Multi-Level Approach to Numerical Solution of Inverse Problems

Mathematical modeling of an engineering system often leads to such formulations for which one can not obtain a closed form solution/analysis, and thus numerical methods are to be used. In the process, we need to transform the system from an infinite dimensional space to a finite dimensional one(discretization). The result is usually a system of linear equations[5] for which the linear least squares method[1,6] is used to compute the solution. The difficulty is that the real engineering problems are ill-posed and consequently their discretization lead to an ill-conditioned system of equations, and thus the method of least squares produce irrelevant solutions. As a result, application of some types of regularization techniques would be necessary for the solution of ill-conditioned systems. Tikhonov regularization method[3,8] is one of the most popular regularization techniques. In this scheme the original ill-conditioned system of,