Title:     A Trust-Region Algorithm for Ill-posed Nonlinear Problems
           Applied to Diffuse Optical Tomography

Speaker:   Eric de Sturler, Dept. of Mathematics, Virginia Tech

Abstract:

In Newton-type methods for ill-posed nonlinear problems we need to deal
effectively with (very) ill-conditioned Jacobians. Ill-conditioned
systems may lead to large steps that violate the local model
on which Newton-type methods are based, and therefore create convergence
problems. However, it is not just the step-size that is problematic and
needs to be controlled. The sensitivity of the linear system may also lead
to poor search directions. Thus, the usual algorithmic extensions
that have been developed to deal robustly with nonlinear problems,
such as line search (damping) and trust-region methods, lead to
algorithms that require many function evaluations.
For many practical problems, however, a function evaluation is very expensive.
In parameter estimation methods, where the parameters define a
partial differential or integral equation on a domain, a function
evaluation typically amounts to the solution of a large nonlinear or linear
system of equations (a.k.a. solving the 'forward problem' in an inverse problem
setting).  It is therefore crucial to find better search directions with
which to update the approximate solution to the nonlinear problem.
Typical regularization schemes for linear ill-posed problems, such as truncated SVD, 
are not directly applicable to solving for the search direction at each step. 
However, with modification such schemes can be made to fit within the framework of the
robust nonlinear solver.

In this presentation, we will propose a trust-region method that uses a
modified, regularized Gauss-Newton solution to update the approximate
solution of the nonlinear problem. After introducing the
main difficulties and the algorithmic solutions for nonlinear and ill-posed
problems, we will describe and analyze our scheme, and demonstrate it for
a nonlinear inverse problem arising in Diffuse Optical Tomography. We anticipate 
this method should also be valuable for more general nonlinear
problems where very ill-conditioned Jacobian arise.
This is joint work with Misha Kilmer (Tufts).