Piecewise Differentiable Minimization for Ill-posed Inverse Problems Ing from the National Science Foundation and Ibm Corporation, with Additional Support from New York State and Members of Its Corporate Research Institute. 1

Based on minimizing a piecewise diierentiable lp function subject to a single inequality constraint, this paper discusses algorithms for a discretized regularization problem for ill-posed inverse problems. We examine computational challenges of solving this regularization problem. Possible minimization algorithms such as the steepest descent method, iteratively weighted least squares (IRLS) method and a recent globally convergent aane scaling Newton approach are considered. Limitations and eeciency of these algorithms are demonstrated using the geophysical traveltime tomographic inversion and image restoration applications. 1. Minimization and Ill-posed Inverse Problems. Minimization algorithms have long been used in regulating an ill-posed inverse problem. Assuming that a desired property of a solution is known a priori, an ill-posed inverse problem can be regulated by solving a constrained minimization problem. In particular, properties expressed in nondiierentiable form have increasingly been found more appropriate in many applications. Discretization of such a regularization problem often leads to minimizing a large-scale piecewise diierentiable function with a single constraint. In this paper, we consider regularization using piecewise diierentiable minimization , possibly with a single inequality constraint. Consider an ill-posed inverse problem, where A is an operator in a Hilbert space. Assume that k k 2 denotes the Euclidean norm and an a priori condition (e.g., continuity and bounded-ness) of the desired solution is given by kBuk 2 for some linear operator This paper is written for the proceedings of the workshop Large-Scale Optimiza