However, a non-linear forward map leads to non-linear inverse problems, the majority of which are also ill-posed. y = ℱx, with x being the model parameters, then the inverse problem of finding x given y and ℱ is also linear. If the forward map, ℱ (from cause to effect) is linear with respect to the model parameters i.e. However, an inverse mapping from the output of such models to a complete set of input parameters, termed an inverse problem, is much more challenging. Given a mathematical model, this mapping from a set of input parameters to an output state is straightforward. For systems that are described by an exact mathematical model, the parameters of the model represent the cause, and the output of the model the effect. Cause and effect are central to every physical measurement in the classical world.
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