The Induced Dimension Reduction method

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 code for IDR(s), and examples of how to use the code.

MATLAB code:

idrs.m Version of August 2010. This is the bi-ortho 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;
- Matrix-vector 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 convection-diffusion-reaction 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 built-in MATLAB routines for (full) GMRES and Bi-CGSTAB. The picture below shows the convergence of the methods for the default parameters, which specify a highly non-symmetric and indefinite problem consisting of about 60,000 equations.

Reports and papers:

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)
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
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, Applied Numerical Mathematics. Vol 60, pp. 1100-1114, 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 Bi-orthogonality Properties. ACM Transactions on Mathematical Software, Vol. 38, No. 1, pp. 5:1-5:19, 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. 2687-2709, 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 multi-shift IDR variants are described in: Martin B. van Gijzen, Gerard L.G. Sleijpen and Jens-Peter M. Zemke, Flexible and Multi-Shift Induced Dimension Reduction Algorithms for solving Large Sparse Linear Systems. Delft University of Technology, Reports of the Department of Applied Mathematical Analysis, Report 11-06, 2011.

News and events:

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, Jens-Peter Zemke, me, Seiji Fujino, Peter Sonneveld, Man-Chung Yeung, Gerard Sleijpen.

June 3, 2009: DCSE symposium "IDR and block Lanczos solvers for large nonsymmetric systems", with as speakers Martin Gutknecht, me, Man-Chung Yeung, Seiji Fujino, Peter Sonneveld, and Jens-Peter 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).

Peter Sonneveld and me in my office, finalising the talk

The first slide of the presentation

An ocean circulation problem:

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 deflation-based preconditioners", SIAM J. Sci. Comp. 23 (2001) pp. 442-462) 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 Stream-Function Problems on Multiply Connected Domains,
J. Comp. Phys., 140, 1998, pp. 30-46. (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 data-file in Matrix-Market 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 right-hand-side vectors. Each file contains 12 vectors, for each month one.
stommel?_P.mtx: map from system degrees-of-freedom to earth degrees-of-freedom. Needed to make a picture of the solution.
bathymetry.mtx: depth information. Needed to make a picture of the solution.

The following matlab-files 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 right-hand-sides and gives the resulting ocean circulation patterns in the form of a movie.
idrs.m.
mv.m: file to perform deflated matrix-vector products.
subdomain-deflation.m: set-up subdomain deflation vectors.
mmread.m: Matrix-Market routine to read the data files.