Well-posed (numerical analysis)

Definition

A mathematical problem is said to be well-posed if:

1. A solution exists
2. The solution is unique
3. It depends continuously on the problem data (ie. the solution isn't volatile)

Condition 3 means that if a small change were to happen in the problem data, it wouldn't cause an abrupt, disproportionate change in the solution.

This well-posed property is highly desirable, but not always achievable. For example, inferring the internal structure of a physical system solely from external observations, as in tomography or seismology, often leads to mathematical problems that are inherently ill-posed in that distinctly different internal configurations may have indistinguishable external appearances.

However, even when a problem is well-posed, the solution can still be sensitive to perturbations in the problem data. Now, this becomes a manner in which we measure the sensitivity of the solution as well.

Thus, we also have stable algorithms.