Well-posed problem

A problem is well-posed when a solution exists, is unique, and depends continuously on the input data. Failure of any condition makes it ill-posed.

Existence and uniqueness: A mathematical question about whether a solution exists and whether exactly one solution satisfies the problem's conditions. Well-posedness requires both conditions before stability is considered.
Ill-posed problem: A problem that lacks existence, uniqueness, or continuous dependence of its solution on the data. It is the direct opposite of satisfying all three well-posedness conditions.
Hadamard's ill-posed problems: Three classical examples showing that physical and mathematical problems can fail existence, uniqueness, or stability. Hadamard used these cases to illustrate how each well-posedness condition can fail.
Numerical ill-conditioning: A situation in which small input or rounding errors can cause large changes in a computed result. It is a practical form of sensitivity that can undermine well-posed solutions.
Continuous dependence on data: A property stating that small changes in input data cause only small changes in a problem's solution. This is the stability condition that distinguishes well-posedness from mere solvability.
Regularization: A method for stabilizing an ill-posed problem by adding constraints or modifying its solution criterion. It makes unstable inverse problems solvable in a controlled, approximate form.
Cauchy problem for the heat equation: The problem of recovering a heat distribution from data specified on a non-spatial boundary, such as a later time slice. Small measurement errors can grow without bound when heat evolution is run backward.
Computability: The study of whether a mathematical function or problem can be solved by an algorithm in finite steps. A problem may be well-posed yet lack an effective method for computing its solution.
Condition number: A numerical measure of how strongly a problem's output responds to changes in its input. It quantifies sensitivity to data perturbations in finite-dimensional settings.
Tikhonov regularization: A regularization method that balances data fit against a penalty on the solution's size or roughness. It is a standard construction for stabilizing ill-posed equations.

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