käänteisongelma

Käänteisongelma, known in English as an inverse problem, is a problem where one seeks to determine the causes of an observed phenomenon, rather than predicting the effects from the causes. In a forward problem, given a set of parameters and a model, one computes the outcome. In an inverse problem, given the outcome and the model, one attempts to determine the parameters that produced the outcome.

Inverse problems are ubiquitous in science and engineering. Examples include determining the internal structure of the Earth from seismic wave measurements, reconstructing an image from a blurred or noisy photograph, identifying the parameters of a physical system from experimental data, or diagnosing a disease from medical imaging.

A key characteristic of many inverse problems is their ill-posedness. This means that a solution may not exist, may not be unique, or may be highly sensitive to small changes in the observed data. For instance, two different internal structures could produce nearly identical seismic wave patterns. Addressing ill-posedness often requires regularization techniques, which introduce additional constraints or assumptions to stabilize the solution.

The mathematical formulation of inverse problems often involves integral equations or systems of linear equations. Techniques used to solve them include iterative methods, statistical inference, and machine learning approaches. The difficulty and complexity of an inverse problem depend heavily on the nature of the model and the quality of the observed data.