SOLVING ILL-POSED LINEAR SYSTEMS WITH GMRES AND A SINGULAR PRECONDITIONER
2012 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 33, 1369-1394 p.Article in journal (Refereed) Published
Almost singular linear systems arise in discrete ill-posed problems. Either because ofthe intrinsic structure of the problem or because of preconditioning, the spectrum of the coefficientmatrix is often characterized by a sizable gap between a large group of numerically zero eigenvaluesand the rest of the spectrum. Correspondingly, the right-hand side has leading eigencomponentsassociated with the eigenvalues away from zero. In this paper the effect of this setting in theconvergence of the generalized minimal residual (GMRES) method is considered. It is shown thatin the initial phase of the iterative algorithm, the residual components corresponding to the largeeigenvalues are reduced in norm, and these can be monitored without extra computation. Theanalysis is supported by numerical experiments. In particular, ill-posed Cauchy problems for partialdifferential equations with variable coefficients are considered, where the preconditioner is a fast,low-rank solver for the corresponding problem with constant coefficients.
Place, publisher, year, edition, pages
2012. Vol. 33, 1369-1394 p.
ill-posed, linear system, GMRES, singular preconditioner, nearly singular
IdentifiersURN: urn:nbn:se:liu:diva-86985DOI: 10.1137/110832793OAI: oai:DiVA.org:liu-86985DiVA: diva2:584116