Talk:Well-posed problem

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I plan to edit the following sentence:

"A measure of well-posedness of a discrete linear problem is the condition number."

I might change it to something like the following two sentences:

"Even if a problem is well-posed, it may still be ill-conditioned, meaning that a small error in the initial data can result in much larger errors in the answers. An ill-conditioned problem is indicated by a small condition number."

Please discuss if you diagree with this change.

Further information:

Either a problem has a unique solution, or it does not. There is no such thing as degree of well-posedness. The quality indicated by the condition number is whether a problem is ill-conditioned or not -- not whether it is ill-posed or not.

Hadamard, 1902: "D'autre part, on a pu trouver des cas très étendus dans lesquels l'un ou l'autre de ces problèmes se présentait comme parfaitement bien posé, je veux dire comme possible et déterminé." (On the other hand, one has been able to find very extensive cases in which one or the other of these problems [the Cauchy problem and the Dirichlet problem] presents itself as perfectly well-posed, I mean as possible and determined.) (Italics are in the original.)

• "Condition: the product of the norm of a matrix and of its inverse."
• "Well-posed problem: A problem that has a unique solution that depends continuously on the initial data."
• "Ill-posed problem: [MATH] A problem which may have more than one solution, or in which the solutions depend discontinuously on the initial data. Also known as improperly posed problem."
• "Ill-conditioned problem: [COMPUTER SCI] A problem in which a small error in the data or in subsequent calculation results in much larger erros in the answers."

Definitions from McGraw-Hill Dictionary of Scientific and Technical Terms, 4th edition 1974, 1989. Sybil B. Parker, editor in chief. McGraw-Hill book company, New York. ISBN 0-07-045270-9

--Coppertwig 02:36, 10 December 2006 (UTC)

You're completely right, please go ahead. I assume that in the proposed text, you mean that an ill-conditioned problem is indicated by a big condition number. -- Jitse Niesen (talk) 04:33, 10 December 2006 (UTC)

Thank you. I went ahead (substituting "big" for "small".) I had that bit wrong but have checked it at the external links given on the condition number page and "big" looks correct. Thanks for saying I was "completely" right in spite of that! --Coppertwig 05:18, 10 December 2006 (UTC)

[edit] Example of ill-posed problem

It says By contrast the backwards heat equation, deducing a previous distribution of temperature from final data is not well-posed in that the solution is highly sensitive to changes in the final data. but this is wrong: the backwards heat equation is ill-conditioned but well-posed. How about this instead: "By contrast, the heat equation without specified boundary conditions is an ill-posed problem with infinitely many solutions." --Coppertwig 14:42, 6 May 2007 (UTC)