inversioonimeetodite

Inverse methods, known in Estonian as inversioonimeetodite, are mathematical techniques used to infer the properties of a system from indirect measurements. Unlike forward modeling, where a model predicts observable data, inverse methods work backward from data to estimate unknown parameters. The approach is common in many scientific disciplines, including geophysics, medical imaging, astrophysics, and machine learning.

A core challenge for inverse methods is that many inverse problems are ill‑posed: small data errors can cause large variations in the solution, and multiple model sets can explain the same observations. Regularization methods, such as Tikhonov regularization or the use of prior information, are applied to stabilize solutions and enforce smoothness or physical plausibility. Bayesian frameworks treat unknowns as random variables and combine data likelihood with prior distributions to obtain posterior probability distributions, offering uncertainty quantification.

Popular inverse techniques include linear inverse methods, where the relationship between data and model parameters is linear, and nonlinear inversion, which uses iterative optimization or Monte Carlo sampling to handle complex relationships. Inverse problems also appear in tomography (e.g., CT and MRI), electrical resistivity surveying, seismic wave inversion, and the reconstruction of atmospheric properties from satellite observations.

Historically, inverse theory developed in the mid‑twentieth century with pioneers like Charles H. Willsky and later advanced through computational power and statistical theory. Modern inverse methods rely on large‑scale matrix computations, parallel processing, and sophisticated regularization strategies, allowing practitioners to extract meaningful information from noisy or incomplete datasets. The field continues to evolve, integrating machine learning approaches to improve accuracy and reduce computational cost.