Knowra Inverse problem Inverse problem An inverse problem infers hidden causes, parameters, or structures from observed effects or measurements. It is often difficult because different hidden states can produce the same observations.
Forward problem : A problem that predicts observable effects from a specified model, cause, or set of parameters. Inverse problems run this mapping backward, from predicted effects toward the unknown inputs.
Mathematical model : A mathematical representation of a system, process, or relationship among quantities. An inverse problem uses a model to connect hidden quantities with measurable outcomes.
Computed tomography : An imaging method that reconstructs cross-sectional images from measurements of transmitted X-rays. It recovers internal structure from projections gathered around an object.
Optimization problem : A mathematical task of minimizing or maximizing an objective over a set of permitted choices. Many inverse problems are solved by optimizing a data-fit objective, often with added penalties.
Hadamard's well-posedness criteria : Three conditions for a well-posed problem: a solution exists, is unique, and depends continuously on the data. These criteria make precise the mathematical failures that complicate inverse recovery.
Ill-posed problem : A problem that fails at least one of existence, uniqueness, or continuous dependence of its solution on the data. Inverse problems often violate these conditions, making small measurement errors produce large changes in solutions.
Observation : A recorded measurement or datum about a system, produced by an instrument or observational procedure. The observations provide the evidence from which unknown causes or parameters are inferred.
Seismic inversion : The estimation of subsurface geological properties from recorded seismic waves. It infers underground structure and material properties from waves reflected or transmitted through Earth.
Least squares : A method that estimates unknowns by minimizing the sum of squared residuals between model predictions and data. It supplies a common data-fit criterion, though alone it may not stabilize an ill-posed inversion.
Uncertainty quantification : The study of representing and analyzing uncertainty in mathematical models and their predictions. A single reconstructed estimate can conceal the range of hidden states consistent with the data.
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