Preconditioning for Regularized Inverse Problems

gwade@math.bgsu.edu

We consider a quasi-Newton method for minimization problems arising in nonlinear, regularized distributed parameter inverse problems. The regularization methods to be considered are Tikhonov and Total Variation; the latter may be viewed as an adaptive Tikhonov regularization. The minimization method to be considered is a quasi-Newton method which on each iteration results in a large, non-sparse linear system in "du", of the form 

   [ H(u) + A(u) ] du = -g(u).
Here, u is approximate solution of the minimization problem at the current iteration, H(u) is a approximate Hessian, A(u) is a self-adjoint elliptic-type operator, and g(u) is the gradient of the objective functional.

The method to to be discussed combines the inverse of A(u) (or a fast approximation thereof) with a low-rank approximation of H(u) via the Sherman-Morrison formula, and a Broyden-type update of the approximation H(u). Numerical examples will be presented.