Talk:Tikhonov regularization

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Mathematics rating:	B Class	Low Priority	Field: Probability and statistics
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. More explanation and motivation would help. Geometry guy 23:06, 9 June 2007 (UTC)

Contents 1 Generalized Tikhonov 2 Generalized Tikhonov - correction 3 Least Squares Solution 4 Textbooks? 5 General Form 6 explain what the variables represent

[edit] Generalized Tikhonov

The formula

x₀+ (A^TP A + Q)^-1A^TP(b- Ax₀)

is from Tarantola, eg (1.93) page 70. There are several other equivalent formulas.

[edit] Generalized Tikhonov - correction

It should be α²Q instead of just Q.

[edit] Least Squares Solution

"For α = 0 this reduces to the least squares solution of an overdetermined problem (m > n)."

This isn't correct. (A^TA)^-1 won't exist unless A has full column rank, and (m>n) does not imply that A has full column rank. The sentence should read "For α = 0 this reduces to the least squares solution provided that (A^TA)^-1 exists."

Isn't this equivalent to saying just that A ^{− 1} exists? —Simetrical (talk • contribs) 21:37, 14 August 2007 (UTC)

No: A doesn't have to be square. If A has full column rank then A^TA is invertible. --Zvika 06:28, 15 August 2007 (UTC)

[edit] Textbooks?

I take it that Tikhonov regularization is not discussed in text books yet? If it is, I would love a reference. 17:50, 16 October 2006 (UTC)

Yes of course. See some of the references in Inverse problemBilllion 08:22, 15 August 2007 (UTC)

[edit] General Form

Why not mentioning the general form incorporating an arbitrary smoothing operator (i.e. derivation operators) instead of just the identity matrix in special? Ref.: Hansen, P. C.: Rank-Deficient and Discrete Ill-posed Problems

[edit] explain what the variables represent

like the "m" and "q" in the bottom section. They just appear out of nowhere. Not very helpful. It's kinda like making up words adn randomly using them without explaining what they mean. Not very keaulnor. 65.26.249.208 (talk) 22:58, 9 December 2007 (UTC)