A block‐preconditioner for a special regularized least‐squares problem

We consider a linear system of the form A₁x₁ + A₂x₂ + η=b₁. The vector ηconsists of independent and identically distributed random variables all with mean zero. The unknowns are split into two groups x₁ and x₂. It is assumed that AA₁ has full rank and is easy to invert. In this model, usually there are more unknowns than observations and the resulting linear system is most often consistent having an infinite number of solutions. Hence, some constraint on the parameter vector x is needed. One possibility is to avoid rapid variation in, e.g. the parameters x₂. This can be accomplished by regularizing using a matrix A₃, which is a discretization of some norm (e.g. a Sobolev space norm). We formulate the problem as a partially regularized least‐squares problem and use the conjugate gradient method for its solution. Using the special structure of the problem we suggest and analyse block‐preconditioners of Schur compliment type. We demonstrate their effectiveness in some numerical tests. The test examples are taken from an application in modelling of substance transport in rivers. Copyright © 2007 John Wiley & Sons, Ltd.