Well-posed problem

The mathematical term well-posed problem stems from a definition given by Hadamard. He believed that mathematical models of physical phenomena should have the properties that

Related Topics:
Mathematical - Hadamard

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• A solution exists
• The solution is unique
• The solution depends continuously on the data, in some reasonable topology.

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 that solve these problems. By contrast the backwards 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. Problems that are not well-posed on the sense of Hadamard are termed ill-posed. Inverse problems are often ill-posed.

Related Topics:
Laplace's equation - Heat equation - Inverse problem

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Such continuum problems must often be discretized in order to obtain a numerical solution. While in terms of functional analysis such problems are typically continuous, they may suffer from numerical instability when solved with finite precision, or with errors in the data. A measure of well-posedness of a discrete linear problem is the condition number.

Related Topics:
Functional analysis - Condition number

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If a problem is well-posed, then it stands a good chance of solution on

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a computer using a stable algorithm. If it is not well-posed, it needs

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to be re-formulated for numerical treatment. Typically this involves including additional assumptions, such as smoothness of solution. This process is known as regularization.

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Well-posed problem: References ►