Related papers: Estimating the error in CG-like algorithms for lea…
Estimation of actual errors from the residue in iterative solutions is necessary for efficient solution of large problems when their condition number is much larger than one. Such estimators for conjugate gradient algorithms used to solve…
In the book [Meurant and Tichy, SIAM, 2024] we discussed the estimation of error norms in the conjugate gradient (CG) algorithm for solving linear systems $Ax=b$ with a symmetric positive definite matrix $A$, where $b$ and $x$ are vectors.…
In practical computations, the (preconditioned) conjugate gradient (P)CG method is the iterative method of choice for solving systems of linear algebraic equations $Ax=b$ with a real symmetric positive definite matrix $A$. During the…
We analyze the convergence of the Conjugate Gradient (CG) method in exact arithmetic, when the coefficient matrix $A$ is symmetric positive semidefinite and the system is consistent. To do so, we diagonalize $A$ and decompose the algorithm…
This paper presents distributed adaptive algorithms based on the conjugate gradient (CG) method for distributed networks. Both incremental and diffusion adaptive solutions are all considered. The distributed conventional (CG) and modified…
The demands of accuracy in measurements and engineering models today, renders the condition number of problems larger. While a corresponding increase in the precision of floating point numbers ensured a stable computing, the uncertainty in…
In practical conjugate gradient (CG) computations it is important to monitor the quality of the approximate solution to $Ax=b$ so that the CG algorithm can be stopped when the required accuracy is reached. The relevant convergence…
Non-stationary approximations of the final value of a converging sequence are discussed, and we show that extremal eigenvalues can be reasonably estimated from the CG iterates without much computation at all. We introduce estimators of…
The adaptive $s$-step CG algorithm is a solver for sparse, symmetric positive definite linear systems designed to reduce the synchronization cost per iteration while still achieving a user-specified accuracy requirement. In this work, we…
Since the development of the conjugate gradient (CG) method in 1952 by Hestenes and Stiefel, CG, has become an indispensable tool in computational mathematics for solving positive definite linear systems. On the other hand, the conjugate…
This paper presents distributed conjugate gradient algorithms for distributed parameter estimation and spectrum estimation over wireless sensor networks. In particular, distributed conventional conjugate gradient (CCG) and modified…
The conjugate gradient (CG) method is an efficient iterative method for solving large-scale strongly convex quadratic programming (QP). In this paper we propose some generalized CG (GCG) methods for solving the $\ell_1$-regularized…
This paper proposes distributed adaptive algorithms based on the conjugate gradient (CG) method and the diffusion strategy for parameter estimation over sensor networks. We present sparsity-aware conventional and modified distributed CG…
We consider three mathematically equivalent variants of the conjugate gradient (CG) algorithm and how they perform in finite precision arithmetic. It was shown in [{\em Behavior of slightly perturbed Lanczos and conjugate-gradient…
We consider estimators obtained by iterates of the conjugate gradient (CG) algorithm applied to the normal equation of prototypical statistical inverse problems. Stopping the CG algorithm early induces regularisation, and optimal…
Many problems encountered in science and engineering can be formulated as estimating a low-rank object (e.g., matrices and tensors) from incomplete, and possibly corrupted, linear measurements. Through the lens of matrix and tensor…
When the CG method for solving linear algebraic systems was formulated about 70 years ago by Lanczos, Hestenes, and Stiefel, it was considered an iterative process possessing a mathematical finite termination property. CG was placed into a…
The problem of optimal precision switching for the conjugate gradient (CG) method applied to sparse linear systems is considered. A sparse matrix is defined as an $n\!\times\!n$ matrix with $m\!=\!O(n)$ nonzero entries. The algorithm first…
This paper describes a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. The proposed estimator…
The paper studies the solution of stochastic optimization problems in which approximations to the gradient and Hessian are obtained through subsampling. We first consider Newton-like methods that employ these approximations and discuss how…