TheMomEnt+Project A Computational Exercise in Partially Finite Optimization

Abstract

The MomEnt+ project is a computational engine intended to implement and expand the applicability of much of the maximum entropy theory that has been developed in the past few years. The project is not a single package or piece of code, but rather a diverse combination of computer programs, theory, and contacts, designed to enhance the participants' collective knowledge of numerical optimization, real world issues, and optimization theory, as well as to offer our mathematical and computational skills to a variety of nonmathematical researchers.

The immediate goal is to design and implement a collection of computer algorithms and codes, and to make contacts in image processing, geophysics, electrical engineering, statistics and wherever else we might find interest in our activities.

We have developed a TOOL that allows one to reconstruct any given number of moments of a user supplied input function. There are a number of types of entropy to chose from. Uniform or Gaussian noise can be added to the recontruction problem and Newton, Conjugate Gradient or Barzilai-Borwein numerical methods can be selected for the computations.

Persons and research groups who believe that the work of the MomEnt+ project may be of interest to them are urged to contact Ron Haynes of the CECM.

Established by Mark A. Limber, the MomEnt+ project is the computational engine behind the convex entropy optimization research at the CECM. The name "MomEnt+" is derived from "Moment problems solved via Entropy maximization with Positivity". However, we are not exclusively working on moment problems, except in a general sense, nor are we restricting ourselves to maximum entropy methods.

The underlying problem we study is Ax = b where A : X -> R^n is a continuous linear operator and X is some function space. Since this is generally an underdetermined problem, we can pick a solution via an optimization criterion, for example, the solution to

inf { f(x) : Ax = b, x >= 0 }

where f:X -> R is a suitable convex functional. We have implemented the theory of convex duality as developed in [2] in both a Maple and C environment [3] to solve such problems. We are also concentrating on other computation methods, including projection [1] and multigrid [4] methods.

We have made contacts with the medical imaging groups at Vancouver General Hospital and TRIUMF, the government research centre located in Vancouver. These and other contacts will keep this project application oriented and force us to address real world issues that arise in applications, in particular, extremely noisy data.

Other interesting entropy/optimization related links are:

• Entropy on the World Wide Web, A comrehensive site maintained by the Department of Mathematics at the University of Washington
• Claude Shannon A short biography of Claud Shannon (no links here but interesting none the less)
• Entropy: Information Theory and Statistical Mechanics Confused by the word "Entropy"? Look at this ASKII document.
• Moment+ is an out-growth of much of the optimization work that has been done at and around the CECM. We would like to recognize the broader optimization community that has provided much of the impetus, both directly and indirectly through their continuing research.
• [1] H. H. Bauschke and J.M. Borwein, "On projection algorithms for solving convex feasibility problem," SIAM Review, submitted.
• [2] J.M. Borwein and A. S. Lewis, "Duality relationships for entropy-like minimization problems," SIAM J. Control and Optim., 29(1991), 325-338.
• [3] J.M. Borwein, R. K. Goodrich, and M. A. Limber, "A comparison of entropies in the underdetermined moment problem," Numerical Functional Analysis and Optimization, submitted.
• [4] M. A. Limber, T. A. Manteuffel, S. F. McCormick and D. S. Sholl, "Optimal Resolution in Maximum Entropy Image Reconstruction from Projections with Multigrid Acceleration," Proceedings of the Sixth Annual Copper Mountain Conference on Multigrid Methods, Eds. S. F. McCormick and T. A. Manteuffel, 1993

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Last updated Wednesday, August 8, 1996, 5:06PM