Very short comment on Tikhonov regularization

Tikhonov regularization method

This method is widely used to resolve discrete ill-posed inverse linear problems (also non-linear). The problem it is usually establish as, \[y=F(x)\]

where F(x) is a linear operator, therefore we may write, \[y=Kx\]

The solution of the inverse discrete problem its usually described with the singular value decomposition (SVD). Then it is defined the pseudoinverse of K as, \[K^{\dagger}=(K^{T}K)^{-1}K^{T}=V\Sigma^{\dagger}U^{T}\]

The Tikhonov regularization propose another definition of the pseudoinverse which includes a regularization matrix \(L\) and a regularization parameter \(\alpha\), \[K^{\dagger}=(K^{T}K+\alpha L^{T}L)^{-1}K^{T}=\bar{V}\Sigma^{\dagger}_{\alpha}\bar{U}^{T}\]

In order to write the last equation the generalized singular value decomposition (GSVD) is used.

Measurements with noise¹

The inversion method has an additional difficulty when the measurements incorporate errors, usually modelled as Gaussian Errors. In that case the vector state solution, written as \(x^{\epsilon}\) is related with the noisy measurements \(y^{\epsilon}\). The problem is formulated now as: \[y^{\epsilon}=Kx^{\epsilon}\]

and the additional problem is not just how to calculate the singular values, but how different are the estimation \(x^{\epsilon}\) and the true value \(x\).

However also there is a methodological problem on the definition of the solution \(x^{\epsilon}\) given that is not true that always \(y^{\delta}\in Im(K)\). Therefore, a new concept of solution is needed and it is usually formulated as an optimization rule: \[x^{\epsilon}: ||y^{\epsilon}-Kx^{\epsilon}||\le||y^{\epsilon}-Kx||\,\,\, \forall x \in \mathcal{X}\]

where \(\mathcal{X}\) is the domain of the Forward Model.