What is the meaning of variational regularization methods for inverse problems?

Question

I know if a inverse problem is ill posed, then we can partially obtain the information about the solution by applying regularization techniques. The most commonly used regularization method is Tikhonov regularization. We can also use iterative regularization.

Now if i am right, Tikhonov regularization is also known as variational regularization. Correct me please if i am wrong. Now i have two doubts.

1) Why Tikhonov regularization is also known as variational regularization?

2) Why Tikhonov regularization is most commonly used regularization method?

3) If i am wrong then what is the meaning of variational regularization and why it is named so?

Answer 1

2. Classic Tikhonov Regularization is very popular as in Inverse Problem usually the Fidelity Term is Least Squares and then using Tikhonov Regularization one could have a closed form solution.

Answer 2

If $A:X\to Y$ is continuous linear operator between Hilbert spaces, then one can use the Borel measurable functional calculus for self adjoint operators to define Tikhonov regularisation as

$$x_{\alpha,\delta}=(A^\ast A+\alpha I)^{-1}A^\ast y_\delta.$$

Your first question may be answered by the fact that the above is equivalent to

$$x_{\alpha,\delta}=\operatorname{argmin}_{x\in X}\|Ax-y_\delta\|+\alpha\|x\|.$$

In particular, if $X$ is a Banach space or $A$ is nonlinear, then the aforementioned spectral theory is no longer available, but one may still compute a regularised solution from the minimisation problem above. Such methods are called "variational". Indeed, the above formulation is easily derived by computing Gateaux derivatives which stem from the calculus of variations, hence the name. I hope then that this answers your third question.