For a problem to be well-posed in the Hadamard sense,
it must meet the following criteria:
- (1)
- For each set of data, there exists a solution.
- (2)
- The solution is unique.
- (3)
- 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.
