An inverse problem seeks an unknown from data related by a forward map . Small singular values of the forward map can amplify measurement noise and make direct inversion unstable.
An inverse scattering problem reconstructs a medium or obstacle from measured scattered waves. Under the Born approximation, fixed-frequency far-field data sample the Fourier transform of the scattering potential on a restricted frequency surface.
For a refractive index and background wavenumber , one sign convention defines the scattering potential by . The total field then satisfies .
The Sommerfeld radiation condition selects outgoing solutions of the exterior Helmholtz equation. In three dimensions it requires as uniformly in direction.
A problem is well posed in the sense of Hadamard when a solution exists for every admissible datum, is unique, and depends continuously on the data. Failure of any condition makes it ill posed.
A regularization is a family of bounded approximate inverses together with a parameter rule such that whenever and lies in the domain of the Moore--Penrose inverse.
Tikhonov regularization minimizesgiving . The spectral filter suppresses unstable division by small singular values.
Iterated Tikhonov regularization repeatedly applies a filtered correction based on . Its singular-system representation exposes the iteration as a family of scalar spectral filters.
Variational regularization balances data fidelity against a lower-semicontinuous penalty, for example by minimizing .
A -minimizing exact solution satisfies the source condition when some obeysIt connects a penalty subgradient to the range of the adjoint forward operator.
For a convex functional and , the Bregman divergence isIt is nonnegative but need not be symmetric or satisfy the triangle inequality.
An exact penalty method uses a nonsquared residual such as . Under a source condition it can recover an exact constrained minimizer for every sufficiently small fixed positive .
The normal equation for is . Its solutions minimize ; when they exist, the unique solution orthogonal to is the Moore--Penrose solution .
A singular system for a compact operator has positive singular values and orthonormal vectors satisfying and . It diagonalizes spectral regularization methods.
For a compact operator with singular system , the datum lies in the domain of exactly when its component in satisfies
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The term "inverse problem" generally refers to a type of problem in various fields (such as mathematics, physics, engineering, and data science) where one aims to infer or reconstruct the inputs or causes from observed outputs or effects. Inverse problems contrast with "forward problems," where the relationship between inputs and outputs is known, and the goal is to predict the results of certain input conditions.