Illposedness

Ill-posedness, also written illposedness, describes a characteristic of mathematical problems in which one or more of Hadamard's criteria for well-posedness fail. A well-posed problem requires existence of a solution, uniqueness of that solution, and continuous dependence of the solution on the input data. If any of these conditions fails, the problem is considered ill-posed.

Ill-posedness is common in inverse problems, integral equations of the first kind, and problems involving smoothing operators. Classic examples include the backward heat equation, deconvolution with noisy data, and analytic continuation from limited measurements. These problems are often unstable and highly sensitive to measurement error.

Distinction: ill-posedness is a property of the problem's formulation, not merely its numerical conditioning. A problem can be well-posed in theory but numerically ill-conditioned due to discretization; conversely, a truly ill-posed problem remains ill-posed regardless of scale.

Regularization methods aim to restore stability by incorporating additional information or constraints. Techniques include Tikhonov regularization, truncated or damped singular value decompositions, variational approaches, and Bayesian inference. The choice of regularization reflects prior knowledge about the solution and balances fit to data with smoothness or other desired features.

History and use: The concept was introduced by Jacques Hadamard in the early 20th century to formalize when inverse problems can be meaningfully solved. Ill-posedness remains a central consideration in fields such as geophysics, medical imaging, astronomy, and signal processing, where robust recovery from incomplete or noisy data is essential.