The SOLA mollifier method is an inversion method which, in terms of both computational efficiency and resolution ability, is competitive to other classes of inversion methods. In its original implementation, however, it is inefficient when many regularization parameters must be used, e.g., in connection with parameter-choice methods such as generalized cross-validation. We have demonstrated how this problem can be overcome using standard ``building blocks'' from numerical linear algebra. The improvement for dense matrices is quite dramatic when using the full bidiagonalization approach in Algorithm 2.3 (click here). For sparse or structured matrices it is convenient to use an iterative Lanczos-based method, Algorithm 2.3 (click here), in which the amount of regularization is controlled by the number of iterations.
Another advantage of using the standard numerical linear algebra ``building blocks'' is that these subroutines are usually available in highly vectorized and parallelized versions on today's supercomputers, thus ensuring good performance without the need for a tedious fine-tuning of the software; see, e.g., Dongarra et al. (1991).
As illustrated in our numerical example, Algorithm 2.3 (click here) computes solutions in relatively few iterations which are very close to the optimal solutions obtained from Algorithm 2.3 (click here). It is quite typical when solving large ill-posed problems, e.g., in inverse helioseismology, that the number of iterations needed in a Lanczos-based method is far less than the dimension of the problem. This makes these iterative methods an interesting alternative to the direct methods such as Algorithms 2.1 (click here) and 2.3 (click here).
Acknowledgements
We thank Jørgen Christensen-Dalsgaard for fruitful discussions during this work.