Well-posed problem
In mathematics, a well-posed problem is one for which the following properties hold:[a]
- The problem has a solution
- The solution is unique
- The solution's behavior changes continuously with the initial conditions
Examples of
Problems that are not well-posed in the sense above 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
in the data.Conditioning
Even if a problem is well-posed, it may still be
If the problem is well-posed, then it stands a good chance of solution on a computer using a
Energy method
The energy method is useful for establishing both uniqueness and continuity with respect to initial conditions (i.e. it does not establish existence). The method is based upon deriving an upper bound of an energy-like functional for a given problem.
Example: Consider the diffusion equation on the unit interval with homogeneous
Multiply the equation by and integrate in space over the unit interval to obtain
This tells us that (
This result is the energy estimate for this problem.
To show uniqueness of solutions, assume there are two distinct solutions to the problem, call them and , each satisfying the same initial data. Upon defining then, via the linearity of the equations, one finds that satisfies
Applying the energy estimate tells us which implies (almost everywhere).
Similarly, to show continuity with respect to initial conditions, assume that and are solutions corresponding to different initial data and . Considering once more, one finds that satisfies the same equations as above but with . This leads to the energy estimate which establishes continuity (i.e. as and become closer, as measured by the norm of their difference, then ).
The maximum principle is an alternative approach to establish uniqueness and continuity of solutions with respect to initial conditions for this example. The existence of solutions to this problem can be established using Fourier series.
See also
- Total absorption spectroscopy – an example of an inverse problem or ill-posed problem in a real-life situation that is solved by means of the expectation–maximization algorithm
Notes
- physical phenomena.
References
- Hadamard, Jacques (1902). Sur les problèmes aux dérivées partielles et leur signification physique. Princeton University Bulletin. pp. 49–52.
- Parker, Sybil B., ed. (1989) [1974]. McGraw-Hill Dictionary of Scientific and Technical Terms (4th ed.). New York: McGraw-Hill. ISBN 0-07-045270-9.
- Tikhonov, A. N.; Arsenin, V. Y. (1977). Solutions of ill-Posed Problems. New York: Winston. ISBN 0-470-99124-0.
- Strauss, Walter A. (2008). Partial differential equations; An introduction (2nd ed.). Hoboken: Wiley. ISBN 978-0470-05456-7.