an ill-posed problem
/ ill-POHZD /
An ill-posed problem is one that fails at least one of Hadamard's three conditions for a sensible question: a solution might not exist, or might not be unique, or — most often and most dangerously — the solution does not depend continuously on the data, so an arbitrarily small change in the input can produce an enormous change in the output. The question is fragile at its root, before any computer touches it.
The unstable kind is the one numerical people meet constantly, especially in inverse problems: trying to recover a cause from its effect. Reconstructing an image from blurred or incomplete measurements, recovering a heat source from later temperatures, or differentiating noisy data are classic examples. Differentiation makes the trap vivid: if measured data carries noise of size epsilon at frequency omega, the naive derivative amplifies it by omega, so high-frequency noise — invisible in the data — explodes in the answer. No clever algorithm fixes this, because the instability lives in the problem itself.
The cure is to change the problem, not the solver. Regularization (such as Tikhonov regularization or truncated-SVD) adds a small dose of prior information or smoothness that tames the instability, trading a little bias for a lot of stability and turning a hopeless question into an answerable nearby one. Recognizing ill-posedness early — by asking whether the answer is even a stable function of the data — saves you from chasing accuracy that the problem can never deliver.
Numerically differentiating f from noisy samples: with data noise of about 1e-3 and step h, the central-difference derivative carries error of order noise/h, so as h shrinks to reduce truncation error the noise term blows up — the problem, not the formula, is unstable.
An effect-to-cause (inverse) problem where small data wobbles cause huge answer swings — the signature of ill-posedness.
Ill-posed is not the same as ill-conditioned. Ill-posed is a yes/no defect of the problem (a condition fails); ill-conditioned is a quantitative warning that a well-posed problem is highly sensitive. In practice discretized ill-posed problems show up as severely ill-conditioned matrices.