Survey and Review

SIAM Review, Volume 66, Issue 2, Page 203-203, May 2024.
Inverse problems arise in various applications---for instance, in geoscience, biomedical science, or mining engineering, to mention just a few. The purpose is to recover an object or phenomenon from measured data which is typically subject to noise. The article “Computational Methods for Large-Scale Inverse Problems: A Survey on Hybrid Projection Methods,” by Julianne Chung and Silvia Gazzola, focuses on large, mainly linear, inverse problems. The mathematical modeling of such problems results in a linear system with a very large matrix $A \in \mathbb{R}^{m\times n}$ and a perturbed right-hand side. In some applications, it is not even possible to store the matrix, and thus algorithms which only use $A$ in the form of matrix-vector products $Ax$ or $A^Tx$ are the only choice. The article starts with two examples from image deblurring and tomographic reconstruction illustrating the challenges of inverse problems. It then presents the basic idea of regularization which consists of augmenting the model by additional information. Two variants of regularization methods are considered in detail, namely, variational and iterative methods. For variational methods it is crucial to know a good regularization parameter in advance. Unfortunately, its estimation can be expensive. On the other hand, iterative schemes, such as Krylov subspace methods, regularize by early termination of the iterations. Hybrid methods combine these two approaches leveraging the best features of each class. The paper focuses on hybrid projection methods. Here, one starts with a Krylov process in which the original problem is projected onto a low-dimensional subspace. The projected problem is then solved using a variational regularization method. The paper reviews the most relevant direct and iterative regularization techniques before it provides details on the two main building blocks of hybrid methods, namely, generating a subspace for the solution and solving the projected problem. It covers theoretical as well as numerical aspects of these schemes and also presents some extensions of hybrid methods: more general Tikhonov problems, nonstandard projection methods (enrichment, augmentation, recycling), $\ell_p$ regularization, Bayesian setting, and nonlinear problems. In addition, relevant software packages are provided. The presentation is very clear and the paper is also readable for those who are not experts in the field. Hence, it is valuable for everyone interested in large-scale inverse problems.