请教ill-conditioned与ill-posed的区别，附well-posed问题相关概念说明

Hey there! Let's clear up the relationship between well-posed problems and well-conditioned/ill-conditioned problems, since the stability aspects of both can feel overlapping at first glance.

First off, your understanding of well-posed problems is spot-on—they must satisfy three non-negotiable conditions:

• Existence of a solution
• Uniqueness of that solution
• Stability: The solution depends continuously on the problem's data (i.e., tiny perturbations to the data only lead to tiny perturbations in the solution)

Now, let's dive into well-conditioned vs ill-conditioned, and how they relate to well-posedness:

1. What they describe #

• Well-posedness is a fundamental property of the problem itself: It answers the question, "Is this problem mathematically valid and solvable in a meaningful way?" Think of it as a minimum bar the problem has to clear to be worth solving.
• Well/ill-conditioning is a property of how sensitive the solution is to data perturbations—for already well-posed problems: It’s a quantitative measure, not just a qualitative one. While well-posedness requires that solutions change somewhat with data changes, conditioning tells you how much they change.

2. A concrete example to illustrate #

Take a linear system Ax = b:

• If it’s well-posed, we know there’s exactly one solution, and small tweaks to b won’t make the solution blow up entirely.
• But if the system has an extremely large condition number (cond(A)), it’s ill-conditioned. Even a tiny error in b (like rounding error from floating-point calculations) will get amplified drastically in the solution x. Importantly, this system is still well-posed (since the solution doesn’t become infinite or undefined)—it’s just hard to solve accurately numerically.

3. The stability connection #

• The "stability" in well-posedness is a qualitative requirement: it just ensures that solution changes are bounded when data changes are small.
• Conditioning is a way to quantify that stability. A well-conditioned problem has low sensitivity to data perturbations (so stability is "strong"), while an ill-conditioned one has high sensitivity (stability is "weak," even though it still meets the well-posedness bar).

内容的提问来源于stack exchange，提问作者Bee