Preconditioning and spectral factorization

In Chapter , I discussed how spectral factorization amounts to the problem of finding an invertible square root of an autocorrelation function. The problem of finding an appropriate preconditioning operator for a linear inverse problem can be considered a generalization of this.

In general, any Hessian operator ${\bf A}' \,{\bf A}$ can be described as a non-stationary autocorrelation filter. If we could find a pair of invertible non-stationary factors, they would be the perfect preconditioning operators. Unfortunately for large problems, however, we cannot actually form the Hessian matrix explicitly, let alone factor it directly.