A role of statistics in inverse methods

Inverse problems arise in many applications such a geophysics, biology, engineering, financial mathematics and medical imaging. The goal is to recover unknown parameters in a mathematical model from observations. Rather than try different parameter sets in the mathematical model until ones are found that produce the observations, inverse methods solve an optimization problem for the best parameter set. The resulting inversion for unknown parameters is typically an ill-posed problem because there is not a unique solution or solutions may not depend continuously on the observations. This is why inverse theory is a robust field of research. This talk will be from an inverse methods perspective and explore the relationship of them to statistical methods for parameter estimation. The relationship gives insight into choice of optimization techniques and methods for reducing ill-posedness.