IDR(s) revisited
Marijn Schreuders

Supervisor: Martin van Gijzen

Site of the project: TU Delft

start of the project: November 2013

In February 2014 the Interim Thesis and a presentation has been given.

Summary of the master project:
Suppose we want to solve a linear system of equations. In 1980 the IDR method was first proposed by Peter Sonneveld to solve such a system. IDR stands for Induced Dimension Reduction. The underlying idea is that the residuals are forced to be in subspaces of reducing dimension. IDR has been long been overshadowed by Bi-CG-type methods and GMRES-type methods, although the underlying mathematical ideas are totally different. In recent years there has been renewed interest in IDR and in 2007 Peter Sonneveld and Martin van Gijzen proposed an extension to IDR called IDR(s). From practical experience we see that the IDR(s) method can perform very well compared in certain situations (see the Figure below).

It turns out that IDR(s) can be seen in the framework of projection methods. A projection method is an iterative method that searches for an approximate solution in a Krylov subspace such that it is orthogonal to another Krylov subspace. One goal of this research is to find out how exactly we can see IDR(s) in this framework. Here we follow Valeria Simoncini and Daniel B. Szyld in their paper 'Interpreting IDR as a Petrov-Galerkin method'. In this same paper Simoncini and Szyld propose a new version of IDR, called Ritz-IDR. For the omega's, one of the parameters in the IDR(s) algorithm, they use the reciprocals of some of the approximations of the eigenvalues (called Ritz values) of the matrix. They generate these Ritz values by using a small number of iterations of a Krylov subspace method for eigenvalue problems, such as the Arnoldi method. However, IDR(s) can also be used to generate Ritz values. Hence, our second goal will be to use IDR(s) itself to generate the Ritz values.

Convergence of various Krylov subspace methods.

## Contact information: Kees Vuik

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