Compact Linear Operators and Krylov Subspace Methods

This thesis deals with linear ill-posed problems related to compact operators, and iterative Krylov subspace methods for solving discretized versions of these. Linear compact operators in infinite dimensional Hilbert spaces will be investigated and several results on the singular values and eigenvalues for such will be presented. A large subset of linear compact operators consists of integral operators and many results will be based on the kernel of such operators. Finite dimensional approximations to these operators will be considered by using Galerkin discretization. Several results will be shown stating how singular values and eigenvalues (and corresponding vectors) of infinite dimensional operators and their finite dimensional approximations are related. Krylov subspace methods, with focus on GMRES, will be investigated in relation to discrete ill-posed problems, that is, linear finite dimensional systems of equations that originate from ill-posed problems. By using the spectral decomposition of the coefficient matrix, results on the convergence of GMRES are derived.