The well posedness of a problem refers to whether or not the problem is stable, as determined by whether it meets the three Hadamard criteria, which tests whether or not the problem has:
- A solution: s exists for all d (for every d relevant to the problem).
- A unique solution: s is unique for all d; for every data point d there is at most one value of s.
- A stable solution: s depends continously on d (a tiny change in d will lead to a tiny change in s; and a large change in d will lead to a proportionally larger change in s).
The Hadamard criteria tells us how well a problem lends itself to mathematical analysis.
Examples of Well Posedness
The majority of problems we work with in statistics, engineering, and math are well-posed. That includes such problems as f( x ) = x2 + x, f( x )= 3 x / 6, and f( x ) = sin ( x ) + 2 x 2
Take f( x ) = x 2 + x. For every real number x, x2 + x is also real and is well defined. There’s no room for ambiguity; every input k will give exactly one solution; k2 + k.
If x = 2, f( x )= 22 + 2 or 6,
If x = -1, f( x ) = (-1)2 – 1 = 0,
and so on.
The function is continuous because a large difference between data points will lead to a large difference in f( x ) values, while a small difference between data points leads to a small difference in f( x ). For every a,
You can test continuity of many functions by graphing; if you don’t have to take your pencil off the paper at any point or leave any ‘ empty holes’ in your lines, the function is continuous. For more ways to test for continuity, see: How to Check the Continuity of a Function.
A problem which is not well-posed is considered ill posed. Many first order differential equations and inverse problems are ill posed.
For example, consider the equation y’ = ( 2 – y ) / x. The solutions of the function are y = C/x + 2, where C is a constant. Since there are an infinite number of possible values of C, there are an infinite number of solutions, and the second Hadamard criteria is not met.
The History of Well Posedness
The Hadamard criteria was proposed by Jacques-Salomon Hadamard, a French mathematician, in 1923. He considered it to be a differentiation between useful problems and those which were not worth anything scientifically. Since then we’ve discovered that many important facets of real life (quantum mechanics, ultrasound testing, and optimal control theory, among other areas) can best be modeled by ill posed problems.
These problems are no longer avoided and their study is an active branch of applied mathematics, but the distinctions and terminology delineating well-posed and ill-posed remains the same as when Hadamard first defined it.
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