The Induced Dimension Reduction method

IDR(s) is a new, highly efficient Krylov subspace method for solving large nonsymmetric systems of linear equations. On this page you can find reports and papers that describe IDR(s), MATLAB code for IDR(s), and examples of how to use the code.

Reports and papers:

  1. IDR(s) is described in: Peter Sonneveld and Martin B. van Gijzen, IDR(s): a family of simple and fast algorithms for solving large nonsymmetric linear systems. SIAM J. Sci. Comput. Vol. 31, No. 2, pp. 1035-1062 (2008). (copyright SIAM)
  2. The original IDR(s) report is: Peter Sonneveld and Martin B. van Gijzen, IDR(s): a family of simple and fast algorithms for solving large nonsymmetric linear systems. Delft University of Technology, Reports of the Department of Applied Mathematical Analysis, Report 07-07
  3. The relation of IDR(s) with Bi-CGSTAB, and how to derive generalisations of Bi-CGSTAB using IDR-ideas can be found in: Gerard L.G. Sleijpen, Peter Sonneveld and Martin B. van Gijzen, Bi-CGSTAB as an induced dimension reduction method, Delft University of Technology, Reports of the Department of Applied Mathematical Analysis, Report 08-07. Accepted for publication in revised form in APNUM.
  4. A slightly more effcient and for large s more accurate variant of IDR(s) is described in: Martin B. van Gijzen and Peter Sonneveld, An elegant IDR(s) variant that efficiently exploits bi-orthogonality properties. Delft University of Technology, Reports of the Department of Applied Mathematical Analysis, Report 08-21
  5. The combination of IDR(s) with BiCGstab(ℓ) is described in: Gerard L.G. Sleijpen and Martin B. van Gijzen, Exploiting BiCGstab(ℓ) strategies to induce dimension reduction, Delft University of Technology, Reports of the Department of Applied Mathematical Analysis, Report 09-02
  6. NEW! A version of IDR(s) that is tuned for parallel and grid computing is described in: T.P. Collignon and M.B. van Gijzen, Fast solution of nonsymmetric linear systems on Grid computers using parallelvariants of IDR(s). Delft University of Technology, Reports of the Department of Applied Mathematical Analysis, Report 10-05

News and events:

October 27, 2009: Mini symposium "Induced Dimension Reduction (IDR) Methods: a Family of Efficient Krylov Solvers" which was part of the SIAM conference on applied linear algebra LA09.

LA09
Photo of the participants (from left to right): Kuniyoshi Abe, Martin Gutknecht, Jens-Peter Zemke, me, Seiji Fujino, Peter Sonneveld, Man-Chung Yeung, Gerard Sleijpen.

January 2010: IDR(s) (the bi-ortho variant described in Report [4]) has been included in IFISS 3.0, an open source Incompressible Flow & Iterative Solver Software by Howard Elman, David Silvester and Alison Ramage.

MATLAB code:

convergence IDR(s)

Ocean circulation test problem:

The data files below correspond to the ocean circulation example that is described in [1]. The numbers in the file names give the grid spacing (e.g. Stommel4 is the ocean problem on a 4 degree grid).
A detailed description of the ocean circulation example can be found in:
M. B. van Gijzen, C. B. Vreugdenhil, and H. Oksuzoglu, The Finite Element Discretization for Stream-Function Problems on Multiply Connected Domains, J. Comp. Phys., 140, 1998, pp. 30-46. (copyright Academic Press)
Please put a reference to this paper if you use this test problem.

Data files (Matrix-Market format, .tar.gz): MATLAB script to test iterative solvers:
Currently, it_solve calls IDR(4) and the MATLAB built-in iterative solvers, with diagonal scaling as preconditioner. It is straightforeward to modify the file and to add your own method/preconditioner. The script needs idrs.m en mmread.m.