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IR Tools: A MATLAB Package of Iterative Regularization Methods and Large-Scale Test Problems
Silvia Gazzola, Per Christian Hansen, James G. Nagy
TL;DR
Large-scale ill-posed linear inverse problems need stable methods and realistic benchmarks, but available MATLAB packages are limited in scope or scale. This paper presents IR Tools, a flexible MATLAB package of iterative regularization solvers and realistic 2D test problems. The package supports multiple regularization paradigms, matrix representations, constraints, configurable defaults, and numerical demonstrations across its methods and problems.
Problem
Large-scale ill-posed inverse problems require regularization, while existing MATLAB packages provide limited coverage of large-scale solvers or realistic current applications.
Method
IR Tools combines iterative solvers spanning semi-convergence, penalization, hybrid projected regularization, and constraints with realistic 2D test problems in a uniform flexible framework.
Results
Numerical examples illustrate the algorithms and test problems, including semi-convergence behavior, discrepancy-principle stopping, and sparse reconstructions.
Takeaways & Limitations
IR Tools provides a common platform for experimenting with iterative regularization methods and realistic large-scale 2D inverse-problem benchmarks.
Takeaways & Limitations
Poisson noise is not included because its data-dependent mean and variance do not conform to the package's noise-scaling procedure.
Abstract
from arXiv · showhide
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.
1 Introduction
IR Tools targets large-scale ill-posed linear inverse problems with flexible iterative regularization methods and realistic 2D test problems. It combines broad solver coverage with a uniform MATLAB framework that supports multiple matrix representations and configurable defaults.
- Large-scale inverse problems are sensitive to errors because singular values decay and cluster near zero, making regularization necessary for stable solutions.
- Regularization is implemented through semi-convergent iterations, explicit penalized formulations such as Tikhonov regularization, and constraint-capable methods.
- Its solvers support sparse matrices, function handles, and objects, accommodating storage-efficient forward and adjoint computations for large problems.
- IR Tools provides implementations of diverse iterative algorithms, including recently proposed methods unavailable elsewhere.
- The package supplies new large-scale test problems for experimentation and benchmarking, including realistic 2D problems usable without application-specific background.
- Compared with other MATLAB packages, IR Tools is designed to cover a broader range of iterative solvers and large-scale applications than packages focused on smaller or narrower problem classes.
2 Overview of the Iterative Solvers
IR Tools organizes iterative regularization algorithms in a common framework spanning semi-convergence, penalization, hybrid projected methods, constraints, and flexible Krylov approaches. Solvers accept common inputs while defaults and options control algorithm behavior.
- The common solver interface uses A, b, optional K and options inputs, and returns approximate solutions together with iteration information in Info.
- A may be a discrete forward operator represented as a matrix, function handle, or object, while Info records residuals and stopping information.
- Default regularization parameters and stopping criteria enable simple calls, whereas options lets users override solver settings and add problem information.
- Methods and constraints: The package includes Krylov subspace methods, enriched subspaces, flexible and restarted approaches, first-order methods, and algorithms supporting box, energy, or nonnegativity constraints.
- Methods and regularization: Semi-convergent methods regularize by early termination, while penalized methods solve full Tikhonov, 1-norm, or total-variation formulations.
3 Overview of the Test Problems
IR Tools provides realistic 2D large-scale inverse-problem test problems designed for benchmarking iterative algorithms without requiring application-specific expertise. The problems share convenient interfaces, support multiple representations of the forward operator, and expose adjustable parameters, visualization, and noise generation.
- Motivation: The package addresses the scarcity of realistic collections by providing new 2D test problems for testing, debugging, demonstrating, and benchmarking large-scale inverse-problem algorithms.The authors contrast these problems with outdated 1D collections in existing software.
- Design objectives: The test problems are intended as black boxes, so users can work with them without specific knowledge of the originating application domains.This design supports algorithm experiments across varied applications.
- Common interface: All test-problem generators use a common calling sequence with problem size n and an options structure, while omitted inputs receive suitable defaults.The options structure controls model parameters, and default settings produce problems of medium difficulty.
- Operator representations: The package represents forward operators as sparse matrices, user-defined objects, or function handles supporting matrix-vector products and, when available, transpose products.Tomography offers a speed-memory trade-off between sparse matrices and function handles; transpose-free methods are needed when A^T is unavailable.
- Problem types: The test-problem suite covers image deblurring, inverse diffusion, inverse interpolation, NMR relaxometry, and tomography, with related functions summarized in Table 3.The deblurring problems include spatially invariant out-of-focus and Gaussian blurs, while tomography provides three problem types.
- Problem severity: Singular-value decay measures problem severity: PRblurgauss, PRdiffusion, and PRnmr are severely ill posed, whereas the remaining problems are mildly ill posed.The classification is reported for all 12 problems at n = 32 with default options; the rapid decay toward N = n^2 = 1024 is noted as a discretization artifact.
4 Examples and Demonstrations
The examples demonstrate IR Tools on image deblurring, inverse interpolation, NMR relaxometry, and sparse reconstruction, showing how stopping rules, regularization matrices, and penalties affect computed solutions.
- Examples and Demonstrations: The package examples apply iterative reconstruction methods to large-scale image deblurring and other inverse problems.The demonstrations include image deblurring, inverse interpolation, NMR relaxometry, and sparse reconstruction.
- Image Deblurring: CGLS exhibits semi-convergence on the deblurring problem, with the smallest relative error at iteration 39 and discrepancy-principle stopping at iteration 33.The best regularized solution is saved separately, while the discrepancy principle uses the known noise level to select a stopping iteration.
- Image Deblurring: IRhybrid lsqr avoids the semi-convergence behavior seen in IRcgls by enforcing regularization at each iteration and terminates at iteration 75 in the example.Its recommended stopping iteration and reconstruction are recorded in the iteration-history output and shown in the example figures.
- Inverse Interpolation: For inverse interpolation, unconditioned CGLS produces a poor approximation, while a zero-boundary Laplacian is smooth but incorrect.The poor or incorrect solutions result from insufficient information or boundary conditions that conflict with the exact solution.
- Sparse Reconstructions: Sparse-reconstruction examples compare CGLS with IRell1 and IRirn, whose approximate 1-norm penalties produce many small elements and can reduce ringing and other artifacts.The heuristic sparse reconstruction works well when small elements are acceptable instead of exact zeros.
5 Conclusion
IR Tools provides a flexible MATLAB framework for large-scale iterative regularization methods and realistic 2D inverse-problem test problems, supporting experimentation and real-data applications.
- IR Tools provides large-scale iterative regularization methods in a flexible, uniform, and well-documented MATLAB framework.
- The package includes realistic large-scale 2D test problems intended to replace outdated problems from earlier regularization software.