Numerical Algorithms

, Volume 81, Issue 3, pp 773–811 | Cite as

IR Tools: a MATLAB package of iterative regularization methods and large-scale test problems

  • Silvia GazzolaEmail author
  • Per Christian Hansen
  • James G. Nagy
Original Paper


This paper describes a new MATLAB software package of iterative regularization methods and test problems for large-scale linear inverse problems. The software package, called IR TOOLS, serves two related purposes: we provide implementations of a range of iterative solvers, including several recently proposed methods that are not available elsewhere, and we provide a set of large-scale test problems in the form of discretizations of 2D linear inverse problems. The solvers include iterative regularization methods where the regularization is due to the semi-convergence of the iterations, Tikhonov-type formulations where the regularization is explicitly formulated in the form of a regularization term, and methods that can impose bound constraints on the computed solutions. All the iterative methods are implemented in a very flexible fashion that allows the problem’s coefficient matrix to be available as a (sparse) matrix, a function handle, or an object. The most basic call to all of the various iterative methods requires only this matrix and the right hand side vector; if the method uses any special stopping criteria, regularization parameters, etc., then default values are set automatically by the code. Moreover, through the use of an optional input structure, the user can also have full control of any of the algorithm parameters. The test problems represent realistic large-scale problems found in image reconstruction and several other applications. Numerical examples illustrate the various algorithms and test problems available in this package.


Iterative regularization methods Semi-convergence Linear inverse problems Test problems MATLAB 

Mathematics Subject Classification (2010)

65F10 65F22 


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The authors are grateful to Julianne Chung for providing an implementation of HyBR, which forms the basis of our IRhybrid_lsqr function. For further details, see [11, 13] and We also thank Germana Landi for providing insight about the NMR relaxometry problem.

The satellite image in our package, shown in Fig. 1, is a test problem that originated from the US Air Force Phillips Laboratory, Lasers and Imaging Directorate, Kirtland Air Force Base, New Mexico. The image is from a computer simulation of a field experiment showing a satellite as taken from a ground based telescope. This data has been used widely in the literature for testing algorithms for ill-posed image restoration problems; see, for example [35].

Our package also includes a picture of NASA’s Hubble Space Telescope as shown in Fig. 6. The picture is in the public domain and can be obtained from

Funding information

This work received funding from Advanced Grant No. 291405 from the European Research Council and US National Science Foundation under grant no. DMS-1522760.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKgs. LyngbyDenmark
  3. 3.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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