TikhonovMorozov

TikhonovMorozov refers to the combined use of Tikhonov regularization and Morozov's discrepancy principle in solving ill-posed inverse problems. The approach couples a stabilizing penalty on the solution with a data-fit criterion that respects the noise level in the measurements.

In the standard linear setting, one models the data as b = Ax + ε, where A is a known operator, x is the unknown, and ε represents noise with a bound δ. Tikhonov regularization seeks xα by minimizing the objective function ||Ax − b||^2 + α||x||^2, where α > 0 is a regularization parameter. The Morozov discrepancy principle then guides the automatic choice of α by requiring that the corresponding residual satisfies ||Axα − b|| ≈ δ, i.e., the data misfit is at the level of the noise rather than overfitting the data.

Practically, α is selected through a search (for example, a monotone relation between α and the residual is exploited via bisection or other root-finding methods) to achieve a residual close to δ. The method is robust to noise and preserves stability by balancing fidelity to the measured data with the smoothing effect of the Tikhonov penalty.

Extensions include nonquadratic regularizers, general Banach-space formulations, and variants like total variation. The Tikhonov–Morozov framework remains widely used across disciplines such as geophysics, medical imaging, and signal processing, where ill-posedness and measurement noise are common.