Editing Well-posed problem (section)

{{short description|Term regarding the properties that mathematical models of physical phenomena should have}}

In [[mathematics]], a '''well-posed problem''' is one for which the following properties hold:{{efn|This definition of a well-posed problem comes from the work of [[Jacques Hadamard]] on [[mathematical model|mathematical modeling]] of [[physical phenomena]].}}
# The problem has a solution
# The solution is [[Uniqueness quantification|unique]]
# The solution's behavior changes [[continuous function|continuously]] with the [[initial condition]]s

Examples of [[archetypal]] 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 in that there are physical processes modelled 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 problem]]s 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 [[discretization|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 [[Accuracy and precision|precision]], or with [[error]]s in the data.