Related papers: The error accumulation in the conjugate gradient m…
In this paper, we consider the conjugate gradient method for solving the problem of minimizing a quadratic function with additive noise in the gradient. Three concepts of noise were considered: antagonistic noise in the linear term,…
A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is…
The conjugate gradient method is a widely used algorithm for the numerical solution of a system of linear equations. It is particularly attractive because it allows one to take advantage of sparse matrices and produces (in case of infinite…
We consider the conjugate gradient algorithm applied to a general class of spiked sample covariance matrices. The main result of the paper is that the norms of the error and residual vectors at any finite step concentrate on deterministic…
We would like to congratulate the authors of "A Bayesian Conjugate Gradient Method" on their insightful paper, and welcome this publication which we firmly believe will become a fundamental contribution to the growing field of probabilistic…
Qualitative properties of non-negative solutions to a quasilinear degenerate parabolic equation with an absorption term depending solely on the gradient are shown, providing information on the competition between the nonlinear diffusion and…
We consider the eigenvalue problem of certain kind of non-compact linear operators given as the sum of a multiplication and a kernel operator. A degenerate kernel method is used to approximate isolated eigenvalues. It is shown that entries…
Incorporating a deep generative model as the prior distribution in inverse problems has established substantial success in reconstructing images from corrupted observations. Notwithstanding, the existing optimization approaches use gradient…
This work proposes a mathematically founded mixed precision accumulation strategy for the inference of neural networks. Our strategy is based on a new componentwise forward error analysis that explains the propagation of errors in the…
In [Meurant, Pape\v{z}, Tich\'y; Numerical Algorithms 88, 2021], we presented an adaptive estimate for the energy norm of the error in the conjugate gradient (CG) method. In this paper, we extend the estimate to algorithms for solving…
Multilevel methods represent a powerful approach in numerical solution of partial differential equations. The multilevel structure can also be used to construct estimates for total and algebraic errors of computed approximations. This paper…
The preconditioned conjugate gradient (PCG) algorithm is one of the most popular algorithms for solving large-scale linear systems Ax = b, where A is a symmetric positive definite matrix. Rather than computing residuals directly, it updates…
We investigate the method of conjugate gradients, exploiting inaccurate matrix-vector products, for the solution of convex quadratic optimization problems. Theoretical performance bounds are derived, and the necessary quantities occurring…
The Bayesian conjugate gradient method offers probabilistic solutions to linear systems but suffers from poor calibration, limiting its utility in uncertainty quantification tasks. Recent approaches leveraging postiterations to construct…
Motivated by applications in machine learning and statistics, we study distributed optimization problems over a network of processors, where the goal is to optimize a global objective composed of a sum of local functions. In these problems,…
The recent article "A Bayesian conjugate gradient method" by Cockayne, Oates, Ipsen, and Girolami proposes an approximately Bayesian iterative procedure for the solution of a system of linear equations, based on the conjugate gradient…
We study the linear convergence of the primal-dual hybrid gradient method. After a review of current analyses, we show that they do not explain properly the behavior of the algorithm, even on the most simple problems. We thus introduce the…
In this paper we describe a systematic procedure to analyze the convergence of degenerate preconditioned proximal point algorithms. We establish weak convergence results under mild assumptions that can be easily employed in the context of…
Corrigendum : An inverse problem in corrosion detection:stability estimates, J. Inv. Ill-posed Problems 12 (4) (2004), 349-367.
We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_{2}$ weight. We obtain representations and boundary traces for solutions…