The solution of large-scale problems by truncated iterative methods has recently received considerable attention. For instance, the conjugate gradient (CG) method has been applied to solve the normal
equations associated with the linear system
(1) Ax = b,
where b is the right-hand side contaminated by an error e. One seeks to terminate the iteration before the error e in b gives rise to a large propagated error in the computed solution. The iteration number can be thought of as a regularization parameter, which determines how accurately
(1) is solved.
In this talk we describe a modification of the conjugate gradient method for the normal equations (CGNR) that allows us to enrich the Krylov subspaces, in which the computed approximate solutions live with available information about the desired solution. Also we propose an enriched CG
method to solve the Tikhonov minimization problem. A few numerical examples
will be presented at the end of this talk to show the performance of the
methods discussed.
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