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.
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.