next up previous

2 Direct and Inverse Problem Formulation     continued...

For a problem to be well-posed in the Hadamard sense, it must meet the following criteria:

For each set of data, there exists a solution.
The solution is unique.
The solution depends continuously on the data.

If a problem does not meet one or more of these criteria the problem is considered to be ill-posed. The bioelectric inverse problem in terms of primary current sources that was just presented lacks two of the three criteria for being well-posed: there is not a unique solution, and the solution doesn't depend continuously on the data. The first property, lack of uniqueness, means that there is a multitude of solutions; the second property means that small errors in measurements may cause large errors in the solution. When Hadamard wrote his 1902 paper defining well- and ill-posed problems, it was with the intent of saving mathematicians and scientists substantial time and trouble. That is, he warned against trying to solve ill-posed problems, as there wasn't any way of getting accurate results from such problems. Unfortunately (or fortunately, depending on your point of view), there exist many important problems in science and engineering that are ill-posed and that are in need of solutions. These include problems in mechanical engineering (inverse kinematics, crack detection), robotics (vision), geophysics (geophysical prospecting), astrophysics, quantum mechanics (inverse scattering), medicine (ultrasound, bioelectric and biomagnetic source problems), electrical engineering (inverse optics), and elsewhere. Since there are so many important ill-posed problems, mathematicians, scientists, and engineers have developed numerous methods to get around the problem of ill-posedness.