Diagonal weighting preconditioning

The convergence rate of the conjugate gradient algorithm depends on the condition number of the matrix to be inverted. For ill-conditioned matrices a preconditioner is often necessary. The design of a good preconditioner depends directly on the structure of the matrix. In the inversion relation equ1, the number of equations is the number of traces in the input data and the number of unknowns is the number of output traces or bins. Due to irregular sampling, the rows and columns of $\bold L$ are badly scaled. Since $\bold L$ is essentially a Kirchhoff-type matrix, its condition can be improved by the diagonal weighting described in the previous chapter. This implies pre- and post-multiplying the operator $\bold L$ by a diagonal matrix whose diagonal entries are the inverse of the sum of the rows or columns of $\bold L$.Similar approaches for diagonal scaling are discussed in the mathematical literature using different norms of the rows and columns. They are often referred to as left and right preconditioners; I prefer to call them data-space and model-space preconditioners. The rationale in the terminology is based on the fact that the scaled adjoint is the first step of the inversion. For left preconditioning, the adjoint operator is applied after the data have been normalized by the diagonal operator. I therefore refer to this weighting as data-space preconditioning. Right preconditioning is equivalent to applying the adjoint operator $\bold L^T$ followed by a scaling of the model by the diagonal operator. Consequently, I refer to this weighting as model-space preconditioning.