A direct method for a regularized least-squares problem
2009 (English)In: NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, ISSN 1070-5325, Vol. 16, no 8, 649-675 p.Article in journal (Refereed) Published
We consider a linear system of the form A(1)x(1)+A(2)X(2)+eta=b1. The vector eta consists of identically distributed random variables all with mean zero. The unknowns are split into two groups x(1) and x(2). In the 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 x(2). We formulate the problem as a partially regularized least-squares problem, and propose a direct solution method based on the QR decomposition of matrix blocks. Further we consider regularizing using one and two regularization parameters, respectively. We also discuss the choice of regularization parameters, and extend Reinschs method to the case with two parameters. Also the cross-validation technique is treated. We present test examples taken from an application in modelling of the substance transport in rivers.
Place, publisher, year, edition, pages
2009. Vol. 16, no 8, 649-675 p.
QR-factorization, block matrices, least squares, regularization parameter, cross-validation, Reinschs method
IdentifiersURN: urn:nbn:se:liu:diva-19993DOI: 10.1002/nla.644OAI: oai:DiVA.org:liu-19993DiVA: diva2:232442