Solving Nonlinear Ill-Posed Problems Using CGLS and GCV

Haber

Institute of Applied Mathematics (UBC) and MIR. Faculty of Medicine, Calgary, Alberta, Canada


Abstract

This work discusses the inversion of nonlinear ill-posed problems. Let x be the model, b the data, and e is the noise which is assumed to be Gaussian. We assume that the connection between the model and the data is:

F[x]+e=b

where F is a nonlinear operator. Our goal is to find the model, x, from the noisy data, b. Such problems are usually solved using Tichonov regularization and Gauss-Newton iteration, i.e. the model, x, minimize the function:

f(x,s) = ||F[x]-b||^2 + s||x||^2

The major computational problem arises because the regularization parameter, s, and the noise which contaminates the data are not known a priori.

Recently it has been shown that for linear inverse problems which originated from integral equations of the first kind, Krylov subspace methods give rise to regularized solutions similar to the Tichonov solutions. Our work extend this methodology to the nonlinear inverse problem. We show that by minimizing:

g(x)= ||F[x]-b||^2

Subject to: x belongs to K(J,b,n)

where K is an n dimensional Krylov space, spanned by J which is the Jacobian matrix. We obtain regularized solutions for the nonlinear inverse problem. We show that the space size, n, can be computed using the GCV principle. This formulation which is a subspace formulation in nature, allows to work with large scale problems.