Priorconditioners for linear systems

The construction of suitable preconditioners for the solution of linear systems by iterative methods continues to receive a lot of interest. Traditionally, preconditioners are designed to accelerate convergence of iterative methods to the solution of the linear system. However, when truncated iterative methods are used as regularized solvers of ill-posed problems, the rate of convergence is seldom an issue, and traditional preconditioners are of little use. Here, we present a new approach to the design of preconditioners for ill-posed linear systems, suitable when statistical information about the desired solution or a collection of typical solutions is available. The preconditioners are constructed from the covariance matrix of the solution viewed as a random variable. Since the construction is based on available prior information, these preconditioners are called priorconditioners. A statistical truncation index selection is also presented. Computed examples illustrate how effective such priorconditioners can be.