**Prof. Curt Vogel**

Montana State University

**"Large-Scale Optimization Algorithms for Inverse
Problems in Atmospheric Imaging"**

www.math.montana.edu/~vogel

**Abstract:**

The physical problem of interest is too estimate both the *phase* (or *wave-front aberration*) and the *object *(or true image)
from atmospheric image data modeled by

noise, (1)
where denotes 2-D convolution product and the *point spread function* . The obvious
nonuniqueness issues arising in (1) can be dealt with by taking new image datta generated by a technique known as *phase
diversity*.

The phase/object estimation problem presents some interesting mathematical and computational challenges. The problem is
ill-posed, so regularization must be incorporated to obtain stable, accurate parameter estimates. We employ a penalty approach
known as *Tikhonov regularization*, which requires the minimization of a function of the form

(2)
where is a nonquadratic fit-to-data function, L is a symmetric positive definite matrix, and and are small
parameters. After discretization, the number of unknows is quite large

We apply a pair of algorithms, (i) the *limited memory BFGS method* (l-BFGS) with line search globalization; and (ii) the
Newton/CG algorithm with trust region globalization due to Steihaug. We will present numerical results and point out some
interesting parallels between preconditioners for Newton/CG and the initial Hessian for l-BFGS.

#### Friday, April 6, 2001

3:00 PM in 335 Jabara Hall

Please join us for refreshments before the lecture at 2:30p.m. in room 353 Jabara Hall.

[ Spring 2001]