Solving large linear least squares problems with linear equality constraints

Abstract

We consider the problem of solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. While some classical approaches are theoretically well founded, they can face difficulties when the matrix of constraints contains dense rows or if an algorithmic transformation used in the solution process results in a modified problem that is much denser than the original one. We propose modifications with an emphasis on requiring that the constraints be satisfied with a small residual. We examine combining the null-space method with our recently developed algorithm for computing a null-space basis matrix for a “wide” matrix. We further show that a direct elimination approach enhanced by careful pivoting can be effective in transforming the problem to an unconstrained sparse-dense least squares problem that can be solved with existing direct or iterative methods. We also present a number of solution variants that employ an augmented system formulation, which can be attractive for solving a sequence of related problems. Numerical experiments on problems coming from practical applications are used throughout to demonstrate the effectiveness of the different approaches.

Penalty‐free discontinuous Galerkin method

“…The constraint problem leads to a linear system of equations that is solved prior to the main discrete problem. The matrix in the constraints problem is rectangular and singular, which admits a nonunique solution, and the null-space method 42,43 is applied to obtain the complete solution. It should be noted here that in Oden at al., 44 a penalty-free DG method is proposed, but this method has not been widely adopted due to its low level of accuracy when compared to penalty-based approaches.…”