Partial differential equations Definition

Inverse problem

Builds on Well-posed problem Normed vector space

Definition

Let $X$ and $Y$ be normed vector spaces, or more generally metric spaces, and let $F \colon X \to Y$ be a map, called the forward map. The elements of $X$ are the unknown parameters, and the elements of $Y$ are the data. The inverse problem for $F$ is: given $y \in Y$ , find $x \in X$ with $F(x) = y$ . In 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 $\delta \to 0$ .

Remarks

The inverse problem is well-posed in the sense of Hadamard when $F$ is a bijection with continuous inverse; otherwise it is ill-posed. Example: let $X = Y = C[0, 1]$ with the maximum norm and $(Fx)(t) = \int_0^t x(s) \, ds$ . The inverse problem is differentiation: $x = y'$ when $y = Fx$ . It is ill-posed, since $y_n(t) = n^{-1} \sin(n^2 t)$ satisfies $\norm{y_n} \le n^{-1} \to 0$ while $\norm{y_n'} = n \to \infty$ . Non-example: computing $y = Fx$ from a given $x$ is the forward problem, not an inverse problem; here it is stable, since $\norm{Fx} \le \norm{x}$ . Standard 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.

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Tag	`GZDAPBG06Q`
URI	`m2://GZDAPBG06Q`
Kind	definition
Writing	W0 stub
MSC	35R30, 47A52, 65J22
Defines	Inverse problem
Uses	Well-posed problem, Normed vector space
Changed	2 November 2026

Inverse problem

Definition

Remarks

Reading path

Further reading

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