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Astron. Astrophys. 321, 634-642 (1997)

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6. Conclusions

We have investigated the feasibility of preprocessing a set of solar mean multiplet frequencies before performing inversions for solar structure. This is achieved through forming a linear combination of the original data, with weights obtained from a singular value decomposition of a discretized form of the inverse problem. The inversion is then carried out on only those combinations of modes which correspond to the largest singular values. This enables a reduction of the computational time required for the inversion by a substantial factor - for inversion with optimally localized averages the reduction factor is of the order [FORMULA], where M is the original number of modes and [FORMULA] the number of combinations included. We have shown that the inversion results, including averaging kernels and propagated errors, obtained with and without the transformations are indeed virtually indistinguishable, provided a sufficient number of combinations are included. In the case considered here the required [FORMULA] is approximately an order of magnitude smaller than M ; thus the saving in computing time is very considerable. The computational effort in setting up the singular value decomposition is modest. Thus for all but the smallest modesets, it is more efficient to preprocess the problem prior to inversion.

The properties of the singular values and associated singular vectors are related to the information content and other aspects of the inverse problem under consideration. For the simple case of rotational inversion in one dimension this was explored in detail by Christensen-Dalsgaard et al. (1993). We have considered a few features of the considerably richer problem of structure inversion, but more detailed investigations of this nature are evidently required. Such investigations would contribute to improving further the efficiency of the inversion procedures, and might provide a deeper insight into the information contents of inversions using different combinations of variables. However, the present investigation has already demonstrated the considerable gain in computational efficiency that can be achieved with SVD preprocessing.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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