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- A solution exists
- The solution is unique
- The solution depends continuously on the data, in some reasonable topology.
Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified initial conditions. These might be regarded as 'natural' problems in that there are physical process that solve these problems. 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. Problems that are not well-posed on the sense of Hadamard are termed ill-posed. Inverse problems are often ill-posed.
Such continuum problems must often be discretized in order to obtain a numerical solution. While in terms of functional analysis such problems are typically continuous, they may suffer from numerical instability when solved with finite precision, or with errors in the data. A measure of well-posedness of a discrete linear problem is the condition number.
If a problem is well-posed, then it stands a good chance of solution on a computer using a stable algorithm. If it is not well-posed, it needs to be re-formulated for numerical treatment. Typically this involves including additional assumptions, such as smoothness of solution. This process is known as regularization.
- Jaques Hadamard (1902): Sur les problèmes aux dérivées partielles et leur signification physique. Princeton University Bulletin, 49--52.
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