arXiv subjects

Lucas Onisk

Publications and source records attributed to Lucas Onisk.

6 recordsLinked to original sources

arXiv2026

Approximating matrix functions by block Krylov methods with randomized vectors

The need to evaluate expressions of the form $f(A)\mathbf{b}$, where $A$ is a square matrix, $f$ is a function, and $\mathbf{b}$ is a vector, arises in several areas of applied mathematics. When the matrix $A$ is very large, it is usually not attractive to evaluate $f(A)$. Instead, $f(A)\mathbf{b}$ often is approximated by computing an estimate in a Krylov subspace that depends on $A$ and $\mathbf{b}$, and only requires that $f$ be evaluated at a small matrix. This paper explores the application of several variants of randomized block Krylov methods to the approximation of $f(A)\mathbf{b}$. Computed examples suggest that block Krylov methods with an initial block vector that contains $\mathbf{b}$ as well as a few randomly generated vectors may require less computing time and reduce the number of Krylov steps than standard Krylov methods.

math.NA↗

arXiv2025

Projected iterated Tikhonov regularization in low precision

We investigate the regularizing behavior of an iterative Krylov subspace method for the solution of linear inverse problems in precisions lower than double. Recent works have considered the projection of iterated Tikhonov methods using Krylov subspaces for both computational efficiency and an additional regularizing effect. To investigate the regularizing behavior of this projected algorithm applied to problems that are naturally severely ill-posed, we formulate the iterates as a filtered solution using the preconditioned Landweber method with a Tikhonov-type preconditioner in a Krylov subspace. Through numerical examples simulating multiple low precision choices, we showcase the filtering properties of the method and the achievement of comparable working accuracy applied to discrete inverse problems (i.e., to within a few decimal places in relative error) compared to results computed in traditional double precision.

math.NA↗

arXiv2025

On the Choice of Subspace for the Quasi-minimal Residual Method for Linear Inverse Problems

Inverse problems arise in various scientific and engineering applications, necessitating robust numerical methods for their solution. In this work, we consider the effectiveness of Krylov subspace iterative methods, including GMRES, QMR, and their range restricted variants for solving linear discrete ill-posed problems. We analyze the impact of subspace selection on solution quality. Our findings indicate that range restricted QMR can outperform standard QMR, and confirm the previously observed behavior that range restricted GMRES can be superior to conventional GMRES in terms of approximation efficacy. Notably, range restricted QMR demonstrates a key advantage over GMRES with respect to range restricted QMR's singular spectrum which can make the method less sensitive to errors that are naturally present making it particularly effective when the noise level in the problem is uncertain.

math.NA↗

arXiv2025

Iterative Refinement and Flexible Iteratively Reweighed Solvers for Linear Inverse Problems with Sparse Solutions

This paper presents a new algorithmic framework for computing sparse solutions to large-scale linear discrete ill-posed problems. The approach is motivated by recent perspectives on iteratively reweighted norm schemes, viewed through the lens of iterative refinement. This framework leverages the efficiency and fast convergence of flexible Krylov methods while achieving higher accuracy through suitable restarts. Additionally, we demonstrate that the proposed methods outperform other flexible Krylov approaches in memory-limited scenarios. Relevant convergence theory is discussed, and the performance of the proposed algorithms is illustrated through a range of numerical examples, including image deblurring and computed tomography.

math.NA↗

arXiv2024

Mixed precision iterative refinement for linear inverse problems

This study investigates the iterative refinement method applied to the solution of linear discrete inverse problems by considering its application to the Tikhonov problem in mixed precision. Previous works on mixed precision iterative refinement methods for the solution of symmetric positive definite linear systems and least-squares problems have shown regularization to be a key requirement when computing low precision factorizations. For problems that are naturally severely ill-posed, we formulate the iterates of iterative refinement in mixed precision as a filtered solution using the preconditioned Landweber method with a Tikhonov-type preconditioner. Through numerical examples simulating various mixed precision choices, we showcase the filtering properties of the method and the achievement of comparable working accuracy of discrete inverse problems (i.e., to within a few decimal places in relative error) compared to results computed in double precision as well as another approximate iterative refinement method which we use as a benchmark.

math.NA↗

arXiv2024

Iterative Reconstruction Methods for Cosmological X-Ray Tomography

We consider the imaging of cosmic strings by using Cosmic Microwave Background (CMB) data. Mathematically, we study the inversion of an X-ray transform in Lorentzian geometry, called the light ray transform. The inverse problem is highly ill-posed, with additional complexities of being large-scale and dynamic, with unknown parameters that represent multidimensional objects. This presents significant computational challenges for the numerical reconstruction of images that have high spatial and temporal resolution. In this paper, we begin with a microlocal stability analysis for inverting the light ray transform using the Landweber iteration. Next, we discretize the spatiotemporal object and light ray transform and consider iterative computational methods for solving the resulting inverse problem. We provide a numerical investigation and comparison of some advanced iterative methods for regularization including Tikhonov and sparsity-promoting regularizers for various example scalar functions with conormal type singularities.

math.NA↗