Numerical Methods for III-Posed Problems and Large Scale Eigenvalue Programs

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This work concerns two topics in numerical linear algebra: 1) Regularization of ill-posed linear systems; 2) Solution of large, sparse eigenvalue problems. These topics share common features: a wide range of application problems; the use of iterative methods for large-scale problems; and interesting problems in matrix perturbation theory. When continuous ill-posed problems are discretized they result in ill-conditioned linear systems which must be regularized to yield accurate solutions. This part of the work has several goals. The first is to prove the folk theorem that if the components of the data vector with respect to the singular vectors of the matrix decay sufficiently fast then the conjugate gradient iteration will produce a regularizing set of solution vectors. The second is to compare the numerous formulations of discrete ill-posed problems to see which are most effective. The third is to further develop preconditioners to speed convergence of solution algorithms. The fourth is to improve data-gathering techniques so that the attainable accuracy from imaging is better, thus, for example, revealing smaller tumors or more information about distant stars. Over the past decade many new algorithms for finding clusters of eigenvalues of large matrices have been proposed. Although the effectiveness of some of these algorithms has been demonstrated empirically, analytic results are sparse. Fortunately, a large number of these methods share a common framework, so that it is possible to develop analytic tools that are widely applicable. As a start toward this goal, attention is focused on a new, promising method---singular vector enhancement---that fits in the framework. A preliminary analysis of a special case has already yielded valuable results on the relation of eigenvalues and singular values. The results of this project will impact the solution of ill-posed problems such as medical image enhancement, astronomical data processing, nondestructive testing, and spectroscopy, as well as eigenvalue problems arising in systems modeling (Markov chains), computational chemistry, and structural analysis.