Inverse Problems by Mathias Richter

Author:Mathias Richter
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Generalization of Tikhonov Regularization

When information as in Assumption 3.16 is not available, heuristics come into play. Our starting point for going into this is a modification of Problem 3.20.

Problem 3.24 (Linear least squares problem with Lagrange parameter)

Let

(3.63)

Let rank(A) = n. Find the minimizer of

(3.64)

where the parameter λ ≥ 0 has to be chosen.

We know that ( 3.64) has a unique minimizer x λ for each λ ≥ 0. If λ is chosen as the Lagrange multiplier corresponding to the optimization Problem 3.20, then we will get the same result as before. By choosing λ ∈ [0, ∞) freely, the minimizer x λ can be shaped to some extent. To see how this works, remember the functions defined by

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