## Wavelet sparse approximate inverse preconditioners.(English)Zbl 0891.65048

The authors are interested in solving large sparse problems of the form $$Ax= b$$ using sparse approximate inverse preconditioners. This amounts to first solving $$AMy= b$$ and then computing $$x= My$$. The aim is to choose $$M$$ such that it has as much sparsity as possible while $$AM$$ is still an approximation to the identity matrix. The main idea in the present paper is to represent $$A$$ in a different basis in which $$A^{-1}$$ has a sparse approximation. Specifically, if $$A^{-1}$$ presents some piecewise smoothness then one uses a wavelet transformation to convert this smoothness into small wavelet coefficients.
The authors compare this idea to that of a hierarchical basis preconditioner (to which it is closely related) and present a large number of interesting numerical experiments. The weak points of the method are also fairly discussed with suggestions for future work.
Reviewer: W.Govaerts (Gent)

### MSC:

 65F35 Numerical computation of matrix norms, conditioning, scaling 65F50 Computational methods for sparse matrices 65T50 Numerical methods for discrete and fast Fourier transforms
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### References:

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