Bayes meets Krylov: preconditioning CGLS for underdetermined systems
For practitioners solving underdetermined inverse problems, this provides a principled way to incorporate prior knowledge into iterative solvers.
The paper addresses underdetermined linear inverse problems by integrating Bayesian priors into the CGLS method via right preconditioning, demonstrating improved null space handling in computed examples.
The solution of linear inverse problems when the unknown parameters outnumber data requires addressing the problem of a nontrivial null space. After restating the problem within the Bayesian framework, a priori information about the unknown can be utilized for determining the null space contribution to the solution. More specifically, if the solution of the associated linear system is computed by the Conjugate Gradient for Least Squares (CGLS) method, the additional information can be encoded in the form of a right preconditioner. In this paper we study how the right preconditioned changes the Krylov subspaces where the CGLS iterates live, and draw a tighter connection between Bayesian inference and Krylov subspace methods. The advantages of a Krylov-meet-Bayes approach to the solution of underdetermined linear inverse problems is illustrated with computed examples.