Tikhonovregularized

Tikhonov regularized, often referred to as Tikhonov regularization, is a method for stabilizing ill-posed linear inverse problems by augmenting the data-fitting objective with a penalty term. It incorporates prior information about the desired solution, such as smoothness or small magnitude, to reduce sensitivity to noise and measurement errors.

In its standard form, one seeks to recover x from Ax approximately equal to b by minimizing a functional that combines a data fidelity term with a regularization term: minimize ||Ax − b||^2 + α||Lx||^2, where A is the forward operator, b is observed data, α > 0 is the regularization parameter, and L is a regularization operator that encodes prior information. The solution satisfies the normal equations (A* A + α L* L)x = A* b, where A* and L* denote adjoints (or transposes in finite-dimensional cases).

A common special case is when L is the identity, reducing to a form analogous to ridge regression: minimize ||Ax − b||^2 + α||x||^2, with the solution (A* A + α I)x = A* b. Generalized Tikhonov regularization uses other operators L to enforce specific properties, such as smoothness via finite-difference operators or more complex priors, while keeping the quadratic penalty structure.

Parameter selection is critical; methods include the L-curve, Morozov discrepancy principle, and generalized cross-validation. Tikhonov regularization is widely used in image deblurring, tomography, geophysics, and other inverse problems where direct inversion is unstable or infeasible. It is named after Andrey Nikolayevich Tikhonov, who introduced the approach in the context of inverse problems in the 1960s.