Regularization

Notebook

### Introduction to Tikhonov Regularization Method

First Steps with Regularization

SCIENCE-BLOG

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,

where F(x) is a linear operator, therefore we may write,

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,

The Tikhonov regularization propose another definition of the pseudoinverse which includes a regularization matrix $L$ and a regularization parameter $\alpha$,

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:

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:

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

1. Doicu, Adrian, Thomas Trautmann, and Franz Schreier. Numerical regularization for atmospheric inverse problems. Springer, 2010.