Solution of linear discrete ill-posed problems by discretized Chebyshev expansion

Abstract

Large-scale linear discrete ill-posed problems are generally solved by Krylov subspace iterative methods. However, these methods can be difficult to implement so that they execute efficiently in a multiprocessor environment, because some of the computations have to be carried out sequentially. This is due to the fact that only one new basis vector of the Krylov solution subspace is generated in each iteration. It is therefore interesting to investigate the performance of other solution methods that use a solution subspace basis that can be generated in parallel and, therefore, more efficiently on many computers. This paper proposes solution methods that use a solution subspace basis that is made up of discretized Chebyshev polynomials. It compares their performance to a Krylov subspace method that is based on partial Golub–Kahan bidiagonalization of the system matrix, and to a randomized method. The application of a solution subspace basis made up of discretized Chebyshev polynomial is found to be competitive when solving linear discrete ill-posed problems in one space-dimension and for some problems in higher space-dimensions.