illposedhet

Illposedhet is an informal term used in discussions of inverse problems to denote a subclass characterized by ill-posedness combined with heterogeneity across data sources or models. The name blends ill-posed with heterogeneous and is not a standard label in formal literature, but it is used to highlight the additional difficulties that arise when multiple sources or modalities with varying noise, resolution, or calibration must be fused or compared.

In the Hadamard sense, a problem is ill-posed if a solution does not exist, is not unique, or does not depend continuously on the data. Illposedhet draws attention to problems where this instability is amplified by heterogeneity: different sensors, measurement conditions, or modeling assumptions that produce nonuniform conditioning across components of the data or parameter space. This leads to challenges in stability analysis and solution interpretation that go beyond classical single-source ill-posedness.

Mathematically, one can view illposedhet as an inverse problem with a composite or multi-block observation model. Observations b are formed from heterogeneous blocks b_i, each with its own noise level and fidelity, and the forward operator may vary across blocks. Regularization approaches that address illposedhet often incorporate cross-source relations, such as weighted or joint regularization, multi-task sparsity, or Bayesian hierarchical formulations, to stabilize the solution while respecting inter-source differences.

Applications appear in geophysics, medical imaging, and remote sensing, where data from different modalities must be fused or jointly analyzed. Limitations include the informal status of the term and the variability of its usage; researchers typically prefer descriptions like multimodal or heterogeneous inverse problems.

See also: ill-posed problem, inverse problem, data fusion, regularization.