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.