Well-posed problem

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a well-posed problem is one for which the following properties hold:

<ol><li>The problem has a solution</li>
<li>The solution is <a href="/facts/Uniqueness_quantification/qBpdIqel">unique</a></li>
<li>The solution's behavior changes <a href="/facts/Continuous_function/sKbl02pB">continuously</a> with the <a href="/facts/Initial_condition/sA4ohFP4">initial conditions</a></li></ol>
Examples of <a href="/facts/Archetypal/FsGNSHUo">archetypal</a> well-posed problems include the <a href="/facts/Laplace%27s_equation/8n1YkmFS">Dirichlet problem for Laplace's equation</a>, and the <a href="/facts/Heat_equation/2JPPr5GG">heat equation</a> 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 <a href="/facts/Global_optimization/nksWlDqU">global optimization</a> 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. <a href="/facts/Inverse_problem/Fe01pQ9E">Inverse problems</a> 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 <a href="/facts/Discretization/JvjSzwyk">discretized</a> in order to obtain a numerical solution. While solutions may be continuous with respect to the initial conditions, they may suffer from <a href="/facts/Numerical_instability/jRDdo8zV">numerical instability</a> when solved with finite <a href="/facts/Accuracy_and_precision/JQ47xXDq">precision</a>, or with <a href="/facts/Error/20QgR4EI">errors</a> in the data.

Well-posed problem open-in-new

Well-posed problem