Well-posed problem

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In mathematics, the solution to a well-posed problem satisfies the following properties: It exists; It is unique; Its behavior changes continuously with the auxiliary conditions, such as initial or boundary values. These criteria were first introduced by Jacques Hadamard in 1902. 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 processes modeled by these problems. Problems that are not well-posed in the sense above are termed ill-posed. A simple example is a global optimization problem, because the location of the optima is generally not a continuous function of the parameters specifying the objective, even when the objective itself is a smooth function of those parameters. Inverse problems are often ill-posed; for example, the inverse heat equation, deducing a previous distribution of temperature from final data...

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