{"paper":{"title":"Analysis of regularized inversion of data corrupted by white Gaussian noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.AP","authors_text":"Hanne Kekkonen, Matti Lassas, Samuli Siltanen","submitted_at":"2013-11-25T14:58:33Z","abstract_excerpt":"Tikhonov regularization is studied in the case of linear pseudodifferential operator as the forward map and additive white Gaussian noise as the measurement error. The measurement model for an unknown function $u(x)$ is \\begin{eqnarray*} m(x) = Au(x) + \\delta\\hspace{.2mm}\\varepsilon(x), \\end{eqnarray*} where $\\delta>0$ is the noise magnitude. If $\\varepsilon$ was an $L^2$-function, Tikhonov regularization gives an estimate \\begin{eqnarray*} T_\\alpha(m) = \\text{argmin}_{u\\in H^r}\\big\\{\\|A u-m\\|_{L^2}^2+ \\alpha\\|u\\|_{H^r}^2 \\big\\}\\end{eqnarray*} for $u$ where $\\alpha=\\alpha(\\delta)$ is the regul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6323","kind":"arxiv","version":3},"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"}