What is Well-posed problem?
In mathematics, a problem is called '''well-posed''', if all of the following conditions are met: * There is a solution to the problem * The solution of the problem is unique * The solution [[Continuous function|continuously]] depends on the input parameters If at least one of the conditions is not met, the problem is called ill-posed. [[Jacques Hadamard]] (1865-1963), a French mathematician, first studied these problems. He believed that [[Mathematical model|mathematical models]] of [[Physical phenomenon|physical phenomena]] should have the properties listed above.<ref>[[Jacques Hadamard]]: ''Sur les problèmes aux dérivées partielles et leur signification physique.'' In: ''Princeton University Bulletin.'' Bd. 13, Nr. 4, 1902, {{ZDB|1282693-5}}, [https://babel.hathitrust.org/cgi/pt?id=chi.095582186;view=1up;seq=65 S. 49–52].</ref> ==Examples== Examples of well-posed problems include the [[Laplace's equation#Boundary conditions|Dirichlet problem for Laplace's equation]], and the [[heat equation]] with specified initial conditions. These might be regarded as 'natural' problems, because they model there are physical processes.{{cn|date=December 2024}} Problems that are not well-posed in the sense of Hadamard are termed '''ill-posed'''. [[Inverse problem|Inverse problems]] are often ill-posed. For example, the inverse heat equation, deducing a previous [[Distribution (mathematics)|distribution]] of [[temperature]] from final data, is not well-posed in that the solution is highly sensitive to changes in the final data.{{cn|date=December 2024}} ==Discussion== Continuum models{{clarification|date=December 2024}} must often be [[Discretization|discretized]] in order to obtain a [[Numerical analysis|numerical solution]]. While solutions may be continuous with respect to the initial conditions, they may suffer from [[numerical instability]] when solved with finite [[Accuracy and precision|precision]], or with [[Error|errors]] in the data. Even if a problem is well-posed, it may still be '''ill-conditioned''', mean