illposed

Illposed (often written ill-posed) is a term used to describe problems that do not meet the criteria for a well-posed problem in the sense of Jacques Hadamard. A well-posed problem is one that has a solution, whose solution is unique, and whose solution depends continuously on the input data. If any of these conditions fails, the problem is considered ill-posed.

Ill-posed problems frequently arise in inverse problems and numerical analysis. Examples include numerical differentiation from noisy measurements, deconvolution in signal processing, and image deblurring. In such cases, small errors or noise in the input can cause large, unstable changes in the computed solution, or multiple solutions may exist. Ill-posedness is often linked to insufficient data, incomplete modeling, or intrinsic non-uniqueness.

Addressing illposedness typically involves regularization and incorporating additional information or constraints. Regularization methods, such as Tikhonov regularization or truncated singular value decomposition, impose penalties or filter unstable components to produce stable, approximate solutions. Bayesian approaches introduce prior information to constrain solutions probabilistically. Other techniques include smoothing, discretization choices, and iterative stabilization. The goal is to transform the original problem into a nearby well-posed one that yields useful, robust results in practice.

In summary, illposed refers to problems that are unstable or underdetermined with respect to data, necessitating extra assumptions or methods to obtain reliable solutions. The term is widely used in mathematics, physics, engineering, and data science, where inverse and reconstruction problems are common. Some writers omit the hyphen, writing illposed as a variant.