IDR(s) is a robust and efficient short recurrence 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 and FORTRAN implementations for IDR(s), and examples of how to use the codes.
idrs.m
Version of August 2010. This is the
biortho variant of IDR(s) (with enhancements) that is described in
[4] (see below).
The most important changes with respect to
version of December 2008 are:
Preconditioner can be passed in decomposed form;
Matrixvector multiplication and preconditioning operations can be defined by functions;
Residual smoothing (optional);
Residual replacements to achieve accuracy close to machine precision (optional).
test_idrs.tgz. A testset of 11 examples, also includes idrs.m. This manual describes idrs.m and the accompanying testset.
example_idrs.m (needs idrs.m).
This MATLAB script defines a 3D discretised convectiondiffusionreaction problem on the unit cube. The parameters can be changed via a user interface to create a different test problem. The problem is solved with IDR(1), IDR(2), IDR(4), IDR(8), and with the builtin MATLAB routines for (full) GMRES and BiCGSTAB. The picture below shows the convergence of the methods for the default parameters, which specify a highly nonsymmetric and indefinite problem consisting of about 60,000 equations.
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. 10351062, 2008 (copyright SIAM)
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 0707
The relation of IDR(s) with BiCGSTAB, and how to derive generalisations of BiCGSTAB using IDRideas can be found in: Gerard L.G. Sleijpen, Peter Sonneveld and Martin B. van Gijzen, BiCGSTAB as an induced dimension reduction method, Applied Numerical Mathematics. Vol 60, pp. 11001114, 2010 (copyright Elsevier)
A very stable and efficient IDR(s) variant (implemented in the MATLAB code idrs.m given above) is described in: Martin B. van Gijzen and Peter Sonneveld, Algorithm 913: An Elegant IDR(s) Variant that Efficiently Exploits Biorthogonality Properties. ACM Transactions on Mathematical Software, Vol. 38, No. 1, pp. 5:15:19, 2011 (copyright ACM)
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. SIAM J. Sci. Comput. Vol. 32, No. 5, pp. 26872709, 2010 (copyright SIAM)
A version of IDR(s) that is tuned for parallel and grid computing is described in: T.P. Collignon and M.B. van Gijzen, Minimizing synchronization in IDR(s). Numerical Linear Algebra with Applications, Vol. 18, No. 5, pp. 805–825, 2011 (Copyright John Wiley & Sons, Ltd.)
New! Flexible and multishift IDR variants are described in: Martin B. van Gijzen, Gerard L.G. Sleijpen and JensPeter M. Zemke, Flexible and MultiShift Induced Dimension Reduction Algorithms for solving Large Sparse Linear Systems. Delft University of Technology, Reports of the Department of Applied Mathematical Analysis, Report 1106, 2011.
November 2011: IDR(s) has been included in the
Collected Algorithms of the ACM as Algorithm 913.
July
8, 2010: Invited talk about IDR(s) at
the ICCAM 2010 conference in
Leuven, Belgium.
January 2010: IDR(s) (the
biortho variant described in [4]) has been included in IFISS
3.0, an open source Incompressible Flow & Iterative Solver
Software by Howard Elman, David Silvester and Alison Ramage.
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.
Photo of the participants (from left to right): Kuniyoshi Abe, Martin Gutknecht, JensPeter Zemke, me, Seiji Fujino, Peter Sonneveld, ManChung Yeung, Gerard Sleijpen.
June 3, 2009: DCSE symposium
"IDR
and block Lanczos solvers for large nonsymmetric systems",
with as speakers Martin Gutknecht, me, ManChung Yeung, Seiji Fujino,
Peter Sonneveld, and JensPeter Zemke.
March 17, 2008:
Mini symposium at the 9th
IMACS conference. Participants: Peter Sonneveld, me, Gerard
Sleijpen, Seiji Fujino, and Y. Onoue.
Extract from
the "IMACS NEWS March 2008" (complete
pdf version):
Although this 9th edition was a little
bit rainy, it has been more than enlightened by the
contributions of more than 90 attendants representing more than 20
countries and a.o. by a remarqued come back of the IDR method of our
friends from The Netherlands, which might bring a true breakthrough
in the ﬁeld of Krylov subspace methods and of their
theoretical support.
March 12, 2007: First
presentation about IDR(s) (TU Delft, numerical analysis group
seminar).


The example illustrates how to use
subdomain deflation in combination with idrs.m.
It exploits
the flexibility of the code, subdomain deflation (J. Frank and C.
Vuik, "On the construction of deflationbased preconditioners",
SIAM J. Sci. Comp. 23 (2001) pp. 442462) is implemented without the
need to modify idrs.m.
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 StreamFunction Problems on Multiply Connected
Domains,
J. Comp. Phys., 140, 1998, pp. 3046. (copyright
Academic Press).
You are free to use this example for academic purposes. Please
put a reference to the above mentioned paper if you use the ocean
test problem in a publication.
The file
idrs_ocean_example.tgz contains
all the matlab and data files needed to run the test problem. You can
uncompress and unzip this file by entering the command tar xzf
idrs_ocean_example.tgz. This creates a directory IDRS_OCEAN_EXAMPLE.
In this directory you will find datafile in MatrixMarket format,
and the matlab code to run the test problem.
The data files
have the following names and meanings:
stommel?.mtx: the system matrices. "?" can be either 1, 2, 3, or 4 and refers to the grid spacing in degrees.
stommel?_b.mtx: the righthandside vectors. Each file contains 12 vectors, for each month one.
stommel?_P.mtx: map from system degreesoffreedom to earth degreesoffreedom. Needed to make a picture of the solution.
bathymetry.mtx: depth information. Needed to make a picture of the solution.
The following matlabfiles are provided:
ocean.m: running this program opens a gui that allows you to choose the grid and set the solver parameters. It solves the systems for the 12 righthandsides and gives the resulting ocean circulation patterns in the form of a movie.
idrs.m.
mv.m: file to perform deflated matrixvector products.
subdomaindeflation.m: setup subdomain deflation vectors.
mmread.m: MatrixMarket routine to read the data files.

idrs_f90.tar: this file contains a simple F90/F95 implementation of IDR(s). It has been written and made available by Arash Ghasemi (National Center for Computational Engineering, University of Tennessee).