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# Accession Number:

## ADA551898

# Title:

## Algorithms for the Equilibration of Matrices and Their Application to Limited-Memory Quasi-Newton Methods

# Descriptive Note:

## Doctoral thesis

# Corporate Author:

## STANFORD UNIV CA

# Report Date:

## 2010-05-01

# Pagination or Media Count:

##
111.0

# Abstract:

## Diagonally scaling a matrix often reduces its condition number. Equilibration scales a matrix so that the row and column norms are equal. We review the existence and uniqueness theory for exact equilibration. Then we introduce a formalization of approximate equilibration and develop its existence and uniqueness theory. Next we develop approximate equilibration algorithms that access a matrix only by matrix-vector products. We address both the nonsymmetric and symmetric cases. Limited-memory quasi-Newton methods may be thought of as changing the metric so that the steepest-descent method works effectively on the problem. Quasi-Newton methods construct a matrix using vectors of two types involving the iterates and gradients. The vectors are related by an approximate matrix-vector product. Using our approximate matrix-free symmetric equilibration method, we develop a limited-memory quasi-Newton method in which one part of the quasi-Newton matrix approximately equilibrates the Hessian. Often a differential equation is solved by discretizing it on a sequence of increasingly fine meshes. This technique can be used when solving differential-equation-constrained optimization problems. We describe a method to interpolate our limited-memory quasi-Newton matrix from a coarse to a fine mesh.

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#