A block-preconditioner for a special regularized least-squares problem
2007 (English)In: Linear Algebra with Applications, ISSN 1070-5325 (print) 1099-1506 (online), Vol. 14, no 6, 469-484 p.Article in journal (Refereed) Published
We consider a linear system of the form A1x1 + A2x2 + =b1. The vector consists 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.
Place, publisher, year, edition, pages
2007. Vol. 14, no 6, 469-484 p.
conjugate gradient, least squares, regularization
IdentifiersURN: urn:nbn:se:liu:diva-14423DOI: 10.1002/nla.533OAI: oai:DiVA.org:liu-14423DiVA: diva2:23478