{"paper":{"title":"The regularization theory of the Krylov iterative solvers LSQR and CGLS for linear discrete ill-posed problems, part I: the simple singular value case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Zhongxiao Jia","submitted_at":"2017-01-20T07:29:05Z","abstract_excerpt":"For the large-scale linear discrete ill-posed problem $\\min\\|Ax-b\\|$ or $Ax=b$ with $b$ contaminated by a white noise, the Lanczos bidiagonalization based LSQR method and its mathematically equivalent Conjugate Gradient (CG) method for $A^TAx=A^Tb$ are most commonly used. They have intrinsic regularizing effects, where the number $k$ of iterations plays the role of regularization parameter. However, there has been no answer to the long-standing fundamental concern by Bj\\\"{o}rck and Eld\\'{e}n in 1979: for which kinds of problems LSQR and CGLS can find best possible regularized solutions? Here a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05708","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}