a well-posed problem
/ well-POHZD /
A problem is well-posed, in the sense Jacques Hadamard set out, when it behaves the way you would want a sensible question to behave: it has a solution, that solution is unique, and the solution depends continuously on the data — meaning a small change in the inputs produces only a small change in the answer. These three conditions — existence, uniqueness, and stable dependence on the data — are the licence to compute.
The third condition is the subtle and most important one for computation. Even if an answer exists and is unique, the problem is treacherous if a tiny wobble in the input — a measurement off in its last digit, or a round-off in the machine — can swing the output wildly. Continuous dependence is the promise that the answer is robust to the small, inevitable imperfections in real data and real arithmetic. Think of it as asking the question itself, before any method is chosen: is the true answer even a stable function of the true inputs?
Well-posedness is a property of the problem, not of any algorithm you use to solve it. It is the right thing to check first: if the underlying problem is well-posed you may go hunting for a good, stable method; if it is not, no algorithm can rescue you, and you must reformulate or regularize the problem. (Note: well-posed is a yes/no notion; among well-posed problems, conditioning then measures how sensitive the answer is quantitatively.)
Solving A x = b for a square matrix A with nonzero determinant is well-posed: a unique x exists, and small changes in b shift x only a little. Differentiating noisy measured data is classically ill-posed — tiny noise produces huge swings in the derivative.
Existence + uniqueness + continuous (stable) dependence on the data — Hadamard's three conditions.
Well-posedness is necessary but not sufficient for an easy computation: a well-posed problem can still be very ill-conditioned (sensitive in degree, though stable in the limit), which costs you accuracy even with a perfect algorithm.