Tikhonov

Tikhonov is a Russian surname derived from the given name Tikhon. It is borne by several notable individuals in science and culture. In mathematics and related fields, the name is most closely associated with Tikhonov regularization, a foundational technique for stabilizing ill-posed inverse problems by introducing a penalty term into the objective function.

Tikhonov regularization addresses the problem of recovering a quantity x from noisy observations b that are related to x through a forward model Ax ≈ b. Direct inversion can be unstable or highly sensitive to noise, so the method solves a modified optimization problem that minimizes the sum of the data misfit and a penalty term. This penalty is typically alpha times the squared norm of Lx, where L is a chosen regularization operator and alpha > 0 is a parameter that balances fidelity to the data with the desired properties of the solution, such as smoothness. When L is the identity, the approach resembles ridge-type regression; other choices of L emphasize different features. The Tikhonov–Morozov framework also provides methods for selecting alpha, often via the discrepancy principle or cross-validation, to achieve a stable and meaningful solution.

Notable bearer: Andrey Nikolayevich Tikhonov, a Soviet mathematician who introduced the regularization concept in the 1960s. The technique has since become a standard tool across disciplines, with numerous extensions and variants for diverse inverse problems, noise characteristics, and modeling assumptions.