Inverse Problems

An inverse problem infers hidden causes from observed effects, such as reconstructing internal state from external measurements.

Working backwards

A forward problem computes effects from known causes: given a plasma state, predict the magnetic signals it produces. An inverse problem does the reverse: given the measured signals, infer the plasma state. Inverse problems are how diagnostics turn indirect measurements into physical pictures.

Why they are hard

Inverse problems are often ill-posed: many different causes can produce nearly the same observations, so the data alone do not pin down a unique answer. Small measurement errors can produce large swings in the inferred cause unless the problem is stabilized.

Regularization

The fix is regularization: adding prior knowledge or a smoothness preference that selects a sensible solution among the many that fit the data. This trades a little agreement with the noisy data for a large gain in stability and physical plausibility.

A fusion example

Reconstructing the magnetic equilibrium and internal current profile of a tokamak from external magnetic and other diagnostics is a classic inverse problem. It is what lets operators know the shape and state of a plasma — including quantities like triangularity — that no instrument measures directly.

Trust in the answer

Because the answer depends on the chosen prior as well as the data, an honest inverse solution reports its assumptions and its uncertainty. Two analysts with different regularization can get different reconstructions from the same data, so stating the method is part of stating the result.