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  "body": "\n## Definition\n\nLet $X, Y$ be [[Hilbert spaces|HM2HHE3J51]], let $T \\colon X \\to Y$ be a bounded linear operator, and let $T^\\dagger$ be its [[Moore–Penrose generalized inverse|0YY8VX28JR]], defined on $\\mathcal{D}(T^\\dagger) = \\mathcal{R}(T) + \\mathcal{R}(T)^\\perp$.\nA *regularization* of $T^\\dagger$ is a family $(R_\\alpha)_{\\alpha \u003e 0}$ of maps $Y \\to X$ together with a *parameter choice rule* $\\alpha \\colon (0, \\infty) \\times Y \\to (0, \\infty)$ such that:\n\n1. each $R_\\alpha$ is continuous;\n2. for every $y \\in \\mathcal{D}(T^\\dagger)$, $\\sup \\set{ \\alpha(\\delta, y^\\delta) : y^\\delta \\in Y, \\ \\norm{y^\\delta - y} \\le \\delta } \\to 0$ as $\\delta \\to 0$;\n3. for every $y \\in \\mathcal{D}(T^\\dagger)$, $\\sup \\set{ \\norm{R_{\\alpha(\\delta, y^\\delta)} y^\\delta - T^\\dagger y} : y^\\delta \\in Y, \\ \\norm{y^\\delta - y} \\le \\delta } \\to 0$ as $\\delta \\to 0$.\n\nThe pair is then called a *convergent regularization method*.\n\n## Remarks\n\nExample: [[Tikhonov regularization|M68V0M5A3R]] $R_\\alpha = (T^* T + \\alpha I)^{-1} T^*$ with an a priori rule $\\alpha = \\alpha(\\delta)$ such that $\\alpha(\\delta) \\to 0$ and $\\delta^2 / \\alpha(\\delta) \\to 0$.\nEach $R_\\alpha$ is bounded with $\\norm{R_\\alpha} \\le 1/(2\\sqrt{\\alpha})$, which gives clause 1.\nThe error splits as $\\norm{R_\\alpha y^\\delta - T^\\dagger y} \\le \\norm{R_\\alpha y - T^\\dagger y} + \\delta/(2\\sqrt{\\alpha})$.\nThe first term tends to $0$ as $\\alpha \\to 0$ by spectral calculus, and the second by the choice of $\\alpha$, which gives clause 3.\nNon-example: $R_\\alpha = T^\\dagger$ for all $\\alpha$ fails clause 1 whenever $\\mathcal{R}(T)$ is not closed, because $T^\\dagger$ is then unbounded.\nThis happens for every injective [[compact operator|M2NEP1K5AE]] with infinite-dimensional range, the typical model of an ill-posed problem.\nOther standard methods are truncated singular value decomposition, [[Landweber iteration|87DV5ZTJ9J]] with the iteration index as parameter, and the discrepancy principle as a parameter choice rule.\n\n## Explanations\n\n### L1 human\n\nA regularization method replaces the unstable inverse by stable approximations, one for each value of a parameter.\nThe parameter is chosen from the noise level.\nThe method is convergent when the answers approach the true solution as the noise goes to zero.\n",
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