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Well Posed and Ill Posed problems

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Contents:

  1. Well posed problems
  2. Ill-posed problems

1. What is a Well Posed Problem?

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:

  1. A solution: a solution (s) exists for all data point (d), for every d relevant to the problem.
  2. A unique solution: s is unique for all d; d there is at most one value of s.
  3. A stable solution: s depends continuously 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 calculus, engineering, and math are well-posed. That includes problems like:

  • 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 following continuous function is an example of a well posed function; 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,
well posed

You can check 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.

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.

2. Ill-Posed Problem

An ill posed problem is one which doesn’t meet the three Hadamard criteria for being well-posed. These criteria are:

  • Having a solution
  • Having a unique solution
  • Having a solution that depends continuously on the parameters or input data.

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.

Examples of Ill Posed Problems

One simple example of an ill-posed problem is given by the equation y’ = (3/2)y1/3 with y(0) = 0. Since the solution is y(t) = ± t3/2, the solution is not unique (it could be plus t3/2 or it could be minus t3/2). As this violates rule 2 of the Hadamard criteria, the problem is ill posed.

Many inverse problems are ill-posed because either they don’t have a solution everywhere, their solution is not unique, or their solution is not stable (continuous).

A classic example is the inverse heat problem, where the distribution of surface temperature of solid is deduced from information on the inner surface area. Although the direct heat equation (with which you can derive the interior heat from surface data) is well defined, the inverse problem is not stable. The smallest changes in surface temperature data can lead to arbitrarily large differences in calculated interior heat distribution.

Hadamard and Well-Posedness

Jacques-Salomon Hadamard, the French mathematician who described the three Hadamard criteria in 1923, believed that any useful mathematical model of any physical problem must satisfy these criteria. At that time it was believed that natural problems should have continuous mathematical solutions; it was thought to be part of the inherent order of things. Since then we’ve discovered that many important scientific and technical problems  are not in fact well-posed in the traditional sense because they do not have continuous solutions.This includes problems in medicine (for instance, in Nuclear Magnetic Resonance topography and ultrasound testing), in physics (quantum mechanics, acoustics, etc.) and in economics (in optimal control theory, among other fields). Today the study of ill-posed problems is a very live branch of applied mathematics. Still,  the differentiation between them and more stable problems remains useful.

Solving an Ill Posed Problem

An ill posed problem will often need to be regularized or re-formulated before one can give it a full numerical analysis using computer algorithms or other computational methods. Reformulation often involves bringing in new assumptions to fully define the problem and narrow it down.

References

Well Posed and Stable Problems
Definition and Examples of Ill Posed and Inverse Problems
http://bayes.wustl.edu/etj/articles/well.pdf
Lecture Notes on Inverse Problems

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Stephanie Glen. "Well Posed and Ill Posed problems" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/well-posed-ill/
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