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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[normed vector spaces|03J5RAKB30]], or more generally metric spaces, and let $F \\colon X \\to Y$ be a map, called the forward map.\nThe elements of $X$ are the unknown parameters, and the elements of $Y$ are the data.\nThe *inverse problem* for $F$ is: given $y \\in Y$, find $x \\in X$ with $F(x) = y$.\nIn practice only noisy data $y^\\delta$ with $\\norm{y^\\delta - y} \\le \\delta$ are known, and one seeks approximations of $x$ that converge to it as $\\delta \\to 0$.\n\n## Remarks\n\nThe inverse problem is [[well-posed|W09B9NZ9B5]] in the sense of Hadamard when $F$ is a bijection with continuous inverse; otherwise it is ill-posed.\nExample: let $X = Y = C[0, 1]$ with the maximum norm and $(Fx)(t) = \\int_0^t x(s) \\, ds$.\nThe inverse problem is differentiation: $x = y'$ when $y = Fx$.\nIt is ill-posed, since $y_n(t) = n^{-1} \\sin(n^2 t)$ satisfies $\\norm{y_n} \\le n^{-1} \\to 0$ while $\\norm{y_n'} = n \\to \\infty$.\nNon-example: computing $y = Fx$ from a given $x$ is the forward problem, not an inverse problem; here it is stable, since $\\norm{Fx} \\le \\norm{x}$.\nStandard regularization methods replace $F^{-1}$ by a family of stable maps, such as Tikhonov regularization $x_\\alpha = \\operatorname{argmin}_x \\left( \\norm{Fx - y^\\delta}^2 + \\alpha \\norm{x}^2 \\right)$ in Hilbert spaces.\n",
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