Splitting Dense Rows In Sparse Least Squares Problems

Splitting Dense Rows in Sparse Least Squares Problems

Numerical Methods for Least Squares Problems

The method of least squares was discovered by Gauss in 1795. It has since become the principal tool to reduce the influence of errors when fitting models to given observations. Today, applications of least squares arise in a great number of scientific areas, such as statistics, geodetics, signal processing, and control. In the last 20 years there has been a great increase in the capacity for automatic data capturing and computing. Least squares problems of large size are now routinely solved. Tremendous progress has been made in numerical methods for least squares problems, in particular for generalized and modified least squares problems and direct and iterative methods for sparse problems. Until now there has not been a monograph that covers the full spectrum of relevant problems and methods in least squares. This volume gives an in-depth treatment of topics such as methods for sparse least squares problems, iterative methods, modified least squares, weighted problems, and constrained and regularized problems. The more than 800 references provide a comprehensive survey of the available literature on the subject.

Incomplete Factorization Preconditioning for Linear Least Squares Problems

When certain conditions are imposed on the sparsity pattern of the preconditioning matrix, incomplete Cholesky (IC) factorization is identical to IMGS. Therefore, a sufficient condition on the sparsity pattern for a stable IC factorization can be derived. Based on this condition, three algorithms for modifying a sparsity pattern for which IC may not succeed have been designed. These methods differ from previously proposed methods to guarantee the existence and improve the stability of the IC factorization since they do not require numerical information. Numerical experiments illustrating the capabilities of the preconditioners are also presented."