Questions: Well-Posed Problems

Question 1

Numerical differentiation of a noisy function produces wildly inaccurate results even when the input noise is tiny. Which Hadamard condition does this violate, and why?

• A. Existence — the derivative may not exist for noisy functions
• B. Uniqueness — there are infinitely many functions that match the noisy input
• C. Continuous dependence — small perturbations in input produce unbounded changes in output
• D. None — numerical differentiation is well-posed; the issue is floating-point precision

Answer: (C)

The derivative problem violates continuous dependence. Adding ε·sin(ωx) to f changes f by ε in sup-norm but changes f' by εω, which is unbounded as ω grows. This is not a floating-point artifact — it is a structural property of the differentiation operator. The problem is ill-posed regardless of numerical precision: the map from f to f' fails to be continuous in the relevant norms. Recognizing this explains why no choice of step size fully solves the problem.

Question 2

A linear system Ax = b is presented where b lies outside the column space of A. Which Hadamard condition does this violate?

• A. Uniqueness — infinitely many solutions exist
• B. Continuous dependence — the solution is highly sensitive to perturbations in b
• C. Existence — no exact solution exists
• D. None — this is a well-posed problem solvable by least squares

Answer: (C)

If b is outside the column space of A, there is no vector x satisfying Ax = b exactly — the existence condition fails. The system is ill-posed in the strictest sense. Least squares finds a best approximation, but that is a different (and modified) problem, not a solution to the original one. Recognizing this as an existence failure rather than a precision or algorithm problem is the diagnostic value of well-posedness analysis.

Question 3

A problem can satisfy both the existence and uniqueness conditions yet still be ill-posed if small changes in the input data produce large changes in the solution.

• T. True
• F. False

Answer: True

This is exactly the continuous dependence condition — the third and often most practically critical of Hadamard's three requirements. A system like Ax = b might have a unique solution for every b, yet if A is nearly singular, tiny perturbations in b cause enormous swings in x. Existence and uniqueness are 'theoretical' conditions; continuous dependence is the 'numerical' condition. All three must hold for a problem to be well-posed.

Question 4

An ill-posed problem can seldom be solved numerically and should simply be abandoned in favor of a different problem formulation.

• T. True
• F. False

Answer: False

Ill-posedness is a diagnosis, not a verdict. Once a problem is identified as ill-posed, regularization techniques — Tikhonov regularization, truncated SVD, smoothness priors — can restore continuous dependence by slightly modifying the problem (adding a constraint or penalty term). The modified problem is well-posed and numerically tractable, at the cost of a slight bias in the solution. Knowing the problem is ill-posed is the first step; regularization is the engineering response.

Question 5

Explain why understanding well-posedness provides more useful diagnostic information than knowing the condition number alone.

Model answer: The condition number measures sensitivity — how much the solution amplifies perturbations in the input — but it presupposes that a unique solution exists and that continuous dependence holds in some form. Well-posedness analysis comes first: it identifies which of the three conditions (existence, uniqueness, continuous dependence) is violated and why. A large condition number diagnoses degree of ill-conditioning within an otherwise well-posed framework; an ill-posed problem may not even have a condition number that is meaningful. More importantly, knowing that continuous dependence is violated tells you *why* the problem is hard structurally and points directly to the cure: regularization, not algorithmic refinement.

This distinction matters practically: if you don't know your problem is ill-posed, you might spend weeks optimizing your algorithm when the issue is in the problem formulation itself. Numerical differentiation is the canonical example — no amount of algorithmic improvement fixes its instability, because the instability is intrinsic to the operator. Well-posedness analysis tells you to either change the problem (regularize) or accept approximate solutions.