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1. Fixed-point iterative linear inverse solver with extended precision

   https://doi.org/10.1038/s41598-023-32338-5  

   Zhu, Zheyuan; Klein, Andrew B.; Li, Guifang; Pang, Sean (December 2023, Scientific Reports)

   Abstract Solving linear systems, often accomplished by iterative algorithms, is a ubiquitous task in science and engineering. To accommodate the dynamic range and precision requirements, these iterative solvers are carried out on floating-point processing units, which are not efficient in handling large-scale matrix multiplications and inversions. Low-precision, fixed-point digital or analog processors consume only a fraction of the energy per operation than their floating-point counterparts, yet their current usages exclude iterative solvers due to the cumulative computational errors arising from fixed-point arithmetic. In this work, we show that for a simple iterative algorithm, such as Richardson iteration, using a fixed-point processor can provide the same convergence rate and achieve solutions beyond its native precision when combined with residual iteration. These results indicate that power-efficient computing platforms consisting of analog computing devices can be used to solve a broad range of problems without compromising the speed or precision. 

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2. Accuracy Controlled Schemes for the Eigenvalue Problem of the Radiative Transfer Equation

   https://doi.org/10.1007/s10208-025-09694-x  

   Dahmen, Wolfgang; Mula, Olga (January 2025, Foundations of Computational Mathematics)

   Abstract The criticality problem in nuclear engineering asks for the principal eigenpair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper, we propose a paradigm that deviates from the common practice of approximately solving the corresponding spectral problem with a fixed, presumably sufficiently fine discretization. Instead, the present approach is based on first contriving iterative schemes, formulated in function space, that are shown to converge at a quantitative rate without assuming any a priori excess regularity properties, and that exploit only properties of the optical parameters in the underlying radiative transfer model. We develop the analytical and numerical tools for approximately realizing each iteration step within judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigenpair. This is carried out in full first for a Newton scheme. Since this is only locally convergent we analyze in addition the convergence of a power iteration in function space to produce sufficiently accurate initial guesses. Here we have to deal with intrinsic difficulties posed by compact but unsymmetric operators preventing standard arguments used in the finite dimensional case. Our main point is that we can avoid any condition on an initial guess to be already in a small neighborhood of the exact solution. We close with a discussion of remaining intrinsic obstructions to a certifiable numerical implementation, mainly related to not knowing the gap between the principal eigenvalue and the next smaller one in modulus. 

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3. Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities

   Diakonikolas, J (January 2020, Proceedings of Thirty Third Conference on Learning Theory)

   null (Ed.)

   We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving monotone inclusion problems. These results immediately translate into near-optimal guarantees for approximating strong solutions to variational inequality problems, approximating convex-concave min-max optimization problems, and minimizing the norm of the gradient in min-max optimization problems. Our analysis is based on a novel and simple potential-based proof of convergence of Halpern iteration, a classical iteration for finding fixed points of nonexpansive maps. Additionally, we provide a series of algorithmic reductions that highlight connections between different problem classes and lead to lower bounds that certify near-optimality of the studied methods. 

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4. Multivariate Monotone Inclusions in Saddle Form

   https://doi.org/10.1287/moor.2021.1161  

   Bùi, Minh N.; Combettes, Patrick L. (May 2022, Mathematics of Operations Research)

   We propose a novel approach to monotone operator splitting based on the notion of a saddle operator. Under investigation is a highly structured multivariate monotone inclusion problem involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators, as well as various monotonicity-preserving operations among them. This model encompasses most formulations found in the literature. A limitation of existing primal-dual algorithms is that they operate in a product space that is too small to achieve full splitting of our problem in the sense that each operator is used individually. To circumvent this difficulty, we recast the problem as that of finding a zero of a saddle operator that acts on a bigger space. This leads to an algorithm of unprecedented flexibility, which achieves full splitting, exploits the specific attributes of each operator, is asynchronous, and requires to activate only blocks of operators at each iteration, as opposed to activating all of them. The latter feature is of critical importance in large-scale problems. The weak convergence of the main algorithm is established, as well as the strong convergence of a variant. Various applications are discussed, and instantiations of the proposed framework in the context of variational inequalities and minimization problems are presented. 

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5. Parameter Selection by GCV and a \(\boldsymbol {\chi^2}\) Test within Iterative Methods for \(\boldsymbol{\ell_1}\)-Regularized Inverse Problems

   https://doi.org/10.1137/24M165805X  

   Sweeney, Brian; Renaut, Rosemary; Español, Malena I (May 2025, SIAM Journal on Scientific Computing)

   l1 regularization is used to preserve edges or enforce sparsity in a solution to an inverse problem. We investigate the split Bregman and the majorization-minimization iterative methods that turn this nonsmooth minimization problem into a sequence of steps that include solving an -regularized minimization problem. We consider selecting the regularization parameter in the inner generalized Tikhonov regularization problems that occur at each iteration in these iterative methods. The generalized cross validation method and chi2 degrees of freedom test are extended to these inner problems. In particular, for the chi2 test this includes extending the result for problems in which the regularization operator has more rows than columns and showing how to use the -weighted generalized inverse to estimate prior information at each inner iteration. Numerical experiments for image deblurring problems demonstrate that it is more effective to select the regularization parameter automatically within the iterative schemes than to keep it fixed for all iterations. Moreover, an appropriate regularization parameter can be estimated in the early iterations and fixed to convergence. 

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