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 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 models of physical phenomena should have the properties listed above.^[1]

Examples of 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, because they model there are physical processes.

Problems that are not well-posed in the sense of Hadamard are termed ill-posed. Inverse problems are often ill-posed. For example, the inverse 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.

Continuum models must often be discretized in order to obtain a numerical solution. While solutions may be continuous with respect to the initial conditions, they may suffer from numerical instability when solved with finite precision, or with errors in the data. 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. Problems in nonlinear complex systems (so-called chaotic systems) provide well-known examples of instability. An ill-conditioned problem is indicated by a large condition number.

If the 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. Tikhonov regularization is one of the most commonly used for regularization of linear ill-posed problems.

References

1. Jacques Hadamard: Sur les problèmes aux dérivées partielles et leur signification physique. In: Princeton University Bulletin. Bd. 13, Nr. 4, 1902, ZDB-ID 1282693-5, S. 49–52.