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)

-- Wrong example: The problem in the example with the energy method is over-determined and not well-posed. The author ignores the method of characteristics. Also energy conservation doesn't imply well-posedness! And the opposite as well: Well-posedness doesn't imply energy conservation. I hope you correct these inaccuracies! Dmitsot (talk) 10:55, 17 August 2021 (UTC) User:Dmitsot

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)

Again on the examples of ill-posed problems

The Dirichlet problem for the Laplace equation is well-posed in the sense of Hadamard: it is the Cauchy problem for the same equation which is ill-posed.Daniele.tampieri (talk) 19:24, 15 April 2009 (UTC)

Great

Now where is the article about problems that are well-posed in the sense that they are meaningfully answerable (at least in principle)? In other words, what some antinomies are not. — Keφr 19:38, 28 December 2014 (UTC)

Well-posedness

I think that the article should also initially mention the corresponding property of well-posedness (also discussed above). However, in its current form I didn't have a clear idea of where to add such notion. — Preceding unsigned comment added by Natematic (talk • contribs) 18:27, 16 March 2020 (UTC)

"Well-poised" listed at Redirects for discussion

An editor has identified a potential problem with the redirect Well-poised and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 July 23#Well-poised until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 11:17, 23 July 2022 (UTC)

Why inequality not equation?

Perhaps it's just me being thick, but the derivation in the "energy method" section appears to me to lead to "$\left|\left|u\left(\cdot ,t\right)\right|\right|_{2}=\left|\left|f\left(\cdot \right)\right|\right|_{2}$. Why the "$\leq$"? Danhatton (talk) 11:38, 7 July 2023 (UTC)

Relating energy-method inequality to concept of well-posedness

As the article stands, it's not at all clear what the inequality $\left|\left|u\left(\cdot ,t\right)\right|\right|_{2}\leq \left|\left|f\left(\cdot \right)\right|\right|_{2}$ has to do with well-posedness. I can see that the primitive $-\alpha \left.{\frac {u^{2}}{2}}\right|_{0}^{1}$ vanishing is somehow related to having enough boundary conditions to make the solution unique, and that placing an upper bound on the mean-square value of $u$ is somehow related to smoothness, but I don't have either of these clear enough in my head to improve the article in this respect myself. Anyone fancy it? Danhatton (talk) 12:13, 7 July 2023 (UTC)

The inequality is stating that the linear operator defined by pushing the solution forward by a fixed amount of time is bounded. By the equivalence of boundedness and continuity for linear maps, the operator is continuous. Uniqueness comes from linearity of the problem statement. Existence of solution is not shown here though, can be found via Fourier transform. Akrodger (talk) 18:10, 6 November 2023 (UTC)

Hyeris Ulam stability

example 46.143.53.209 (talk) 13:24, 26 April 2024 (UTC)