x} \\ {\bf m}& =& {\bf W}_{\rm m}^{n-1}\; {\bf x},\end{eqnarray}$$ (102) (103)

where $0\leq n \leq 1$ is an adjustable parameter. Chemingui (1999) provided no advice on the choice of n, but for the problem he was solving, he observed that applying both ${\bf W}_{\rm m}$ and ${\bf W}_{\rm d}$ with n=1/2 converged to a solution more rapidly than either end member ($n = 0 \mbox{ or } 1$).

The first alternative 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,

$$\tilde{\bf W}_{\rm d}^{2} = \frac{ <{\rm\bf diag} ({\bf d}_{... ...epsilon_{\rm d} {\bf I}} \approx \frac{1}{{\bf B}\, {\bf B}'}.$$ (104)

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,

$$\begin{eqnarray} \tilde{\bf W}_{\rm d} \; {\bf d} &=& \tilde{\bf W}_{\rm d} \; {... ... {\rm with} \hspace{0.15in} {\bf m}& =& {\bf W}_{\rm m}\; {\bf x},\end{eqnarray}$$ (105) (106)

with appropriate model-space and data-space preconditioning operators. The adjoint solution to this system is given by

$${\bf m} = {\bf W}_{\rm m}^2 \; {\bf A}' \; \tilde{\bf W}_{\rm d}^2 \; {\bf d}.$$ (107)

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. The first step is to calculate ${\bf W}_{\rm d}$ with equation (), and set up a new system of equations,

$${\bf W}_{\rm d} \; {\bf d} = {\bf W}_{\rm d}\; {\bf A}\; {\bf m} = {\bf C}\; {\bf m}.$$ (108)

The second step is to calculate a model-space weighting function based on this new operator,

$$\tilde{\bf W}_{\rm m}^{2} = \frac{ <{\rm\bf diag} ({\bf m}_... ..., {\bf C} \; {\bf m}_{\rm ref})\gt + \epsilon_{\rm m} {\bf I}}.$$ (109)

The preconditioned composite system of equations is now

$$\begin{eqnarray} {\bf W}_{\rm d} \; {\bf d} &=& {\bf W}_{\rm d} \; {\bf A}\; \ti... ...with} \hspace{0.15in} {\bf m}& =& \tilde{\bf W}_{\rm m}\; {\bf x}.\end{eqnarray}$$ (110) (111)