适定性问题
zh.wikipedia.org/wiki/適定性問題
数学术语适定性问题来自于哈达玛所给出的定义。他认为物理现象中的数学模型应该具备下述性质：
- 存在着解
- 解是惟一的
- 解连续地取决于初边值条件
适定性问题的原型范例包括对于拉普拉斯方程的狄利克雷问题，以及给定初始条件的热传导方程式。在物理过程中解决的这些问题，也许被视为“自然”问题。相较之下，反向热导方程，推演来自最终数据的温度的稍早分布就不是适定的，因为这个解对最终数据极为敏感。一个问题如果不是适定的，哈达玛就将其视为不适定。逆问题通常是不适定的。
这些连续问题必须使其离散，以取得数值解。泛函分析问题通常是连续的，当以有限精度或存有错误的资料求解时，它可以承受这些数值的不稳定性。
即使一个问题是适定的，它也可能仍是病态的；即在初始资料中的一个微小错误，可以造成很大错误的答案。病态问题以大的条件数表示。
如果某一个问题是适定的，它就有机会在使用了稳定算法的电脑上取得解。如果问题是不适定的，就需要为数值处理重新以公式表示。这通常包含了额外的假设，例如：解的平滑性。这个过程称为正则化（Regularization）。
en.wikipedia.org/wiki/Well-posed_problem
The mathematical term well-posed problem stems from a definition given by Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that:
- a solution exists,
- the solution is unique,
- the solution's behavior changes continuously with the initial conditions.
Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified initial conditions. These might be regarded as 'natural' problems in that there are physical processes modelled by these problems.
Problems that are not well-posed in the sense of Hadamard are termed ill-posed. Inverse problems are often ill-posed. For example, the inverse heat equation, deducing a previous distribution of temperature from final data, is not well-posed in that the solution is highly sensitive to changes in the final data.
Continuum models must often be discretized in order to obtain a numerical solution. While solutions may be continuous with respect to the initial conditions, they may suffer from numerical instability when solved with finite precision, or with errors in the data. Even if a problem is well-posed, it may still be ill-conditioned, meaning that a small error in the initial data can result in much larger errors in the answers. An ill-conditioned problem is indicated by a large condition number.
If the problem is well-posed, then it stands a good chance of solution on a computer using a stable algorithm. If it is not well-posed, it needs to be re-formulated for numerical treatment. Typically this involves including additional assumptions, such as smoothness of solution. This process is known as regularization. Tikhonov regularization is one of the most commonly used for regularization of linear ill-posed problems.