Efficient Isolation of the nonlinearity of reconstruction problems

A method for the solution of nonlinear differential equations, describing, e. g., large signals and inverse problems, using nonlinear renormalization techniques is presented. The method allows the solution of nonlinear problems in two steps: An auxiliary linear differential equation describing a virtual variable is first introduced. The virtual variable is related to the actual (looked for) variable via an algebraic nonlinear renormalization. Second the linear auxiliary differential equation obtained is solved by an appropriate linear algorithm. The availability of this approach is shown on the example of reconstructing a one-dimensional permittivity profile from passband measurements. Although some approximations have to be made to solve the nonlinear problem, very good results could be achieved by proper choice of the renormalization.