Problems with Inverse Problems

Inverse problems are very often ill-posed. Solutions may not exist, may not be unique, or may not behave well with respect to changing initial conditions. Even well-posed problems can be ill-conditioned, as the solutions may be numerically unstable. In this case, small errors in data can lead to solutions that contain no useful information. Regularization techniques are used to combat these problems. I will discuss Tikhonov regularization that is developed through analysis of the singular value decompositions. Then special consideration will be given to a higher order method where an approximation of the inverse covariance matrix is used to account for the variance in initial parameter estimates. The goal of this analysis is to find a relationship that will allow for the simplification of the inverse solutions for this special case.