Conclusions

We observe a significant (order-of-magnitude) speed-up in the optimization convergence when preconditionining interpolation problems with inverse recursive filtering. Since inverse filtering takes almost the same time as forward convolution, this speed-up translates straightforwardly into computational time savings.

The savings are hardly noticeable for simple test problems, but they can have a direct impact on the mere feasibility of iterative least-square inversion for large-scale (seismic-exploration-size) problems.

In the multidimensional case, recursive filtering is enabled by Claerbout's helix transform. The Wilson-Burg spectral factorization method allows us to construct stable recursive filters. By analyzing the role of B-spline interpolation in data regularization, I have introduced a method of constructing B-spline discrete regularization operators from continuous differential equations.

In the next chapter, I discuss possible choices of the regularization operator $\bold{D}$ and the preconditioning operator $\bold{P}$ in data regularization problems.