On numerical solution of arbitrary symmetric linear systems by approximate orthogonalization

A very important class of inverse problems are those modelled by integral equations of the first kind. These equations are usually ill-conditioned, such that any discretization technique will produce an ill-conditioned system, in classical or least-squares formulation. For such kind of symmetric problems, we propose in this paper a stable iterative solver based on an approximate orthogonalization algorithm introduced by Z. Kovarik. We prove convergence of our algorithm for general symmetric least-squares problems and present some numerical experiments ilustrating its good behaviour on problems concerned with the determination of charge distribution generating a given electric field and gravity surveying, both modelled by first kind integral equations.